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What Are Factorial Experiments and Why Can They Be Helpful?

1 Johnson & Johnson Global Epidemiology, Titusville, New Jersey

Kaplan and colleagues 1 report the findings of 2 studies that were part of a randomized clinical trial that aimed to determine whether a combined intervention for adolescents, using both light exposure during sleep and cognitive behavioral therapy, would encourage them to get to sleep earlier than usual, thereby increasing total sleep time. They conducted 2 similar, but somewhat different, studies.

In the first study, participants were randomly assigned to receive either 3 weeks of light flashes (light alone) or a sham light intervention. The light flashes were brief light pulses administered in 3-millisecond bursts delivered 20 seconds apart, starting 3 hours before the targeted wake time for the individual participant. In the second study, participants were randomized to a combination of light plus cognitive behavioral therapy or sham light therapy plus cognitive behavioral therapy. The main outcomes were self-reported sleep times, momentary ratings of evening sleepiness, and subjective measures of sleepiness and sleep quality.

Kaplan et al 1 found that, in study 1, light therapy alone did not change sleep timing. However, in the second study, light plus behavioral therapy significantly moved sleep onset approximately 50 minutes earlier, on average, and increased nightly sleep time by approximately 43 minutes. There were also improvements in several secondary outcomes.

This combination of studies is similar in some respects to a factorial experiment but lacks certain unique advantages of factorial studies. The purpose of this commentary is to elaborate on those potential advantages of factorial studies, referring back to the article by Kaplan et al 1 for context.

Factorial designs are used to test more than 1 experimental factor (whence the name) in the context of a single study. In the studies by Kaplan and colleagues, 1 the 2 experimental factors were light and cognitive therapy. The first study addressed the efficacy of light in the absence of behavioral therapy. The second study tested the efficacy of light in the presence of behavioral therapy. The 2 studies were conducted sequentially. In a traditional factorial experiment, participants would have been randomized to 1 of 4 groups: light alone, cognitive therapy alone, both light and cognitive therapy, and neither. Another way to think of the design is that participants would be randomized to light vs sham; then, within each of those groups, they would be randomized again to cognitive behavioral therapy or no therapy.

The potential for statistical interaction, ie, when the effect of an intervention depends on the presence or absence of another intervention, is both a strength and a limitation. The strength is that the question of differential effects may actually be of scientific and clinical interest. The limitation is that if the interest of the investigators is solely on the effects of each intervention, assuming that there is no interaction, then the statistical power to address those individual questions is potentially compromised. 2 Again, in the studies by Kaplan et al, 1 the question is whether light flashes have different effects in the presence or absence of behavioral therapy.

In this situation, the investigators were uninterested in the effects of behavioral therapy alone. Nonetheless, the conducted studies confounded several design features that slightly complicate the interpretation of their findings. Just by virtue of doing separate studies, the opportunity for increased variability is introduced. The expectation in conducting randomized trials is that results can vary from study to study even when the designs are similar. The fact that the studies were conducted sequentially allows for further variability due to calendar time. It may be that neither of these sources of variability would be meaningful for these studies, but the design does not allow disaggregation of these extraneous features. The timing of the administration of light flashes (starting 3 hours before wake time in study 1 vs 2 hours before wake time in study 2) also changed. In this case, different findings about the efficacy of light therapy could be (and probably are) due to the presence of behavioral therapy, but it is also possible, especially given the small sample size in the second study, that unrelated study-to-study variability could have produced the findings (not to mention the timing of the administration of light flashes).

The use of factorial designs in medicine is not new. A very famous example was the Physicians’ Health Study, the design of which is discussed in a methodological article. 3 Their goal was to test 2 hypotheses in the same study: efficacy of aspirin in prevention of cardiovascular disease and of beta carotene in the prevention of cancer among US physicians.

For beta carotene, 4 with follow-up over nearly 13 years, the trial found no significant decrease in cancer risk overall, or in risk of prostate, colon, or lung cancer. For myocardial infarction, 5 there was a 44% reduction in risk (relative risk, 0.56; 95% CI, 0.45-0.70). There was no reduction in overall mortality (relative risk, 0.96; 95% CI, 0.60-1.54).

Given the potential advantages of factorial experiments, one might wonder why they are not used more frequently. As noted above, there can be reduced statistical power for testing the effects of each treatment when there is a statistical interaction between treatments characterized by 1 treatment (either presence or absence) reducing the effect of the other. An example of this is provided in the study by Jaki and Vasileiou. 2 For any study with more than 1 treatment, there may also be logistical complexities introduced by the need for multiple placebos, with the associated burden on trial participants, who are asked to take multiple medications. My own view is that factorial studies are underused, for the reasons I have described. In particular, the ability to eliminate confounding between study, calendar time, and the 2 factors should be very appealing. Furthermore, if we are confident about the lack of interaction, the factorial design can be very efficient (because we have 2 studies done at the same time). When we are actually interested in testing the interaction, the factorial design offers a great opportunity. That said, Jaki and Vasileiou 2 suggest the use of an alternative design that accommodates multiple treatments and note that these other designs can be equally, or more, statistically efficient. The theme is the same, though, that testing multiple treatments in the same study is advantageous compared with doing separate studies.

In summary, the authors’ conclusions about the efficacy of light flashes are likely to be correct (within the limits of self-reported outcomes, which they note). That said, when designing studies that address more than 1 question, when possible, one should consider the potential advantages of unconfounding those factors by conducting a factorial experiment.

Published: September 25, 2019. doi:10.1001/jamanetworkopen.2019.11917

Corresponding Author: Jesse A. Berlin, ScD, Johnson & Johnson Global Epidemiology, 1125 Trenton-Harbourton Road, J3, Zone 1, Titusville, NJ 08560 ( [email protected] ).

Conflict of Interest Disclosures: None reported.

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Berlin JA. What Are Factorial Experiments and Why Can They Be Helpful? JAMA Netw Open. 2019;2(9):e1911917. doi:10.1001/jamanetworkopen.2019.11917

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What Is a Factorial Design? Definition and Examples

Categories Dictionary

A factorial design is a type of experiment that involves manipulating two or more variables. While simple psychology experiments look at how one independent variable affects one dependent variable, researchers often want to know more about the effects of multiple independent variables.

Table of Contents

How a Factorial Design Works

Let’s take a closer look at how a factorial design might work in a psychology experiment:

The independent variable is the variable of interest that the experimenter will manipulate.
The dependent variable is the variable that the researcher then measures.

By doing this, psychologists can see if changing the independent variable results in some type of change in the dependent variable.

For example, imagine that a researcher wants to do an experiment looking at whether sleep deprivation hurts reaction times during a driving test. If she were only to perform the experiment using these variables–the sleep deprivation being the independent variable and the performance on the driving test being the dependent variable–it would be an example of a simple experiment.

However, let’s imagine that she is also interested in learning if sleep deprivation impacts the driving abilities of men and women differently. She has just added a second independent variable of interest (sex of the driver) into her study, which now makes it a factorial design.

Types of Factorial Designs

One common type of experiment is known as a 2×2 factorial design. In this type of study, there are two factors (or independent variables), each with two levels.

The number of digits tells you how many independent variables (IVs) there are in an experiment, while the value of each number tells you how many levels there are for each independent variable.

So, for example, a 4×3 factorial design would involve two independent variables with four levels for one IV and three levels for the other IV.

Advantages of a Factorial Design

One of the big advantages of factorial designs is that they allow researchers to look for interactions between independent variables.

An interaction is a result in which the effects of one experimental manipulation depends upon the experimental manipulation of another independent variable.

Example of a Factorial Design

For example, imagine that researchers want to test the effects of a memory-enhancing drug. Participants are given one of three different drug doses, and then asked to either complete a simple or complex memory task.

The researchers note that the effects of the memory drug are more pronounced with the simple memory tasks, but not as apparent when it comes to the complex tasks. In this 3×2 factorial design, there is an interaction effect between the drug dosage and the complexity of the memory task.

Understanding Variable Effects in Factorial Designs

So if researchers are manipulating two or more independent variables, how exactly do they know which effects are linked to which variables?

“It is true that when two manipulations are operating simultaneously, it is impossible to disentangle their effects completely,” explain authors Breckler, Olson, and Wiggins in their book Social Psychology Alive .

“Nevertheless, the researchers can explore the effects of each independent variable separately by averaging across all levels of the other independent variable . This procedure is called looking at the main effect.”

Examples of Factorial Designs

A university wants to assess the starting salaries of their MBA graduates. The study looks at graduates working in four different employment areas: accounting, management, finance, and marketing.

In addition to looking at the employment sector, the researchers also look at gender. In this example, the employment sector and gender of the graduates are the independent variables, and the starting salaries are the dependent variables. This would be considered a 4×2 factorial design.

Researchers want to determine how the amount of sleep a person gets the night before an exam impacts performance on a math test the next day. But the experimenters also know that many people like to have a cup of coffee (or two) in the morning to help them get going.

So, the researchers decided to look at how the amount of sleep and caffeine influence test performance.

The researchers then decided to look at three levels of sleep (4 hours, 6 hours, and 8 hours) and only two levels of caffeine consumption (2 cups versus no coffee). In this case, the study is a 3×2 factorial design.

Baker TB, Smith SS, Bolt DM, et al. Implementing clinical research using factorial designs: A primer . Behav Ther . 2017;48(4):567-580. doi:10.1016/j.beth.2016.12.005

Collins LM, Dziak JJ, Li R. Design of experiments with multiple independent variables: a resource management perspective on complete and reduced factorial designs . Psychol Methods . 2009;14(3):202-224. doi:10.1037/a0015826

Haerling Adamson K, Prion S. Two-by-two factorial design . Clin Simul Nurs . 2020;49:90-91. doi:10.1016/j.ecns.2020.06.004

Watkins ER, Newbold A. Factorial designs help to understand how psychological therapy works . Front Psychiatry . 2020;11:429. doi:10.3389/fpsyt.2020.00429

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Factorial Design

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21 Aug, 2024

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Edited by :

Ashish Kumar Srivastav

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Dheeraj Vaidya

What Is A Factorial Design?

Factorial design is a statistical experimental design used to investigate the effects of two or more independent variables (factors) on a dependent variable. By manipulating the levels of the characteristics and measuring the resulting impact on the dependent variable, researchers can identify each element's unique contributions and their combined or interactive effects.

These are beneficial when investigating interactions between variables. They allow researchers to explore how one factor's effects may depend on another element's levels. This can provide valuable insights into the underlying mechanisms. It further drives the observed impacts and helps identify potential moderators or mediators of the relationship between variables.

Factorial design explained.

Advantages and disadvantages

Frequently Asked Questions (FAQs)

Advantages And Disadvantages

The advantages are as follows:

Ability to investigate multiple factors : These allow researchers to investigate the effects of various independent on a dependent variable in a single experiment, which can save time and resources.
Identification of main effects and interactions : These enable researchers to identify the main products of each independent variable and any interaction effects between them, providing a more nuanced understanding of the relationships between variables.
Increased statistical power : By manipulating multiple independent variables, factorial designs can increase the statistical power of a study and improve the likelihood of detecting meaningful effects.
Flexibility : These adapt to various research questions and uses in multiple fields, including psychology, education, medicine, and engineering.

The disadvantages are as follows:

Increased complexity : Using multiple independent variables can interpret results more complexly, mainly when interaction effects are present.
Increased sample size requirements : A larger sample size is preferable to a more straightforward design with fewer independent variables to power such a design adequately.
Potential for confounding : The presence of interaction effects can make it challenging to determine which independent variable is responsible for observed effects, potentially confounding the results.
Limited generalizability : The specific conditions of it may not be generalizable to other contexts.

A main effect refers to the impact of a single independent variable on the dependent variable. In contrast, an interaction effect refers to the combined effect of two or more independent variables on the dependent variable. Here, both direct and interaction effects can be present.

In a between-subjects design, participants are with only one independent variable level. In such scenarios, each participant is open to all possible combinations of stories of the independent variables.

Yes, these can be helpful in observational and experimental research. In observational studies, the independent variables are not manipulated, but their effects on the dependent variable can still be investigated using a factorial design.

This article has been a guide to what is Factorial Design. Here, we explain the topic in detail with its examples, advantages, disadvantages, and types. You may also find some helpful articles here -

Permutation
Poisson Distribution
Binomial Distribution Formula

Teach yourself statistics

What is a Full Factorial Experiment?

This lesson describes full factorial experiments. Specifically, the lesson answers four questions:

What is a full factorial experiment?
What causal effects can we test in a full factorial experiment?
How should we interpret causal effects?
What are the advantages and disadvantages of a full factorial experiment?

What is a Factorial Experiment?

A factorial experiment allows researchers to study the joint effect of two or more factors on a dependent variable . Factorial experiments come in two flavors: full factorials and fractional factorials. In this lesson, we will focus on the full factorial experiment, not the fractional factorial.

Full Factorial Experiment

A full factorial experiment includes a treatment group for every combination of factor levels. Therefore, the number of treatment groups is the product of factor levels. For example, consider the full factorial design shown below:

	A			A
	B	B	B	B	B	B
C	Grp 1	Grp 2	Grp 3	Grp 4	Grp 5	Grp 6
C	Grp 7	Grp 8	Grp 9	Grp 10	Grp 11	Grp 12
C	Grp 13	Grp 14	Grp 15	Grp 16	Grp 17	Grp 18
C	Grp 19	Grp 20	Grp 21	Grp 22	Grp 23	Grp 24

Factor A has two levels, factor B has three levels, and factor C has four levels. Therefore, this full factorial design has 2 x 3 x 4 = 24 treatment groups.

Full factorial designs can be characterized by the number of treatment levels associated with each factor, or by the number of factors in the design. Thus, the design above could be described as a 2 x 3 x 4 design (number of treatment levels) or as a three-factor design (number of factors).

Fractional Factorial Experiments

The other type of factorial experiment is a fractional factorial. Unlike full factorial experiments, which include a treatment group for every combination of factor levels, fractional factorial experiments include only a subset of possible treatment groups.

Causal Effects

A full factorial experiment allows researchers to examine two types of causal effects: main effects and interaction effects. To facilitate the discussion of these effects, we will examine results (mean scores) from three 2 x 2 factorial experiments:

Experiment I: Mean Scores

	A	A
B	5	2
B	2	5

Experiment II: Mean Scores

	C	C
D	5	4
D	4	1

Experiment III: Mean Scores

	E	E
F	5	3
F	3	1

Main Effects

In a full factorial experiment, a main effect is the effect of one factor on a dependent variable, averaged over all levels of other factors. A two-factor factorial experiment will have two main effects; a three-factor factorial, three main effects; a four-factor factorial, four main effects; and so on.

How to Measure Main Effects

To illustrate what is going on with main effects, let's look more closely at the main effects from Experiment I:

Assuming there were an equal number of observations in each treatment group, we can compute the main effect for Factor A as shown below:

Effect of A at level B 1 = A 2 B 1 - A 1 B 1 = 2 - 5 = -3

Effect of A at level B 2 = A 2 B 2 - A 1 B 2 = 5 - 2 = +3

Main effect of A = ( -3 + 3 ) / 2 = 0

And we can compute the main effect for Factor B as shown below:

Effect of B at level A 1 = A 1 B 2 - A 1 B 1 = 5 - 2 = +3

Effect of B at level A 2 = A 2 B 2 - A 2 B 1 = 2 - 5 = -3

Main effect of B = ( 3 - 3 ) / 2 = 0

In a similar fashion, we can compute main effects for Experiment II (see Problem 1 ) and Experiment III (see Problem 2 ).

Warning: In a full factorial experiment, you should not attempt to interpret main effects until you have looked at interaction effects. With that in mind, let's look at interaction effects for Experiments I, II, and III.

Interaction Effects

In a full factorial experiment, an interaction effect exists when the effect of one independent variable depends on the level of another independent variable.

When Interactions Are Present

The presence of an interaction can often be discerned when factorial data are plotted. For example, the charts below plot mean scores from Experiment I and from Experiment II:

Experiment I

Experiment II

In Experiment I, consider how the dependent variable score is affected by level A1 versus level A2. In the presence of B1, the dependent variable score is bigger for A1 than for A2. But in the presense of B2, the reverse is true - the dependent variable score is bigger for A2 than for A1.

In Experiment II, level C1 is associated with a little bit bigger dependent variable score in the presence of D1; but a much bigger dependent variable score in the presence of D2.

In both charts, the way that one factor affects the dependent variable depends on the level of another factor. This is the definition of an interaction effect. In charts like these, the presence of an interaction is indicated by non-parallel plotted lines.

Note: These charts are called interaction plots. For guidance on creating and interpreting interaction plots, see Interaction Plots .

When Interactions Are Absent

Now, look at the chart below, which plots mean scores from Experiment III:

Experiment III

In this chart, E1 has the same effect on the dependent variable, regardless of the level of Factor F. At each level of Factor F, the dependent variable is 2 units bigger with E1 than with E2. So, in this chart, there is no interaction between Factors E and F. And you can tell at a glance that there is no interaction, because the plotted lines are parallel.

Number of Interactions

The number of interaction effects in a full factorial experiment is determined by the number of factors. A two-factor design (with factors A and B) has one two-way interaction (the AB interaction). A three-factor design (with factors A, B, and C) has one three-way interaction (the ABC interaction) and three two-way interactions (the AB, AC, and BC interactions).

A general formula for finding the number of interaction effects (NIE) in a full factorial experiment is:

where k C r is the number of combinations of k things taken r at a time, k is the number of factors in the full factorial experiment, and r is the number of factors in the interaction term.

Note: If you are unfamiliar with combinations, see Combinations and Permutations .

How to Interpret Causal Effects

Recall that the purpose of conducting a full factorial experiment is to understand the joint effects (main effects and interaction effects) of two or more independent variables on a dependent variable. When a researcher looks at actual data from an experiment, small differences in group means are expected, even when independent variables have no causal connection to the dependent variable. These small differences might be attributable to random effects of unmeasured extraneous variables .

So the real question becomes: Are observed effects significantly bigger than would be expected by chance - big enough to be attributable to a main or interaction effect rather than to an extraneous variable? One way to answer this question is with analysis of variance. Analysis of variance will test all main effects and interaction effects for statistical significance. Here is how to interpret the results of that test:

If no effects (main effects or interaction effects) are statistically significant, conclude that the independent variables do not affect the dependent variable.
If a main effect is statistically significant, conclude that the main effect does affect the dependent variable.
If an interaction effect is statistically significant, conclude that the interaction factors act in combination to affect the dependent variable.

Recognize that it is possible for factors to affect the dependent variable, even when the main effects are not statistically significant. We saw an example of that in Experiment I.

In Experiment I, both main effects were zero; yet, the interaction effect is dramatic. The moral here is: Do not attempt to interpret main effects until you have looked at interaction effects.

Note: To learn how to implement analysis of variance for a full factorial experiment, see ANOVA With Full Factorial Experiments .

Advantages and Disadvantages

Analysis of variance with a full factorial experiment has advantages and disadvantages. Advantages include the following:

The design permits a researcher to examine multiple factors in a single experiment.
The design permits a researcher to examine all interaction effects.
The design requires subjects to participate in only one treatment group.

Disadvantages include the following:

When the experiment includes many factors and levels, sample size requirements may be excessive.
The need to include all treatment combinations, regardless of importance, may waste resources.

Test Your Understanding

The table below shows results from a 2 x 2 factorial experiment.

Assuming equal sample size in each treatment group, what is the main effect for both factors?

(A) -2 (B) 3.5 (C) 4 (D) 7 (E) 14

The correct answer is (A). We can compute the main effect for Factor C as shown below:

Effect of C at level D 1 = C 2 D 1 - C 1 D 1 = 4 - 5 = -1

Effect of C at level D 2 = C 2 D 2 - C 1 D 2 = 1 - 4 = -3

Main effect of C = ( -1 + -3 ) / 2 = -2

And we can compute the main effect for Factor D as shown below:

Effect of D at level C 1 = C 1 D 2 - C 1 D 1 = 4 - 5 = -1

Effect of D at level C 2 = C 2 D 2 - C 2 D 1 = 1 - 4 = -3

Main effect of D = ( -1 + -3 ) / 2 = -2

(A) -12 (B) -2 (C) 0 (D) 3 (E) 4

The correct answer is (B). We can compute the main effect for Factor E as shown below:

Effect of E at level F 1 = E 2 F 1 - E 1 F 1 = 3 - 5 = -2

Effect of E at level F 2 = E 2 F 2 - E 1 F 2 = 1 - 3 = -2

Main effect of E = ( -2 + -2 ) / 2 = -2

And we can compute the main effect for Factor F as shown below:

Effect of F at level C 1 = E 1 F 2 - E 1 F 1 = 3 - 5 = -2

Effect of F at level C 2 = E 2 F 2 - E 2 F 1 = 1 - 3 = -2

Main effect of F = ( -2 + -2 ) / 2 = -2

Consider the interaction plot shown below. Which of the following statements are true?

(A) There is a non-zero interaction between Factors A and B. (B) There is zero interaction between Factors A and B. (C) The plot provides insufficient information to describe the interaction.

The correct answer is (B). At every level of Factor B, the difference between A1 and A2 is 3 units. Because the effect of Factor A is constant (always 3 units) at every level of Factor B, there is no interaction between Factors A and B.

Note: The parallel pattern of lines in the interaction plot indicates that the AB interaction is zero.

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5 Reasons Factorial Experiments Are So Successful

Last week we began an experimental design trying to get at how to drive the golf ball the farthest off the tee by characterizing the process and defining the problem. The next step in our DOE problem-solving methodology is to design the data collection plan we’ll use to study the factors in the experiment.

We will construct a full factorial design, fractionate that design to half the number runs for each golfer, and then discuss the benefits of running our experiment as a factorial design.

advantages of factorial experimental design

The four factors in our experiment and the low / high settings used in the study are:

Club Face Tilt (Tilt) – Continuous Factor : 8.5 degrees & 10.5 degrees
Ball Characteristics (Ball) – Categorical Factor : Economy & Expensive
Club Shaft Flexibility (Shaft) – Continuous Factor : 291 & 306 vibration cycles per minute
Tee Height (TeeHght) – Continuous Factor : 1 inch & 1 3/4 inch

To develop a full understanding of the effects of 2 – 5 factors on your response variables, a full factorial experiment requiring 2 k runs ( k = of factors) is commonly used. Many industrial factorial designs study 2 to 5 factors in 4 to 16 runs (2 5-1 runs, the half fraction, is the best choice for studying 5 factors) because 4 to 16 runs is not unreasonable in most situations. The data collection plan for a full factorial consists of all combinations of the high and low setting for each of the factors. A cube plot, like the one for our golf experiment shown below, is a good way to display the design space the experiment will cover.

There are a number of good reasons for choosing this data collection plan over other possible designs. The details are discussed in many excellent texts . Here are my top five.

1. Factorial and fractional factorial designs are more cost-efficient.

Factorial and fractional factorial designs provide the most run efficient (economical) data collection plan to learn the relationship between your response variables and predictor variables. They achieve this efficiency by assuming that each effect on the response is linear and therefore can be estimated by studying only two levels of each predictor variable.

After all, it only takes two points to establish a line.

2. Factorial designs estimate the interactions of each input variable with every other input variable.

Often the effect of one variable on your response is dependent on the level or setting of another variable. The effectiveness of a college quarterback is a good analogy. A good quarterback can have good skills on his own. However, a great quarterback will achieve outstanding results only if he and his wide receiver have synergy. As a combination, the results of the pair can exceed the skill level of each individual player. This is an example of a synergistic interaction. Complex industrial processes commonly have interactions, both synergistic and antagonistic, occurring between input variables. We cannot fully quantify the effects of input variables on our responses unless we have identified all active interactions in addition to the main effects of each variable. Factorial experiments are specifically designed to estimate all possible interactions.

3. Factorial designs are orthogonal.

We analyze our final experiment results using least squares regression to fit a linear model for the response as a function of the main effects and two-way interactions of each of the input variables. A key concern in least squares regression arises if the settings of the input variables or their interactions are correlated with each other. If this correlation occurs, the effect of one variable may be masked or confounded with another variable or interaction making it difficult to determine which variables actually cause the change in the response. When analyzing historical or observational data, there is no control over which variable settings are correlated with other input variable settings and this casts a doubt on the conclusiveness of the results. Orthogonal experimental designs have zero correlation between any variable or interaction effects specifically to avoid this problem. Therefore, our regression results for each effect are independent of all other effects and the results are clear and conclusive.

4. Factorial designs encourage a comprehensive approach to problem-solving.

First, intuition leads many researchers to reduce the list of possible input variables before the experiment in order to simplify the experiment execution and analysis. This intuition is wrong. The power of an experiment to determine the effect of an input variable on the response is reduced to zero the minute that variable is removed from the study (in the name of simplicity). Through the use of fractional factorial designs and experience in DOE, you quickly learn that it is just as easy to run a 7 factor experiment as a 3 factor experiment, while being much more effective.

Second, factorial experiments study each variable’s effect over a range of settings of the other variables. Therefore, our results apply to the full scope of all the process parameter settings rather than just specific settings of the other variables. Our results are more widely applicable to all conditions than the results from studying one variable at a time.

5. Two-level factorial designs provide an excellent foundation for a variety of follow-up experiments.

This will lead to the solution to your process problem. A fold-over of your initial fractional factorial can be used to complement an initial lower resolution experiment, providing a complete understanding of all your input variable effects. Augmenting your original design with axial points results in a response surface design to optimize your response with greater precision. The initial factorial design can provide a path of steepest ascent / descent to move out of your current design space into one with even better response values. Finally, and perhaps most commonly, a second factorial design with fewer variables and a smaller design space can be created to better understand the highest potential region for your response within the original design space.

I hope this short discussion has convinced you that any researcher in academics or industry will be well rewarded for the time spent learning to design, execute, analyze, and communicate the results from factorial experiments. The earlier in your career you learn these skills, the … well, you know the rest.

For these reasons, we can be quite confident about our selection of a full factorial data collection to study the 4 variables for our golf experiment. Each golfer will be responsible for executing only one half of the runs, called a half fraction, of the full factorial. Even so, the results for each golfer can be analyzed independently as a complete experiment.

In my next post, I’ll answer the question: How do we calculate the number of replicates needed for each set of run conditions from each golfer so that our results have a high enough power that we can be confident in our conclusions? Many thanks to Toftrees Golf Resort and Tussey Mountain for use of their facilities to conduct our golf experiment.

Catch Up with the other Golf DOE Posts:

Part 1: A (Golf) Course in Design of Experiments Part 3: Mulligan? How Many Runs Do You Need to Produce a Complete Data Set? Part 4: ANCOVA and Blocking: 2 Vital Parts to DOE Part 5: Concluding Our Golf DOE: Time to Quantify, Understand and Optimize

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METHODS article

Factorial designs help to understand how psychological therapy works.

College of Life and Environmental Sciences, University of Exeter, Exeter, United Kingdom

A large amount of research time and resources are spent trying to develop or improve psychological therapies. However, treatment development is challenging and time-consuming, and the typical research process followed—a series of standard randomized controlled trials—is inefficient and sub-optimal for answering many important clinical research questions. In other areas of health research, recognition of these challenges has led to the development of sophisticated designs tailored to increase research efficiency and answer more targeted research questions about treatment mechanisms or optimal delivery. However, these innovations have largely not permeated into psychological treatment development research. There is a recognition of the need to understand how treatments work and what their active ingredients might be, and a call for the use of innovative trial designs to support such discovery. One approach to unpack the active ingredients and mechanisms of therapy is the factorial design as exemplified in the Multiphase Optimization Strategy (MOST) approach. The MOST design allows identification of the active components of a complex multi-component intervention (such as CBT) using a sophisticated factorial design, allowing the development of more efficient interventions and elucidating their mechanisms of action. The rationale, design, and potential advantages of this approach will be illustrated with reference to the IMPROVE-2 study, which conducts a fractional factorial design to investigate which elements (e.g., thought challenging, activity scheduling, compassion, relaxation, concreteness, functional analysis) within therapist-supported internet-delivered CBT are most effective at reducing symptoms of depression in 767 adults with major depression. By using this innovative approach, we can first begin to work out what components within the overall treatment package are most efficacious on average allowing us to build an overall more streamlined and potent therapy. This approach also has potential to distinguish the role of specific versus non-specific common treatment components within treatment.

Introduction: The Need to Understand How Psychological Therapies Work

Psychological treatments for mental health disorders have been robustly established as proven and evidence-based interventions through multiple clinical trials and meta-analyses ( 1 – 3 ). Nonetheless, there is a pressing need to further improve psychological interventions: even the best treatments do not work for everyone. Many patients do not have sustained improvement, and treatments need to scaled up to tackle the global burden of mental health ( 4 ). For example, psychological treatments for depression only achieve remission rates of 30%–40% and have limited sustained efficacy (at least 50% relapse and recurrence) ( 1 , 5 ). Further, it is estimated that current treatments, if delivered optimally, would only reduce the burden of depression by one third ( 6 ). As such, psychological treatments for depression need to be significantly enhanced.

One pathway to improving the efficacy and effectiveness of therapies is to develop our understanding of how complex psychological interventions work. Despite determining that a number of psychological treatments are effective, for example, cognitive-behavioral therapy (CBT), we still do not know how psychological treatments work. There is little evidence on the precise mechanisms through which psychological treatments work or what are the active ingredients of treatments ( 7 – 10 ), especially for disorders involving general distress such as depression and generalized anxiety disorder. Historically, there has been little progress in specifying the active ingredients of CBT for depression, and as a consequence, there have been no significant gains in the effectiveness of CBT for depression for over 40 years.

Resolving the active mechanisms and active ingredients of psychological interventions has been repeatedly identified as a major priority for research ( 4 , 7 , 10 , 11 ). For example, the Institute of Medicine (2015) highlighted the need to identify the key elements of psychosocial interventions that casually drive its effects ( 11 ).

To be clear, we distinguish between the active components of therapy, operationalized as the active elements or ingredients within a therapy that produce clinical benefit, which could be therapist-based, client activities, specific techniques, or related to therapy structure and delivery, versus the active mechanisms of the therapy, operationalized as the underlying change processes that causally underpin therapeutic benefit. While active components will necessarily impact on one or more active mechanisms, knowing the most effective components of a therapy is distinct from knowing how this component leads to symptom change [i.e., its underlying mechanism(s)]. For example, in CBT, identifying behavioral activation as an active therapy component does not necessarily confirm that the mechanism-of-action is behavioral as behavioral activation may work through changing cognitions.

Understanding the mechanisms or the active components of psychological treatments are important because either potentially enables the development of more direct, precise, potent, simpler, briefer, and effective treatments. Understanding the active components of a psychological therapy is necessary in order to parse and distil the therapy to focus on what is essential and most engaging to patients.

Psychological treatments are complex interventions, typically made up of multiple elements and components, including the particular content and techniques of the therapy, the interaction between the therapist and patient, the structure of the therapy, and the mode and organization of delivery, each of which potentially acts via distinct mechanisms. Therapy is thus a complex multifactorial process. Any or none of these factors could contribute to the efficacy of an intervention, alone or in interaction with the other factors. It is therefore critical to determine the beneficially active, inactive, or inert, and iatrogenic components within an intervention so that the intervention can be honed to become optimally effective, by focusing on the active elements and by removing irrelevant or unhelpful elements ( 12 ).

Relatedly, if we know the active mechanisms of an intervention, we may be able to adapt the intervention or develop novel approaches to more directly target this mechanism and, thereby, increase the efficacy of the intervention.

Because of the high prevalence of common mental health problems, there is also a scalability gap because there are not sufficiently available therapists to tackle the global burden of poor mental health ( 13 ). It is therefore critical that ways are found to make treatments more efficient, scalable, and easier to train and disseminate. Understanding the underlying components of therapy and being able to remove unnecessary elements may make psychological therapies more effective and more cost-effective by streamlining and simplifying the treatment. For example, the same treatment benefit could be achieved from fewer sessions, enabling a greater volume of patients to be treated for the same volume of therapists. Understanding the critical active components of therapy will also help to adapt treatments for the alternative delivery means that are necessary for increased scalability (for example, to convert for self-help, lay provision, or digital interventions), without losing the core elements needed for efficacy. Understanding how therapy works will also make it easier to effectively train and disseminate therapies, facilitating wider treatment coverage. This understanding may also help to identify moderators of treatment outcome and more effectively personalize therapy to each individual.

Common Versus Specific Treatment Factors

One key issue with respect to resolving the underlying mechanisms underpinning the efficacy of psychological treatments concerns the question of whether treatment works through specific versus non-specific common factors ( 8 , 14 ). Specific factors are procedures or techniques arising from the particular therapy approach, such as those typically described in structured treatment manuals, for example, cognitive restructuring in CBT; exposure in CBT for anxiety disorders. Common (or non-specific) factors are those that are hypothesized to be common across all psychological interventions. The most important of these include a positive and genuine relationship between the therapist and patient, engendering positive expectancies and hope in the patient, and a convincing rationale that explains the symptoms experienced and gives credible reasons for the treatment to be helpful ( 15 ). There is a long-standing and still unresolved debate between those who propose that psychotherapies mainly work through specific factors versus those who propose that psychotherapies mainly work through common factors.

One argument made in support of common factors is that different specific psychotherapies are generally not found to differ in efficacy, although this does not logically rule out that treatments may work via different mechanisms ( 16 ). A recent review concludes that there is as yet no conclusive evidence that either common or specific factors can be considered a validated working mechanism for psychotherapy, in other words, the evidence is insufficient to determine the role of either ( 8 ).

The relative contribution of common versus specific factors in the efficacy of psychological interventions has important implications for how therapists should be trained, how therapies should be delivered, and for how treatment services should be organized. If the substantive part of the treatment effect is due to common factors, then therapy training should predominantly emphasize therapists learning how to develop a strong therapeutic relationship, develop a rationale etc. In parallel, therapy research should focus on understanding how to strengthen positive common factor effects. However, if specific factors are important then these also need to be emphasized in training and delineated in further research. Furthermore, the increasing importance of specific factors indicates a potentially greater need for discriminating and selecting therapy to match the individual clinical presentation.

Methodologies to Examine the Mechanisms of Psychotherapy

Comparative randomized controlled trials.

One reason for limited progress in understanding the mechanisms of psychological treatments is the focus on parallel group comparative randomized controlled trials (RCTs). Parallel group RCTs are the gold standard for establishing if an intervention works more than another intervention or against a control and the best means for establishing the relative efficacy of one treatment intervention versus another. However, they are not designed for investigating the specific mechanisms of how interventions work or identifying the active components of therapy. Because comparative RCTs can only compare the overall effects of each intervention package, they are not intended to and unable to provide information about the performance of the individual elements within complex multifactorial interventions. In standard comparative RCTs, all of the multiple treatment components and factors in an intervention package and their hypothetical mechanisms are aggregated and confounded together in the comparison of one treatment versus another. As a consequence, this design is unable to test specific main effects of treatment components nor any possible synergistic or antagonistic interactions between individual treatment components, limiting advances in mechanistic understanding. If an RCT finds one treatment better than another, we do not know which components made a difference; if there is no difference, we do not know whether there are any components that effected an improvement.

This limitation of standard comparative RCTs also applies to their ability to resolve the relative contribution of specific versus common factors. One major issue concerns the difficulty in finding an adequate control arm to compare against a putative active treatment to distinguish the role of specific versus non-specific factors. Some comparative RCTs and meta-analyses have found that one therapy has outperformed another therapy ( 17 , 18 ), which proponents of specific factors have argued as evidence for specific treatment effects. However, proponents of the common factors model have counter-argued that sometimes the comparison treatments used are not bona fide therapies, defined as viable treatments that are based on psychological principles and delivered by trained therapists, and thus that this is not a fair comparison. When comparisons are made between bona fide therapies, no differences in efficacy are found ( 19 ).

Relatedly, other designs have compared an active treatment to a psychotherapy placebo or attentional control on the argument that any differential beneficial effect observed for the active treatment will then be due to specific factors as the effects of the attentional control can only be due to common factors. However, most psychotherapy placebos do not control for all the potential common factors hypothesized in therapy, and thus, any difference found between a placebo and an active treatment could be due to either specific or common factors or some combination thereof ( 20 ). For example, it is hard to generate psychotherapy placebos that are exactly matched to active treatments in therapy rationale and credibility, without the placebo itself becoming a bona fide treatment. Similarly, psychotherapy placebos tend to differ from active treatments with respect to the structure of the therapy, for example, the number and duration of sessions, training of therapist, format of therapy, and range of topics covered. A meta-analysis of comparative trials found that there were larger effect sizes found between active treatments and structurally inequivalent placebos than between active treatments and structurally equivalent placebos, for which there were negligible differences ( 20 ). These difficulties in finding matched placebo controls or bona fide interventions have limited the conclusions that can be reached about the relative contribution of specific or common factors examined in parallel RCTs.

Attempts have also been made in RCTs to determine mechanisms by examining changes in putative mediators. For example, in trials of CBT, measures of change in negative thinking are examined as a mediator of symptom change. However, these mediational approaches are necessarily limited because they are still indirect and correlational ( 7 ). Even if an intervening variable is found to statistically account for the relationship between the treatment and its outcome, this does not provide strong evidence of a mechanism of change, because it does not support a strong causal inference that the mediator influences outcome. In such associations, the mediator may be a proxy to another variable(s) and there may be another unknown or unmeasured variable that is related to both the outcome and the mediator. Ultimately, direct experimental manipulation of the relevant factor is required for strong causal inference, and this is not possible for multiple elements of psychological interventions within a parallel group comparison RCT.

Component Study Designs

One experimental approach that has been used to examine the specific elements of psychological interventions is the component study ( 9 ), in which the full intervention is compared with the intervention with at least one component removed (a dismantling study) or in which a component is added to an existing intervention to test whether it improves outcomes (an additive study) ( 21 ). In principle, this approach can enable a strong causal inference that a component has a direct effect on outcome if there is a significant difference in outcomes between the variant of the therapy with a component and the variant without that component.

Nonetheless, there are limitations of component designs. First and critically, the component design does not necessarily test the main effect of a component, that is, the difference between the mean response in the presence of a particular component and the mean response in the absence of the particular component collapsing over the levels of all remaining factors. This can be illustrated with reference to one of the seminal dismantling studies—the dismantling study of CBT for depression by Jacobson and colleagues ( 22 ). In this study, patients with depression were randomized to either the full CBT treatment package including behavioral activation, cognitive restructuring to modify negative automatic thoughts, and work on core schema, or to behavioral activation plus cognitive restructuring or to just behavioral activation element alone, with 50 patients in each arm. No significant difference was found between the three versions, leading some observers to suggest that behavioral activation alone is sufficient for the effects of CBT on depression. However, it is important to realize that all versions of the treatment involved behavioral activation: as a consequence, for example, the trial is testing the effect of cognitive restructuring in the context of behavioral activation versus behavioral activation alone. It can only tell us the effect of that component in the context of the other component. Thus, the effects estimated are only the simple effects of each component with the remaining component set to one specific level. For example, for cognitive restructuring, this design only reveals the effect of cognitive restructuring in the presence of behavioral activation. It does not test the main effect of cognitive restructuring, i.e., does the presence of cognitive restructuring have a treatment effect relative to the absence of cognitive restructuring. Similarly, because there is no condition without behavioral activation, it is not possible to estimate the direct main effect of behavioral activation.

Second, the component design assumes that there is no interaction between the components, that is, that the effect of one component is independent of the presence or absence of other components. This may not always be a realistic assumption. For example, it is possible that behavioral activation and cognitive restructuring either complement each other or are antagonistic to each other.

Third, there is a concern that most component studies are not sufficiently powered to detect a difference between two potentially active treatment arms. For example, it has been estimated based on the assumption that a minimally clinically important difference for depression is d=0.24 that a trial would need 274 participants in each condition.

The Factorial Approach

We propose the use of factorial and fractional factorial designs as an alternative methodological approach to standard comparative RCTs and component designs, which has advantages over both for resolving the active components of psychotherapy. Factorial experiments allow one to explore main effects of factors and interactions among factors ( 23 – 27 ).

Factorial designs systematically experimentally manipulate multiple components or factors of interest. Indeed, factorial designs are commonly used to test the role of different factors simultaneously in experimental psychology. As such, they meet the requirement for delineating active components raised by multiple commentators ( 8 , 10 , 14 ). For example, the Institute of Medicine (2015, p3-10) recently proposed that “determination of which elements are critical depends on testing of the presence or absence of individual elements in rigorous study designs,” which is exactly what a factorial design delivers.

To give a clinical example, if the Jacobson and colleagues dismantling study of CBT for depression was redesigned as a full factorial study, patients would be randomized across three factors [presence or absence of behavioral activation (BA + vs BA - ); presence or absence of cognitive restructuring (CR + vs CR - ); presence or absence of work on core schema (CS + vs CS - )]. This means that patients would be randomized to be balanced across 8 treatments cells reflecting all of the possible combinations: all three elements (BA + : CR + :CS + ); 2 of the 3 elements (BA + : CR + :CS - ; BA + : CR - :CS + ; BA - : CR + :CS + ); 1 of the 3 elements (BA + : CR - :CS - ; BA - : CR + :CS - ; BA - : CR - :CS + ); or none of these elements (BA - :CR - :CS - ). This design can test the main effect of each factor as well as their interactions by comparing the mean effects of combined sets of cells against each other. For example, comparing all 4 cells with BA versus all 4 cells without BA tests the main effect of behavioral activation. The difference from the dismantling design is clear because the dismantling design only has 3 of these 8 combinations (BA + : CR - :CS - ; BA + : CR + :CS - ; BA + : CR + :CS + ), which limits it to only testing simple effects.

Factorial designs have been used extensively in engineering to optimize processes. In the last decade, they have been used to good effect in behavioral health, for example, in enhancing interventions for HIV care and prevention ( 28 ) and smoking cessation ( 29 , 30 ). This approach seems well-suited to expanding to the further understanding of psychological treatments and has been recently adopted in several recent trials ( 31 , 32 ). We believe that factorial designs have advantages for investigating how psychotherapy works that overcome many of the disadvantages noted earlier for comparative RCTs and component trials, as we will outline throughout this paper.

A fractional factorial design is a variation on the factorial design that employs a systematic approach to reduce the number of experimental conditions to allow a more manageable study, at the cost of allowing only main effects and a pre-specified set of interactions to be tested. Fractional factorial designs require the assumption that higher-order interactions are negligible in size, because they are confounded, or aliased, with lower-order effects.

The IMPROVE-2 Study as an Example of a Factorial Design

We illustrate the use of a fractional factorial design to identify the active ingredients and mechanisms of an intervention, with respect to a specific example - the IMPROVE-2 study (Implementing Multifactorial Psychotherapy Research in Online Virtual Environments) [see ( 32 ) for further detail). The IMPROVE-2 study is a Phase III randomized, single-blind balanced fractional factorial trial based in England and conducted on the internet. Adults with depression (operationalized as Patient Health Questionnaire-9 scores ≥ 10) recruited directly from the internet and from an UK National Health Service Improving Access to Psychological Therapies service were randomized across seven experimental factors, each reflecting the presence versus absence of specific treatment components within internet-delivered CBT, guided by an online therapist (activity scheduling, functional analysis, thought challenging, relaxation, concreteness training, absorption, self-compassion training) using a 32 condition balanced fractional factorial design (2 iv 7-2 ) (see Table 1 ).

Table 1 Experimental groups of the IMPROVE-2 fractional factorial design.

All components involved brief prescribed therapist online support to improve retention and adherence, in which secure online written feedback was provided at the end of each completed module (typically fortnightly), with the option for additional secure messaging between therapist and patient. Therapist feedback highlighted positive steps made, encouraged participants to continue to practice previously introduced components, addressed questions and homework, and pointed out areas to focus on in the next module. Therapists were low-intensity Psychological Wellbeing Practitioners and an experienced clinical psychologist.

The IMPROVE-2 trial used a fractional factorial design to retain the benefits of a factorial design while making the study more logistically manageable and feasible to deliver: this fractional factorial design reduces the total number of conditions from 128 to 32. Each component has two “levels” to be compared in the fractional factorial design: either present or absent, i.e., the respective treatment modules are either provided or not provided in the internet platform. IMPROVE-2 therefore tests the main effects and selected interactions for these 7 components within internet CBT for depression to determine the active ingredients of internet CBT. We first outline the general framework used for this study—the Multiphase Optimization Strategy (MOST)—and then explore the particular benefits and methodological issues of using the factorial design to study psychotherapy.

The Multiphase Optimization Strategy (MOST)

Within IMPROVE-2, the factorial design is used as one stage within a wider framework for improving interventions—the Multiphase Optimization Strategy (MOST) ( 33 – 38 ) approach. MOST, rooted in engineering, agriculture, and behavioral science, is a principled and comprehensive framework for optimizing and evaluating behavioral interventions ( 33 – 38 ).

MOST consists of three stages: a preparation stage in which the relevant factors and components to be investigated are identified; an optimization stage in which a factorial experiment is used to evaluate the main effects and interactions of each factors; and then an evaluation stage, in which an optimized intervention based on the results of the previous trial is tested in a RCT. MOST has been established to enhance treatments for smoking cessation, with earlier factorial designs identifying active components ( 29 ), which were then combined into a novel intervention which outperformed recommended standard care in a RCT ( 39 ). MOST is well-validated ( 29 , 30 , 34 , 40 ) and recommended within the Medical Research Council Complex Intervention guidelines ( 41 , 42 ). A key advantage is greater experimental efficiency, with a focus on identifying “active ingredients” versus “inactive” or extraneous components before moving onto large-scale comparative trials, resulting in fewer overall resources required to answer the research questions in the long run than with the traditional approach ( 43 ). However, to date, MOST has not been applied to psychological interventions for mental health.

The IMPROVE-2 trial is one of the first attempts to apply the MOST approach to psychological interventions, building on the preparation and optimization phases so far. It incorporates the MOST approach with an internet delivery format for CBT to build in treatment reach, scalability, and increased treatment coverage for the optimized treatment from the start, as the goal is to develop an optimized and scalable evidence-based treatment. Another benefit of using such an internet-delivered therapy is that treatment content can be standardized and fixed, and written therapist responses can be closely demarcated, reducing unwanted “drift” from treatment protocols. This helps prevent potential contamination between different treatment components, which is an important consideration for a factorial design.

The Preparation Stage in MOST

During the preparation stage, a conceptual model for the intervention is developed, and discrete and distinct intervention components are selected. These components are then pilot tested for acceptability, feasibility, evidence of effectiveness, and ease of implementation, and refined as needed. MOST also involves the identification of the optimization criterion, which is the operational definition of the target change sought that is used to judge the optimal intervention, subject to resource or other constraints. For example, this might be greatest symptom improvement that could be obtained for a particular cost or for a particular duration of treatment.

With respect to the IMPROVE-2 study, a previous feasibility study (IMPROVE-1) established that it was feasible to maintain treatment integrity and fidelity across randomization into multiple treatment conditions and to avoid contamination across treatment conditions. Because the IMPROVE-2 study is focused on determining the ingredients of internet-CBT that are most effective for treating major depression in adults, the operational definition for the optimization criterion was the largest reduction in depressive symptoms, as indexed by using change in scores on the Patient Health Questionnaire-9 score (PHQ-9) ( 44 ) as the primary outcome.

Components Within the Psychological Intervention

A key step within this preparation phase is to identify the components that are to be targeted. When planning a factorial study, the best components to choose are those that are: related to a specific conceptual model; distinct from each other in content, approach or delivery method; have some evidence of efficacy, that can be independently administered, i.e., one component is not dependent on another for delivery; and that are hypothesized to address one or two theoretical mediators. In essence, it is important that components can be distinguished from each other in a meaningful way and that they are conceptually related to different mechanisms.

The elements or components selected can be at different levels of analysis and abstraction. The level selected will depend on the specific question or conceptual model. For example, for CBT, the components chosen could relate to the main hypothesized theoretical mechanisms of change and their associated elements, such as activity monitoring and scheduling and detecting and testing automatic thoughts. Alternatively, the components could relate to lower-level, more discrete elements within the treatment techniques such as the behavioral change techniques outlined in a recent taxonomy ( 45 ). These behavioral change techniques include behaviors such as self-monitoring, goal-setting, and feedback, which are common across different CBT components as well as other psychotherapy modalities. Alternatively, the components could relate to process-related aspects of therapy such as whether the intervention is therapist-supported versus unsupported, or structural aspects, such as the frequency of treatment sessions.

IMPROVE-2 illustrates the selection of components to be examined. Consistent with the principles above, the IMPROVE-2 study chose treatment components that were conceptually and operationally distinct from each other, so that each can be evaluated independently. As the first attempt to disentangle the active components within CBT for depression, components were chosen that were clearly distinct and that could be linked to the main theorized mechanisms of action in CBT. These components were operationalized at a relatively high-level (e.g., thought challenging to reflect cognitive theories of change; activity scheduling to reflect behavioral theories of change) rather than in terms of the more localized behavioral change taxonomy because the goal was to determine the core components relating to key theoretical conceptualizations of CBT and to maximize the likelihood of finding a positive effect. If, for example, thought challenging was found to be a strong active ingredient, then further studies could dissect which elements including more specific behavioral change techniques are critical to the effects of thought challenging. Three of the components chosen had been identified as elements for CBT for depression, using a Delphi technique ( 46 ): applied relaxation; activity monitoring and scheduling; detecting and reality testing automatic thoughts. A further component—functional analysis—is a mainstay of behavioral approaches to depression including behavioral activation ( 47 ). Three components related to recent treatment innovations in CBT derived from experimental research ( 48 , 49 ), with each hypothesized to specifically target distinct mechanisms arising from different theoretical models: self-compassion, concreteness training, and absorption. The components selected relate to three theoretical accounts of how CBT might work: a behavioral account, a cognitive account, and a self-regulation account.

Three components related to behavioral models of depression and of how CBT works. Depression has been hypothesized to result from a reduction in response-contingent positive reinforcement ( 50 ), in which the individual with depression experiences less reward and sense of agency as a consequence of changing circumstances (e.g., loss), poor skills, or avoidance and withdrawal. Within the behavioral conceptualization, activity scheduling is hypothesized to increase response-contingent positive reinforcement by increasing frequency of positive reinforcement thorough building up positive activities. This treatment component provides psychoeducation about the negative effects of avoidance, includes questionnaires to help patients identify their own patterns of avoidance, provides guidance on activity scheduling to build up positive activities and reduce avoidance (e.g., breaking plans into smaller steps; specifying when and where to implement activities), and exercises in which participants generate their own activity plans.

In parallel, functional analysis seeks to determine the functions and contexts under which desired and unwanted behaviors do and don't occur and, thereby, find ways to systematically increase or reduce these behaviors, by exploring their antecedents, consequences, and variability, and then either alter the environment to remove antecedent stimuli that trigger unwanted behaviors and/or practice incompatible and constructive alternative responses to these antecedents. This approach is based on Behavioral Activation (BA) ( 51 ) and rumination-focused CBT ( 49 ) approaches to depression. More specifically, functional analysis is proposed to target habitual avoidance and rumination by identifying antecedent cues, controlling exposure to these cues, and practicing alternative responses to them ( 52 ).

Absorption training is also hypothesized to increase response-contingent positive reinforcement by increasing direct contact with positive reinforcers. Absorption training is focused on teaching an individual to mentally engage and become immersed in what he or she is doing in the present moment to improve direct connection with the experience and enhance contact with positive reinforcers. It is designed to overcome the effects of detachment and rumination which can prevent an individual experiencing the benefits of doing positive activities. When delivered within the internet treatment, patients complete a behavioral experiment using audio-recorded exercises to compare visualizations of memories of being absorbed versus not being absorbed in a task, practice generating a more absorbed mind-set using downloadable audio exercises, and identify absorbing activities.

Two components within the factorial design are based on a cognitive conceptualization of depression, in which the negative thinking characteristic of depression, is hypothesized to play a causal role in the onset and maintenance of depression, and, thereby, reducing negative thinking is hypothesized to be an active mechanism in treating depression ( 53 , 54 ). Central within CBT for depression is the use of thought challenging or cognitive restructuring to reduce negative thinking ( 55 ), and this forms one component in the IMPROVE-2 trial. The internet treatment module that delivers the thought challenging component involves psychoeducation about negative automatic thoughts and cognitive distortions, vignettes of identifying and challenging negative thoughts, and written exercises in which patients practice identifying and then challenging negative thoughts using thought records.

The other cognitive-based component involves concreteness training, based on an intervention found to reduce symptoms of depression in a previous RCT ( 48 ) and derived from experimental research indicating the benefits of shifting into a concrete processing style ( 56 , 57 ). Within the IMPROVE-2 trial, the internet treatment module that delivers this component involves psycho-education about depression, rumination, and overgeneralization, a behavioral experiment using audio-recorded exercises to compare abstract versus concrete processing styles, and downloadable audio exercises to practice thinking about negative events in a concrete way. Unlike thought challenging, concreteness training does not test the accuracy or veridicality of negative thoughts but rather trains patients to focus on the specific and distinctive details, context, sequence (“How did it happen?”), and sensory features of upsetting events to reduce overgeneralization and improve problem-solving. Concreteness training is therefore hypothesized to specifically reduce the overgeneralization cognitive bias identified as important in depression ( 53 , 58 ).

The remaining treatment components are hypothesized to directly improve emotional regulation. Relaxation is hypothesized to improve self-regulation by targeting physiological arousal and tension. In IMPROVE-2, a variant of progressive muscle relaxation and breathing exercises was used to reduce physiological arousal and tension in response to warning signs, based on trial evidence that this intervention alone reduces depression ( 48 ). The treatment component introduces a rationale for relaxation, provides an online relaxation exercise as a behavioral experiment to test if it reduces tension, and a downloadable relaxation exercise.

Self-compassion training is proposed to activate the soothing and safeness emotional system, hypothesized to be downregulated in depression ( 59 ). Recent research has highlighted the potential benefit of increasing self-compassion in treatments for depression ( 49 , 60 – 62 ), although self-compassion has not yet been directly tested within a full-scale clinical trial for patients with major depression. Within this treatment component, patients read psychoeducation about compassion including useful self-statements to encourage and support oneself, complete a behavioral experiment that compares their own self-talk to how they talk to others, try an audio-recorded exercise visualizing past experiences of self-compassion to activate this mind-set and test its benefits, which is downloadable for further practice, and identify activities they would do more of and activities they would do less of to be kinder to themselves.

The Optimization Stage of MOST: Factorial Experiments and Their Benefits

The second stage of MOST involves optimization of the intervention, typically through a component selection experiment (sometimes called a component screening experiment), using a factorial or fractional factorial design. This factorial experiment is used to specifically determine the individual effects of each component and any interactions between components. It is important to note that this step could involve multiple experiments and an iterative process of further refining the intervention. For example, if the first component screening experiment observed statistically significant moderators of treatment outcome, such as mode of treatment delivery or location of treatment, a further experiment could be conducted in which the moderators are introduced as factors into the factorial experiment so that they are directly manipulated to enable stronger causal inference about their potential contribution to outcome.

Advantages of Factorial Design

There are at least four advantages to the use of a factorial design in resolving how therapy works and what its active mechanisms are.

Advantage 1: Directly Testing Individual Components and Their Interactions

The factorial experiment provides direct evidence about the effects and interactions of individual components within a treatment package, which is necessary for methodically enhancing and simplifying complex interventions ( 41 ). It can test each individual component and determine its main effect. Critically, it can also determine possible interactions between components, which other experimental designs are unable to do. Thus, a factorial design has distinct advantages when one needs to determine whether the presence of one component enhances or reduces the effect of another. This approach enables us to identify the active components of therapy and to select active and reject inactive/counter-productive components or elements. By comparing the presence versus absence of each component, this factorial design can examine the main effect of each component on the primary outcome, for example, testing whether thought challenging reduces symptoms of depression.

With respect to the IMPROVE-2 study, it is important to note that despite the many trials of CBT for depression, no trials have directly tested the main effect of each of the selected treatment components—for example, does thought challenging have a direct effect on reducing depression relative to no thought challenging? This design therefore provides the first fully-powered test of the main effects of these ingredients of CBT for depression. Table 1 describes the specific combinations of the two-level intervention factors in the experimental design.

To illustrate how the factorial design works, consider Table 1 . Main effects and interactions are estimated based on aggregates across experimental conditions. For each main effect, half of the study population are randomized to one level of the factor (e.g., in conditions 9–16, 25–32, presence of concreteness training) and half will be randomized to the other level of the factor (e.g., in conditions 1–8, 17–24, absence of concreteness training). Therefore, the main effect of concreteness training can be determined by comparing the average effect of conditions 9–16, 25–32 versus conditions 1–8, 17–24.

Technically, the IMPROVE-2 study is an internet-delivered component selection experiment with seven experimental factors evaluated, each at two levels ((presence, coded as +1 versus absence, coded as -1 of component, effect coded), using a 32-condition balanced fractional factorial design (2 IV 7-2 ). Effect coding is used because it ensures that main effects and interactions are independent.

A full factorial design of seven factors would have required 2 7 = 128 conditions, which was deemed to be impractical and too complex to program and administer, and thus a fractional factorial design was chosen. For IMPROVE-2, a 2 7-2 fractional factorial design was chosen, which reduces the number of experimental conditions by a factor of four, down to 32 conditions. While the full factorial design necessarily includes all possible combinations of all factors, within a fractional factorial design the researcher has to strategically and carefully select a subset of the experimental conditions available.

The first consideration when selecting the subset of the experimental conditions is statistical, with a need to maintain a balanced design in which every factor occurs at an equal number of times at each of the two levels, and in which all factors are orthogonal to each other. This necessarily limits the potential configurations of subsets available. These designs can be mapped out using factorial design tables ( 63 ) or statistical packages (e.g., PROC FACTEX in SAS).

The second key consideration is to select the subset of experimental conditions that maximizes the ability to estimate the main effects and interactions that are of highest priority for the research question. Typically, estimating the main effects of the intervention components is a priority. For a fractional factorial design, some of the main effects are going to be confounded (typically referred to as “aliased” within the factorial literature) with higher-order interactions, and thus the subset of experimental conditions needs to be carefully selected so that the main effects are only aliased with higher-order interactions that are judged to be less likely to be significant (e.g., 3-way or 4 way-interactions) or of less theoretical interest.

For IMPROVE-2, the selected design allows the estimation of all main effects and several pre-specified 2-factor interactions among the seven intervention factors; in statistical terminology, it is a Resolution IV design because main effects are only aliased with 3-way and higher interactions. This means that if a potential effect is observed for a particular component, technically the observed effect is due to the sum of the main effect itself and the specific aliased higher-order interactions, i.e., the estimated lower-order effect may include contribution from these higher-order effects. For example, the main effect of concreteness is aliased with the 4-way interaction of functional analysis by compassion by absorption by thought challenging, and the 4-way interaction of functional analysis by compassion by relaxation by activity scheduling and the 5-way interaction of absorption by concreteness by relaxation by thought challenging by activity scheduling. Thus, the actual effect observed is due to the sum of the main effect plus the 4-way and 5-way interactions. If this comparison is significant, the most likely explanation is that the presence of concreteness training produces better treatment outcomes than the absence of concreteness training although we cannot rule out in the fractional design that configurations of 4 and 5 components, albeit unlikely, could contribute to this effect. In interpreting the results, the assumption is that the 3-way and higher interactions are highly likely to be negligible, based on extensive research and principles within factorial experiment research ( 27 , 63 ). Although in most cases this assumption is reasonable, it may not always apply.

In designing the study, several 2-way interactions were pre-specified as being of particular interest, where it was hypothesized that components might interact with each other, and the design was explicitly chosen so that these 2-way interactions were only aliased with 3-way or 4-way interactions, which we typically expect to be negligible. For example, it was hypothesized that activity scheduling and absorption treatment components may have a positive synergistic effect because the former increases the number of positive activities engaged in, whereas the latter increases the potential absorption and connection with these activities. Similarly, it was hypothesized that thought challenging and self-compassion components may have a positive synergistic effect because thought challenging helps individuals to look logically for evidence against and alternatives to negative self-critical thoughts, while self-compassion encourages a more kindly and tolerant approach to tackle self-criticism.

One choice within the design of the fractional factorial is whether or not it includes the experimental condition in which all intervention components are set to the low level or absent, i.e., a no-treatment control. For the purposes of investigating the active ingredients of therapy, this condition is not necessarily required, since the logic of the factorial experiment is not to compare all the conditions directly with each other, as we would in a comparative RCT, but rather to identify the active components by aggregating mean effects across each factor.

For IMPROVE-2, the fractional factorial design explicitly excluded the condition in which participants receive no treatment components. This has several potential advantages. First, it means that there is not a no-treatment or treatment-as-usual condition, so that the design and trial was suitable for use in a clinical service, where it would not be possible or ethical to randomize patients to not receive any active treatment. Second, because all participants are randomized to active treatment, they are more likely to remain engaged in the trial and to not judge that they are receiving the “inferior option” as can sometimes occur for control conditions.

Within the IMPROVE-2 fractional factorial design, all participants were randomized to receive at least one component of CBT and in the majority of cases 3 or 4 components of CBT. Based on the experience of the IMPROVE-1 feasibility study, in which many patients only completed their first few treatment modules, the IMPROVE-2 counter-balanced the order in which the treatment modules delivering each treatment component were received in the internet platform to ensure that each component was received equally often across all participants as patients progressed through the therapy. In this way, the number and order of treatment components was equivalent between the high (presence) and low levels (absence) of each factor. Of course, this leaves open the question of whether the order of receiving treatment components might be important or not: given the iterative nature of the MOST approach, the effect of sequencing treatment components on efficacy could be a further question for a subsequent component screening experiment.

Advantage 2: Manipulation of Hypothesized Mechanisms and Examination of Individual Mediators

The factorial design allows research on the working mechanisms and mediators that allows strong causal inference because each factor associated with a hypothesized specific mechanism is manipulated and the effect of manipulating this factor can be tested directly on secondary measures indexing the putative mediator. The design also enables examination of the mediators of each individual intervention component, because each factor is manipulated independently. For example, this design can test whether the presence of a thought challenging component has a main effect on reducing self-reported negative thinking relative to the absence of thought challenging, and whether this change in thinking mediates change in depression.

To maximize this opportunity to test mediators, the IMPROVE-2 trial required all patients to complete a series of self-report questionnaires at baseline and at each follow-up assessment (at 12 weeks and 6 months post-randomization), as well as after each completed treatment module that index all the putative mediators across all the treatment components. For each treatment component, the putative mediator was related to the primary mechanism which each treatment component is hypothesized to most strongly influence, including rumination (5-item Brooding scale) ( 64 ) for the functional analysis component, overgeneralization (adapted Attitudes to Self Scale – Revised) ( 58 ) for the concreteness component, self-compassion scale ( 65 ) for the self-compassion component, negative thinking (Automatic Thoughts Questionnaire) ( 66 ) for the thought challenging component; increased behavioral activity and reduced avoidance (Behavioral Activation for Depression Scale Short-form) for the activity scheduling component ( 67 ), and absorption and engagement in positive activities, adapted from measures of “flow” for the absorption component ( 68 ). Mediational analyses can then be used to test the hypotheses that each treatment component primarily works through the hypothesized mediator, using the analytical approach outlined by Kraemer et al. ( 69 ) and modern causal inference methods. In addition, IMPROVE-2 will investigate potential moderation of the treatment components by site, age, sex, severity of depression, co-morbid illness, and antidepressant use. This design enables us to test whether manipulating a particular component influences the underlying process it is hypothesized to change, and whether that process in fact mediates symptom change. By assessing all putative mediators for all components, we can also test whether components influence other processes, e.g., whether components tackling behavior change cognition or vice versa.

Advantage 3: Improved Delineation of Specific Versus Common Treatment Factors

The factorial design provides a stronger test of the relative contribution of specific versus non-specific common treatment factors than existing designs. As noted earlier, the majority of control comparisons are inadequate for disentangling specific from non-specific treatment effects because of the difficulty in creating psychotherapy placebos (attentional controls) that match a bona fide psychotherapy for credibility, rationale, and structure. However, the factorial design overcomes this limitation because for any treatment component (e.g., the relaxation component in IMPROVE-2), the aggregate of the conditions where it is present (i.e., Table 1 , conditions 17–32) are equivalent for treatment credibility, structure, delivery, rationale, therapist contact, therapist content and techniques and therapist allegiance with the aggregate of the 16 conditions where it is absent (i.e., Table 1 , conditions 1–16), except for the specific treatment component itself. Moreover, these conditions are also matched in aggregate for all the other six treatment components, since these are balanced in the design. The evaluation of the main effect of relaxation involves the comparison of the average effect for the conditions where relaxation is present versus for the conditions where relaxation is absent. This design therefore provides the strongest control condition available and one that is able to disentangle specific from non-specific common treatment factors. More specifically, this approach is a rigorous test of whether there are specific treatment effects arising from particular treatment components in addition to any non-specific factors common across the treatment components. If there is a significant main effect for any component in IMPROVE-2, then this is strong evidence for a specific treatment effect above and beyond all the non-specific common therapy factors present in CBT. The nature of the non-specific factors tested will depend on the specific components compared in the trial design: because IMPROVE-2 exclusively examines components within internet-CBT, it confounds non-specific factors common across therapies (e.g., therapeutic alliance, rationale) and those specific to internet-CBT and common to all components (e.g., self-monitoring; homework). A different study that took components from different treatment interventions could better delineate non-specific effects common to all therapies. This approach would not rule out some contribution of common factors to treatment outcome, as common factors would be matched across the two levels of the factor, but would be definitive evidence for a specific treatment effect. Conversely, if none of the components were found to have a significant main effect (assuming sufficient power), this would suggest that any treatment benefit was due to common factors.

Advantage 4: Factorial Designs Are Efficient and Economical

Factorial designs are efficient and economical compared to alternative designs such as individual experiments and single factor designs because they often require substantially fewer trials and participants to achieve the same statistical power for component effects, producing significant savings in recruitment, time, effort and resources ( 23 , 43 ).

For example, as an alternative to the factorial design used in IMPROVE-2, a research program could investigate each of the components separately in seven individual experiments or conduct a comparative RCT or a component trial (dismantling or additive design). For IMPROVE-2, it was assumed that the smallest Meaningful Clinical Important Difference (MCID) would be a small effect size (Cohen's d or standardized mean difference=.2) for the main effect of an individual treatment component or interaction between components on pre-to-post change in depression. An alpha of 0.1 was chosen as this is recommended for component selection experiments to decrease the relative risk of Type II to Type I error when selecting treatment components; i.e., to avoid prematurely ruling out potentially active treatment components ( 23 , 36 ). In order to detect a MCID of d = 0.20 with 80% power at α = 0.10 per treatment, a sample size of N=632 was required (NQuery 7.0). Because participants provide at least five repeated measures on the primary outcome, latent growth curve modeling can be used, which was conservatively estimated to reduce sample size by 30% relative to only using first and last time-point as in an Analysis of Covariance, but then numbers were increased to account for estimated 40% dropout attrition post-treatment, giving a required total sample of N= 736 for the fractional factorial design.

However, the same MCID, power and attrition issues apply for all other trial designs. Thus, each individual experiment would need 736 participants to be adequately powered to examine each component: conducting seven separate experiments to investigate each of the seven components would require N= 5,152, or seven times as many participants as the factorial experiment. A parallel comparative RCT to compare each of the components against each other and against a no-treatment control would have 8 arms and require 368 participants per arm, thus requiring N=2,944, or four times as many participants as the factorial experiment. Similar calculations apply for component experiments – for example a dismantling study that compares a full treatment package (all seven treatment components combined), with incrementally dismantled packages, each with a component removed (i.e., all components minus compassion; all components minus compassion and absorption, etc.) would have 7 arms (assuming there is not a no-treatment control), each requiring 368 participants per arm, requiring N=2,576, or 3.5 times as many participants as the factorial design.

Factorial and fractional factorial designs are efficient and economical because rather than making direct comparisons between experimental conditions as in the other designs, the factorial design compares means based on aggregate combinations of experimental conditions. To illustrate within IMPROVE-2, as indicated in Table 1 , the estimate of the main effect of concreteness training is based on comparing the aggregate of conditions 9–16, 25–32 where it is present, versus aggregate of conditions 1–8, 17–24 where it is absent; the estimate of the main effect of relaxation is based on comparing sum of conditions 1-16 versus sum of conditions 17–32; the estimate of the main effect of thought challenging is based on comparing sum of conditions 1, 4, 6, 7, 10, 11, 13, 16, 17, 20, 22, 23, 26, 27, 29, 32 versus the sum of conditions 2, 3,5,8, 9, 12, 14, 15, 18, 19, 21, 24, 25, 28, 30, 31, etc. In this way all participants are involved in every effect estimate—it effectively recycles each participant by placing each participant in one of the levels of every factor. As such, the full sample size can be used to determine each of the main effects, making this design efficient for power and sample size.

The Evaluation Stage of MOST

The third stage in MOST is the evaluation of the optimized intervention. An optimized intervention is systematically built from the results of the factorial experiment by including the most active components with strongest effect sizes relative to the pre-specified optimization criterion, but excluding and eliminating weak inert or antagonistic components. This optimized intervention is tested against the standard evidence-based treatment in a parallel comparative RCT. Thus, to be clear, the MOST approach still retains the parallel comparative RCT as the best method to evaluate one treatment package against another, but adds the factorial design as the most efficient means to investigate the treatment components. In this way, the MOST framework uses rigorous design to identify active elements of a treatment, build a potentially better therapy and then test whether it is an improvement on existing active treatments.

IMPROVE-2 has not yet reached the optimized intervention and evaluation stage. Nonetheless, the logic is clear: based on the results of the IMPROVE-2 factorial experiment, a refined internet CBT treatment package would be produced by retaining those treatment components that had the largest effect sizes for depression, and by removing those components that had minimal or even negative effect sizes. Both the Pareto principle and prior MOST studies suggest that there will be variability in the treatment effect sizes of different components and their interactions, that not all components will be active in the therapeutic benefit of CBT, and indeed, that many will have insignificant effect sizes ( 30 ). As such, it should be possible to concentrate the therapy elements to make CBT more potent, and as a minimum more effective.

This process also considers any potential interactions between components. For example, if there was a significant positive two-way interaction between two components, such that adding one component to the another produced larger treatment effects than either on their own, then these factors may be added to the treatment package. In contrast, if there was a significant negative antagonistic interaction between two components, such that together the treatment benefit was less than either on their own, the component with the weakest positive main effect would be probably removed from the treatment package.

If an examination of the estimated effect size of the optimized intervention from the component selection experiment looked favorable, then this optimized intervention would then be tested against an established internet CBT for depression treatment package, to test whether these modifications improved treatment outcome. If the optimized intervention looked unlikely to out-perform existing treatments in the modeling of the treatment estimates, or was found to not be superior in a subsequent comparative RCT, then the MOST logic is that further iterations through the three phases are needed. If this approach indicates that some but not all components within internet CBT for depression have a significant effect size in reducing depression, it will lead to the building of better therapies that focus on the active ingredients and discard inert or iatrogenic elements.

Potential Limitations

The IMPROVE-2 trial is only one illustration of how the factorial approach could be used to delineate the active components of psychological therapies. As is true for any single study, it has specific limitations. First, it is relatively complex in utilizing seven components. This has the advantage of testing multiple putative active ingredients at once but the risk that with this complex design main treatment effects may be diluted. Adequate testing of treatment components in the factorial design requires each component to be delivered with sufficient difference between the presence and absence of the component to provide a fair test of its main effect. Because the components in IMPROVE-2 each reflect exposure to specific treatment content and techniques, this means that participants need to receive a sufficient dose of the respective content and techniques, that is, complete the relevant modules and practice the relevant behaviors. We sought to achieve this by having each component as a distinct module that is completed over several weeks, and whose content and techniques are then referenced and checked and practised in all subsequent modules and explicitly referred to in the subsequent written feedback from the therapist, to maintain their ongoing use. This meant that the “dose” of treatment elements should be comparable to proven internet CBT treatments and sufficient for testing the main effects.

Nonetheless, there are alternative approaches to tackling this issue. One alternative way to increase treatment dose would be to have a simpler design with fewer treatment components that each run over multiple modules. Another alternative is to test process-focused components such as the degree or nature of therapist support (e.g., support versus no support), or structural components such as the frequency of treatment sessions (e.g., weekly or twice weekly), both of which involving keeping therapy content constant. Such designs straightforwardly deliver a sufficient difference between the presence and absence of the treatment component. Of course, the selection of different components necessarily tests different hypotheses as to the active ingredients of therapy. At this point, it remains an empirical question which of these different components most contributes to treatment outcome. Each approach is equally valid. This is why we strongly advocate for multiple factorial trials to test these different dimensions so that we can systematically enhance therapy.

Related to this limitation, IMPROVE-2 used a fractional factorial design, which raises the potential risk of main effects being confounded with higher-order effects. While this risk is deemed to be very low because 3-way and 4-way interactions are unlikely to be significant, a full factorial design would avoid this assumption. A full factorial would be more suitable for designs utilising fewer components.

A further limitation of the IMPROVE-2 design is that all the components utilize a CBT framework and include generic CBT elements such as self-monitoring, planning, homework and homework review, Socratic review, building new activities, collaboration with the therapist, and a common CBT rationale focusing on thoughts and behavior. As such, if we were to find no main effects for any of the treatment components, we could not determine to what extent any treatment benefit observed was due to non-specific effects common across therapies (such as therapist alliance, remoralization) or due to non-specific effects particular to CBT. Nonetheless, this design still provides a better matched control to investigate specific main effects than prior designs and to test if there any specific main effects. Either pattern of findings (identifying one or more specific main effects of treatment components versus no main effects observed) would still be an advance on our current knowledge and could then be further explored further within the MOST framework.

We have reviewed the importance of better understanding the mechanisms and active ingredients of psychological treatments in order to refine, condense, and strengthen the potency and effectiveness of these treatments. We have shown that standard comparative RCTs and component trials have limitations for determining the specific treatment contributions of individual treatment components within a psychological treatment package and for inferring causality concerning treatment mechanisms. We have shown how factorial and fractional factorial trials can overcome these limitations and have the particular advantages of directly testing individual components and their interactions, of examination of individual mediators and experimental manipulation of hypothesized mechanisms, of being able to distinguish specific factors from common treatment factors, and of being economical and efficient with respect to sample size and resources.

This approach has been illustrated with respect to the IMPROVE-2 trial ( 32 ), which will provide the first examination of the underlying active treatment components within internet CBT for depression. Understanding the active components of therapy will enhance our understanding of therapeutic mechanisms and potentially enable the systematic building of more effective interventions. The IMPROVE-2 trial has completed the recruitment, treatment and follow-up stages, with 767 adult patients with depression recruited, and statistical analyses underway. It is anticipated that these analyses will significantly extend our understanding of how CBT works. We believe that this innovative approach may provide a useful means to address recent requests for rigorous study designs to determine which elements within psychological interventions are core active components ( 4 , 7 , 10 , 11 ).

Data Availability Statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Ethics Statement

The study protocol for IMPROVE-2 was reviewed and approved by the South West National Research Ethics Committee, NHS National Research Ethics Committee SW Frenchay (reference number, 14/SW/1091, 30/4/2015). The trial sponsor is the University of Exeter, contact person Gail Seymour, Research Manager.

Author Contributions

EW and AN both designed, prepared, and delivered the IMPROVE-2 study. EW prepared the first draft of the manuscript, AN commented on the draft, and both EW and AN finalized the manuscript.

Funding for the IMPROVE-2 study was provided by grants from the Cornwall NHS Partnership Foundation Trust and South West Peninsula Academic Health Research Network. Funding sponsors did not participate in the study design; collection, management, analysis, and interpretation of data; or writing of the report. They did not participate in the decision to submit the report for publication, nor had ultimate authority over any of these activities.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

The authors wish to thank profusely all the staff within Cornwall NHS Partnership Foundation Trust who supported this research and all the patients who volunteered to participate in the IMPROVE-2 study.

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Keywords: factorial, mechanism, psychotherapy, specific factors, common factors, cognitive behavioral therapy

Citation: Watkins ER and Newbold A (2020) Factorial Designs Help to Understand How Psychological Therapy Works. Front. Psychiatry 11:429. doi: 10.3389/fpsyt.2020.00429

Received: 19 June 2019; Accepted: 27 April 2020; Published: 14 May 2020.

Reviewed by:

Copyright © 2020 Watkins and Newbold. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Edward R. Watkins, [email protected]

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

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Factorial Experimental Design: A Comprehensive Guide For Researchers

September 30, 2021

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Factorial design is a statistical method used in experimental research that helps you study the effects of multiple factors simultaneously. It allows you to investigate the influence of multiple variables on survey outcomes in a systematic manner.

By monitoring interactions between factors, factorial experimental designs provides you insights into how these variables may interact to produce certain outcomes, which can help inform more nuanced interpretation of your survey findings.

Additionally, factorial designs are efficient in data collection. The method allows you to examine multiple factors within the same study, thus reducing the need to conduct separate experiments for each intended variable.

In this blog, we will understand the application of this experimental design in real world and explore its advantages.

What is Factorial Experimental Design?

Some experiments involve the study of the effects of multiple factors. For such studies, the factorial experimental design is very useful. A full factorial design, also known as fully crossed design, refers to an experimental design that consists of two or more factors, with each factor having multiple discrete possible values or “levels”.

Using this design, all the possible combinations of factor levels can be investigated in each replication. Although several factors can affect the variable being studied in factorial experiments, this design specifically aims to identify the main effects and the interaction effects among the different factors.

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Example of factorial design in real-world.

Let’s assume a company wants to launch a new product and is conducting a market research survey to understand the preferences of its target consumers. The objective of MR survey is to identify the factors that may potentially influence consumers’ purchase decisions and to use the insight to optimize the product features accordingly.

You can design the survey using factorial design principles by identifying the key factors that may influence purchase decisions, such as price, product features, marketing/advertising channels, brand reputation, and brand awareness.

You can manipulate each factor at different levels, such as high or low prices, marketing/advertising platforms, and product features, resulting in multiple conditions.

After gathering the survey responses, you can implement statistical data analysis to examine the main effects of each factor and any interactions between factors.

This provides valuable insights into which factors result in the greatest impact on consumer preferences.

Components of factorial experimental design

To understand the factorial experimental design, you must be well-acquainted with the following terms:

1. Factors:

This is a broad term used to describe the independent variable that is manipulated in the experiment by the researcher or through selection.

2. Main Effects:

The main effect of a factor refers to the change produced in response to a change in the level of the factor. Therefore, the effect of factor A is the difference between the average response at A1 and A2.

3. Interaction:

An Interaction between factors occurs when the difference in response between the levels of one factor is not the same at all the levels of the other factor.

There are three main types of interactions:

Antagonistic Interaction: When the main effect is non-significant, and interaction is significant. In this interaction, the two independent variables are likely to reverse the effect of each other.
Synergistic Interaction: When the higher level of one independent variable enhances the effect of the other.
Ceiling Effect Interaction: When the higher level of one independent variable lowers the differential effect of another variable.

When there is a large interaction, the main effects have little practical meaning, as a significant interaction often masks the significance of the main effects.

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Types of factorial design

There are three main types of factorial designs, namely “Within Subject Factorial Design”, “Between Subject Factorial Design”, and “Mixed Factorial Design”.

1. Within Subject Factorial Design: In this factorial design, all of the independent variables are manipulated within subjects.

2. Between Subject Factorial Design: In the Between Subject Factorial Design, the subjects are assigned to different conditions and each subject only experiences one of the experimental conditions.

3. Mixed Factorial Design: This design is most commonly used in the study of psychology. It is named the ‘Mixed Factorial Design’ because it has at least one Within Subject variable and one Between Subject variable.

Advantages of factorial experimental design

The following are a few advantages of using the factorial experimental design:

1. Efficient:

When compared to one-factor-at-a-time (OFAT) experiments, factorial designs are significantly more efficient and can provide more information at a similar or lower cost. It can also help find optimal conditions quicker than OFAT experiments can.

2. Comprehensive results:

Researchers can employ the factorial design to calculate the effects of a factor as an estimate at several levels of other factors. This can yield conclusions that are valid over a range of different experimental conditions.

3. Flexibility:

The factorial design offers flexibility in experimental design by allowing you to manipulate multiple variables simultaneously and evaluate various combinations of factor levels. The flexibility enables you to explore complex relationships between factors in a controlled setting.

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4. Increased statistical efficiency:

This experimental design is statistically efficient, as it helps yield more precise estimates of main effects and interactions with fewer observations. The shared variance among factors allows you the sensitivity to detect effects within the same sample size.

5. Enhanced external validity:

Factorial experimental design enhances the external validity or generalizability of research findings by systematically manipulating multiple factors. When you employ factorial design in surveys, it provides insights that represent the complexities and nuances present in the natural environment. This increases the applicability of the survey findings to a broader population.

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Factorial design is a powerful tool for survey research. By incorporating its principles into your survey, you can enhance the quality of the research findings, leading to more informed and actionable insights.

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Full Factorial Design: Comprehensive Guide for Optimal Experimentation

June 4th, 2024

The full factorial design distinguishes itself as a robust, enlightening experimentation approach across industries.

At its core, this systematic method examines multiple metrics’ collective effects on an outcome simultaneously.

Considering all factor level combinations furnishes holistic comprehension beyond individual impacts—illuminating intricate relationships shaping complex systems’ behaviors.

The factorial design’s strength lies in realistically emulating dynamics’ nuances where variables interact intricately nonlinearly. Accounting for interplays guards against oversimplification. This casts light on underlying realities profoundly, priming informed resolutions and refinement pursuits.

From manufacturing to research frontiers, ponder occasions where exhaustively investigating factor relations meaningfully addresses specific problems or opportunities.

Key Highlights

A comprehensive exploration of factor effects and interactions
Robust methodology for process understanding and optimization
Applications across diverse industries, including manufacturing, pharmaceuticals, and marketing
Rigorous statistical analysis techniques, such as ANOVA and regression modeling
Insights into factor-level optimization for enhanced process performance
Evaluation of alternative designs, including fractional factorial and response surface methodologies
Best practices for experimental design, data collection, and analysis

Introduction to Full Factorial Design

The full factorial design stands out as a comprehensive and robust approach, enabling researchers and practitioners to unlock valuable insights and drive meaningful change across diverse industries.

The full factorial design is a systematic way to investigate the effects of multiple factors on a response variable simultaneously.

By considering all possible combinations of factor levels, this experimental strategy provides a holistic understanding of not only the individual factor effects but also the intricate interactions that can shape outcomes in complex systems.

A full factorial design is an experimental design that considers the effects of multiple factors simultaneously on a response variable.

It involves manipulating all possible combinations of the levels of each factor, enabling researchers to determine the main effects of individual factors as well as their interactions statistically.

This comprehensive approach ensures that no potential interaction is overlooked, providing a complete picture of the system under investigation.

The key benefits of employing a full factorial design include:

Main effects : Researchers can identify which factors have the most significant impact on the response variable, allowing them to focus their efforts on the most influential factors.
Interaction effects : By accounting for interactions between factors, full factorial designs reveal how the effect of one factor depends on the level of another factor, providing insights into the complex relationships within the system.
Optimization: With a comprehensive understanding of the main effects and interactions, researchers can estimate the optimal settings for the independent variables, leading to the best possible outcome for the response variable.

The versatility of the full factorial design makes it a valuable tool across various industries, including:

Manufacturing : Optimizing processes, improving product quality, and reducing defects by identifying key variables and their interactions.
Pharmaceuticals : Formulating and developing drugs by assessing factors such as excipient concentrations, drug particle size, and processing conditions on bioavailability, stability, and release profiles.
Marketing: Optimizing promotional strategies by evaluating the effects of factors like ad content, media channels, target audience segments, and pricing on consumer response.

Fundamentals of Design of Experiments (DOE)

Before delving into the intricacies of full factorial design, it is essential to understand the fundamental principles of Design of Experiments (DOE) , a systematic approach to investigating the relationships between input variables (factors) and output variables (responses) .

DOE provides a structured framework for planning, executing, and analyzing experiments, ensuring reliable and insightful results.

Understanding Factors and Levels

In a DOE study, the variables that are manipulated or controlled by the experimenter are known as independent variables or factors.

These can be further classified into:

Numerical factors : Variables that can take on a range of numerical values, such as temperature, pressure, or time.
Categorical factors : Variables that have distinct, non-numerical levels, such as material type or production method.

The outcome or characteristic of interest that is measured and analyzed in an experiment is referred to as the response variable or dependent variable.

Common examples include product yield, strength, purity, or customer satisfaction.

Each independent variable in a DOE study can be set at different levels or values.

The choice of factor levels is crucial, as it determines the range of conditions under which the experiment is conducted.

Carefully selecting factor levels ensures that the study captures the relevant region of interest and provides meaningful insights into the system’s behavior.

Principles of DOE

Replication refers to the practice of repeating the same experimental run multiple times under identical conditions.

This allows researchers to estimate the inherent variability in the experimental process and ensures the reliability of the results by providing a measure of experimental error.

Randomization is the process of randomly assigning experimental runs to different factor level combinations.

This helps to mitigate the potential impact of nuisance variables and ensures that any observed effects can be attributed to the factors under investigation, rather than uncontrolled sources of variation .

Blocking is a technique used to account for known sources of variability in an experiment, such as differences in equipment, operators, or environmental conditions.

By grouping experimental runs into homogeneous blocks, researchers can isolate and quantify the effects of these nuisance variables, ensuring more precise estimates of the factor effects.

Types of Full Factorial Designs

Full factorial designs can be classified into different types based on the number of levels for each factor and the nature of the factors themselves.

Understanding the various types of full factorial designs is crucial for selecting the appropriate experimental strategy to address specific research questions or process optimization objectives.

2-Level Full Factorial Design

The 2-level full factorial design, where each factor has two levels (typically labeled as “low” and “high”), is commonly employed in screening experiments.

These experiments aim to identify the most significant factors influencing the response variable, allowing researchers to focus their efforts on the most promising factors in subsequent, more in-depth investigations.

By evaluating the main effects and interactions in a 2-level full factorial design, researchers can determine which factors have a statistically significant impact on the response variable.

This information is invaluable in prioritizing factors for further optimization or confirming their negligible influence, thereby streamlining the overall experimental process.

3-Level Full Factorial Design

Unlike the 2-level design, which assumes a linear relationship between factors and the response variable, the 3-level full factorial design allows for the investigation of quadratic effects.

These nonlinear effects can be important in scenarios where the response variable exhibits curvature or a peak/valley behavior within the explored factor ranges.

By incorporating three levels for each factor, researchers can model the curvature in the response surface more accurately.

This enhanced understanding of the system’s behavior enables more precise optimization and provides insights into potential optimal operating regions or factor level combinations.

Mixed-Level Full Factorial Design

In many real-world applications, experiments may involve a combination of categorical factors (e.g., material type, production method) and continuous factors (e.g., temperature, pressure).

The mixed-level full factorial design accommodates this scenario by allowing researchers to investigate the effects of both types of factors simultaneously, providing a comprehensive understanding of the system.

Analyzing Full Factorial Design Experiments

Once the experimental data has been collected, the next step is to analyze the results to gain insights into the main effects, interactions, and optimal factor level combinations.

Several statistical techniques are employed in the analysis of full factorial experiments, each serving a specific purpose and providing valuable information for process understanding and optimization.

Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a powerful statistical tool used to determine the significance of main effects (individual factor effects) and interaction effects (combined effects of multiple factors) on the response variable.

By partitioning the total variability in the data into components attributable to each factor and their interactions, ANOVA enables researchers to identify the most influential factors and their relationships.

ANOVA also provides a framework for hypothesis testing, allowing researchers to assess whether the observed effects are statistically significant or simply due to random variability.

This rigorous approach ensures that conclusions drawn from the experimental data are statistically valid and reliable.

Regression Analysis

Regression analysis is another essential tool in the analysis of full factorial experiments.

It involves fitting a mathematical model to the experimental data , relating the response variable to the independent variables (factors) and their interactions.

This model can be used to predict the response variable for any combination of factor levels within the experimental region.

Once a suitable regression model has been obtained, researchers can employ optimization techniques to identify the factor level combinations that maximize or minimize the response variable.

These techniques often involve solving the regression equation subject to relevant constraints, enabling the determination of optimal operating conditions for the process under investigation.

Graphical Analysis & Full Factorial Design

Graphical analysis is a powerful tool for visualizing and interpreting the results of full factorial experiments.

Interaction plots are particularly useful for examining the presence and nature of interactions between factors.

These plots display the response variable as a function of one factor at different levels of another factor, allowing researchers to identify and understand complex relationships within the system.

Main effects plots , on the other hand, illustrate the individual impact of each factor on the response variable, providing a visual representation of the main effects.

These plots can aid in quickly identifying the most influential factors and assessing the relative importance of each factor in the experimental domain.

Advantages and Limitations of Full Factorial Design

While the full factorial design offers numerous advantages in terms of comprehensiveness and insight, it is important to recognize its limitations and potential drawbacks.

Understanding both the strengths and limitations of this experimental approach is crucial for making informed decisions and optimizing the trade-offs between resource allocation and the desired level of process understanding.

Advantages of Full Factorial Design

One of the primary advantages of the full factorial design is its ability to provide comprehensive insights into the system under investigation.

By considering all possible factor combinations, researchers can obtain a complete picture of the main effects, interactions, and potential curvature in the response surface, leading to a thorough understanding of the process dynamics.

Unlike some experimental designs that may overlook or confound interactions, the full factorial design explicitly accounts for interactions between factors.

This capability is particularly valuable in complex systems where the effect of one factor may depend on the level of another factor, allowing researchers to unravel these intricate relationships and optimize processes accordingly.

With a comprehensive understanding of the main effects and interactions, full factorial experiments enable researchers to estimate the optimal settings for the independent variables, leading to the best possible outcome for the response variable.

This optimization potential is invaluable in various industries, where process efficiency , product quality, and cost-effectiveness are paramount.

Limitations of Full Factorial Design

One of the primary limitations of the full factorial design is its resource-intensive nature.

As the number of factors and levels increases, the number of experimental runs required grows exponentially, leading to higher costs, longer experimental durations, and greater logistical challenges.

Related to the resource-intensive aspect, full factorial designs often require large sample sizes to ensure statistical validity and reliable estimates of main effects and interactions.

This can be particularly challenging in situations where resources are limited or experimental conditions are difficult to replicate.

The comprehensiveness of the full factorial design can also lead to an overwhelming amount of data, especially when dealing with numerous factors and levels.

Analyzing and interpreting such large datasets can be a daunting task, requiring advanced statistical techniques and computational resources.

Alternative Designs and Extensions

While the full factorial design is a powerful and comprehensive experimental strategy, some alternative designs and extensions can be employed depending on the specific requirements and constraints of the research or industrial application.

These alternative approaches can offer trade-offs between experimental complexity, resource requirements, and the level of information obtained.

Fractional Factorial Designs

Fractional factorial designs are a class of experimental designs that involve studying only a carefully chosen fraction of the full factorial design.

By sacrificing the ability to estimate certain higher-order interactions, fractional factorial designs can significantly reduce the number of experimental runs required, making them more resource-efficient.

Fractional factorial designs are particularly useful in screening experiments, where the primary goal is to identify the most influential factors before conducting more detailed investigations.

These designs can help researchers prioritize their efforts and allocate resources more effectively.

Response Surface Methodology

Response Surface Methodology (RSM) is a collection of statistical techniques used to model and optimize processes with multiple input variables.

The Central Composite Design (CCD) is a widely used RSM design that combines a factorial design with additional axial and center points, allowing for the estimation of quadratic effects and potential curvature in the response surface.

Another popular RSM design is the Box-Behnken Design, which is a spherical, rotatable, or nearly rotatable design.

This design is particularly efficient for exploring quadratic response surfaces and optimizing processes with three or more factors.

Unlike the Central Composite Design, the Box-Behnken Design does not include any points at the vertices of the cubic region defined by the upper and lower limits of the factors.

Taguchi Methods

Developed by Genichi Taguchi, the Taguchi methods are a set of techniques for robust parameter design and quality improvement.

One of the key elements of the Taguchi approach is the use of orthogonal arrays, which are a special class of fractional factorial designs.

Orthogonal arrays allow for the simultaneous investigation of multiple factors with a minimal number of experimental runs, making them an attractive option when resources are limited.

The Taguchi methods emphasize the concept of robust parameter design, which aims to identify factor-level combinations that minimize the variability in the response variable while achieving the desired target value.

This approach is particularly valuable in manufacturing and product development, where robustness to environmental and operational variations is critical for maintaining consistent performance and quality.

The full factorial design stands as a potent experimental strategy.

Considering every factor combination furnishes holistic comprehension of effects, interplays, and nonlinearities.

This lights pathways empowering better comprehension and optimization across varied sectors.

Advancing capabilities and statistical techniques envision expanding factorial applications.

Machine learning may analyze data efficiently while adaptive designs responsively calibrate based on real-time insights.

Additionally, sustainability priorities could drive factor prioritizations maximizing resource optimization and lessening environmental impacts.

Full factorials offer rigorous yet flexible methods unlocking enriched wisdom from intricate systems.

While exhaustive, generated learnings inspire noteworthy refinements throughout performance, quality, and workflows.

By following best practices, judiciously leveraging other designs, and tracking innovations, researchers and specialists harness factorials’ full gifts in driving innovations remarkably within respective realms.

Their gifts in illuminating intricate relations deserve recognition and prudent application wherever circumstances permit comprehensive experimentation.

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Front Psychiatry

Factorial Designs Help to Understand How Psychological Therapy Works

Associated data.

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Introduction: The Need to Understand How Psychological Therapies Work

Common Versus Specific Treatment Factors

Methodologies to Examine the Mechanisms of Psychotherapy

Comparative randomized controlled trials.

Component Study Designs

The Factorial Approach

The IMPROVE-2 Study as an Example of a Factorial Design

Experimental groups of the IMPROVE-2 fractional factorial design.

Condition	Functional analysis	Concrete training	Compassion	Absorption	Relaxation	Activity scheduling	Thought challenging
1	no	no	no	no	no	yes	yes
2	yes	no	no	no	no	no	no
3	no	no	yes	no	no	no	no
4	yes	no	yes	no	no	yes	yes
5	no	no	no	yes	no	yes	no
6	yes	no	no	yes	no	no	yes
7	no	no	yes	yes	no	no	yes
8	yes	no	yes	yes	no	yes	no
9	no	yes	no	no	no	no	no
10	yes	yes	no	no	no	yes	yes
11	No	yes	yes	no	no	yes	yes
12	yes	yes	yes	no	no	no	no
13	no	yes	no	yes	no	no	yes
14	yes	yes	no	yes	no	yes	no
15	no	yes	yes	yes	no	yes	no
16	yes	yes	yes	yes	no	no	yes
17	no	no	no	no	yes	no	yes
18	yes	no	no	no	yes	yes	no
19	no	no	yes	no	yes	yes	no
20	yes	no	yes	no	yes	no	yes
21	no	no	no	yes	yes	no	no
22	yes	no	no	yes	yes	yes	yes
23	no	no	yes	yes	yes	yes	yes
24	yes	no	yes	yes	yes	no	no
25	no	yes	no	no	yes	yes	no
26	yes	yes	no	no	yes	no	yes
27	no	yes	yes	no	yes	no	yes
28	yes	yes	yes	no	yes	yes	no
29	no	yes	no	yes	yes	yes	yes
30	yes	yes	no	yes	yes	no	no
31	no	yes	yes	yes	yes	no	no
32	yes	yes	yes	yes	yes	yes	yes

Every factor occurs an equal number of times at high and low levels (i.e. balanced) and all factors are orthogonal to each other. Each effect estimate involves all 32 of the conditions in Table 1 , thereby maintaining the power associated with all participants. This Resolution IV design means that all main effects are aliased with 3-way and higher interactions, and all 2-way interactions are aliased with 2-way and higher interactions, on assumption that non-negligible 3-way interactions are unlikely. In contrast, a standard RCT is aliased for all main effects and interactions of treatment components.

The Multiphase Optimization Strategy (MOST)

The Preparation Stage in MOST

Components Within the Psychological Intervention

The Optimization Stage of MOST: Factorial Experiments and Their Benefits

Advantages of Factorial Design

There are at least four advantages to the use of a factorial design in resolving how therapy works and what its active mechanisms are.

Advantage 1: Directly Testing Individual Components and Their Interactions

Advantage 2: Manipulation of Hypothesized Mechanisms and Examination of Individual Mediators

Advantage 3: Improved Delineation of Specific Versus Common Treatment Factors

Advantage 4: Factorial Designs Are Efficient and Economical

The Evaluation Stage of MOST

Potential Limitations

Data Availability Statement

Ethics statement.

Author Contributions

EW and AN both designed, prepared, and delivered the IMPROVE-2 study. EW prepared the first draft of the manuscript, AN commented on the draft, and both EW and AN finalized the manuscript.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

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Design and implementation of a novel uav-assisted lorawan network.

1. Introduction

1.1. background and significance, 1.2. literature review, 1.2.1. uav-assisted lora network without lorawan specification, 1.2.2. uav-assisted lorawan network.

UAV Carrying End-Device as the Payload
UAV Carrying Gateway as the Payload

1.3. Motivation and Contribution

The UAVs only serve as flight carriers, and their communication capabilities are not used to assist LoRaWAN networks.
The altitude advantage of UAVs was leveraged to facilitate the deployment of LoRaWAN networks in complex environments. However, the integrated solution of “UAV + Remote Controller + Server”, which can enhance communication reliability and further expand the coverage of LoRaWAN network in an efficient way, was not considered.
Information interaction between UAVs and LoRaWAN gateways was not realized, including transparent data forwarding, clock synchronization and GPS location acquisition, resulting in a limited application scalability of the existing UAV-assisted LoRaWAN network.
Based on the analysis of the specific requirements of the UAV-assisted LoRaWAN network system, a UAV-assisted LoRaWAN network system architecture is proposed, expanding the LoRaWAN network coverage through the integrated solution of “UAV + remote control + server” effectively.
A LoRaWAN gateway prototype, which is highly integrated with the UAV, has been developed to enhance the LoRaWAN network through UAV altitude and line-of-sight advantages. Various UAV resources were provided to the LoRaWAN gateway through the multiple interfaces provided by a PSDK adapter board, including UART, PPS signal, and USB type-C. In addition, a forwarding program called the UAV packet forwarder was designed to manage the forwarding task and UAV resources.
A relay solution based on remote controller was proposed to further extend the coverage of UAV. With the Wi-Fi or cellular network and UAV communication resources, a MSDK-based APP was developed to realize the relay features. Through the monitoring and transparent forwarding for both the UAV and LoRaWAN server, the coverage of the LoRaWAN network was effectively extended with a single relay.
A performance evaluation and a positioning demonstration were carried out in the real world with the proposed network. The superiority of the UAV-assisted LoRaWAN network is verified in the typical LoRaWAN applications of both data collection and positioning.

2. LoRaWAN Network and UAV Payload Development Technology

2.1. lorawan network, 2.1.1. lorawan end-devices, 2.1.2. lorawan gateway, 2.1.3. lorawan server.

ChirpStack Gateway Bridge
MQTT broker

2.2. UAV Payload Development Technology

2.2.1. payload software development kit (psdk), 2.2.2. mobile software development kit (msdk), 2.2.3. integrated solution of “uav + remote controller + server”, 3. system design, 3.1. system requirement analysis, 3.2. system architecture.

LoRaWAN End-Devices
UAV Gateway
Remote Controller
Cloud Center Server
Application Server

3.3. System Workflow

4. system implementation, 4.1. uav gateway design, 4.1.1. hardware design.

PSDK adapter board
LoRaWAN gateway

4.1.2. Software Design

Thread Up, Thread Down and GWMP-based data forwarding
Thread GPS and clock synchronization

4.2. Remote Controller Relay Design

5. performance evaluation, 5.1. network coverage and communication reliability evaluation, 5.1.1. experimental setting.

Node_1 is deployed about 400 m southeast of the ground gateway, representing the urban propagation environment.
Node_2 is deployed about 400 m southwest of the ground gateway, representing the vegetation propagation environment.
Node_3 is deployed about 1.3 km southwest of the ground gateway, representing a complex propagation environment over long distances.
In group 1, the ground gateway is utilized to demonstrate the performance of the ground LoRaWAN network.
In group 2, the ground gateway is replaced by the UAV gateway and remote controller to construct a UAV-assisted LoRaWAN network. In addition, three scenarios have been designed to analyze the effect of gateway altitude on network performance, where the UAV gateway is deployed at heights of 30 m, 60 m, and 90 m, respectively.
In group 3, the UAV gateway is deployed at the center of three end-devices, which is 800 m away from the remote controller, aiming to verify the relay feature of the remote controller.

5.1.2. Evaluation Indicators

5.1.3. experimental results and discussion, 5.2. positioning demonstration, 5.2.1. tdoa positioning configuration, 5.2.2. tdoa algorithm implementation.

Positioning Method Based on TDOA (Time Difference of Arrival)

timestamps of a packet arriving at 3 different gateways, the GPS coordinates of gateways;

Calculated End-Device location;

(1) Acquire information from Center Server.
(2) Randomly choose a gateway as reference gateway.
(3) Set the coordinates of reference gateway to (0, 0) at a two-dimensional plane.
(4) Project the GPS coordinates of other gateways onto the two-dimensional plane.
(5) Calculate the distance difference of gateways to End-Device based on timestamps difference, with the aim of formulating hyperbolic equations.
(6) the calculated distance difference of gateways to End-Device is outlier
to step 1

calculated End-Device location based on hyperbolic equations.

5.2.3. Positioning Result and Discuss

6. conclusions, author contributions, data availability statement, conflicts of interest.

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Zadorozhny, A.M.; Doroshkin, A.A.; Gorev, V.N.; Melkov, A.V.; Mitrokhin, A.A.; Prokopyev, V.Y.; Prokopyev, Y.M. First Flight-Testing of LoRa Modulation in Satellite Radio Communications in Low-Earth Orbit. IEEE Access 2022 , 10 , 100006–100023. [ Google Scholar ] [ CrossRef ]

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Literature	Architecture/Protocol of UAV-Assisted LoRa Network			Integrated Design of UAV with LoRa End-Device/Gateway
Literature	Point-to-Point	Custom	LoRaWAN	Carrying LoRaWAN Module	Carrying LoRaWAN End-Devices	Carrying LoRaWAN Gateway
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[ ]		Key-Value		√
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[ ]			√			√
[ ]			√			√
[ ]			√			√
[ ]			√			√
[ ]			√			√ (Cellular)
[ ]			√			√ (Cellular)
[ ]			√			√ (Wi-Fi)
[ ]			√			√ (Wi-Fi, cellular)
[ ]			√			√ (Wi-Fi, cellular)
[ ]			√			√ (Local storage)
[ ]			√			√ (Local storage)
[ ]			√			√ (Local storage)
Our Work			√			√ (Integrated solution of “UAV + Remote Controller + Server”)

	PSDK	MSDK
Development Device	UAV payload	Mobile device
Hardware Platform	Mainstream Embedded Hardware Platforms, such as STM32, Raspberry Pi, etc.	Nexus Devices (connected to Remote Controller) or Remote Controller
Software Platform	Linux, ROS and RTOS	iOS and Android
UAV channel resources	Bidirectional wired communication between User payload and UAV	Bidirectional wireless communication between Remote Controller and UAV
Specific function	Power supply Time synchronization	Networking capability UAV flight control
General function	GPS information subscription Camera data acquisition and control Flight parameter subscription...

Configuration	Value
Bandwidth	125 KHz
Code Rate	CR_4_5
Spreading Factor	11
Transmission power	17 dbm
Payload	28 Byte
Mode	Class A
Antenna gain	2 dbi
Transmission period	10 s

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Zhao, H.; Tang, W.; Chen, S.; Li, A.; Li, Y.; Cheng, W. Design and Implementation of a Novel UAV-Assisted LoRaWAN Network. Drones 2024 , 8 , 520. https://doi.org/10.3390/drones8100520

Zhao H, Tang W, Chen S, Li A, Li Y, Cheng W. Design and Implementation of a Novel UAV-Assisted LoRaWAN Network. Drones . 2024; 8(10):520. https://doi.org/10.3390/drones8100520

Zhao, Honggang, Wenxin Tang, Sitong Chen, Aoyang Li, Yong Li, and Wei Cheng. 2024. "Design and Implementation of a Novel UAV-Assisted LoRaWAN Network" Drones 8, no. 10: 520. https://doi.org/10.3390/drones8100520

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Analysis, modelling, and optimization of force in ultra-precision hard turning of cold work hardened steel using the CBN tool

Technical Paper
Open access
Published: 24 September 2024
Volume 46 , article number 624 , ( 2024 )

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Ogutu Isaya Elly ORCID: orcid.org/0009-0000-9857-1431 1 ,
Ugonna Loveday Adizue 1 , 2 ,
Amanuel Diriba Tura 1 ,
Balázs Zsolt Farkas 1 &
M.Takács 1

The machinability of high-performance materials such as superalloys, composites, and hardened steel has been a big challenge due to their mechanical, physical, and chemical properties, which give them inherent complex machining characteristics. Additionally, majority of machinability tests conducted on these materials have been carried out on conventional and less precise lathes based on Taguchi, composite, and other designs of experiments that do not exploit all the possible combinations of cutting parameters. This work reports an investigation on ultra-precision hard turning (UHT) of cold work hardened AISI D2 steel of HRC 62, based on the full factorial design of experiment, carried out on an ultra-precision lathe. A theoretical analysis of the force components generated is reported. Modelling of the process, based on the resultant force, is also reported through a machine learning model. The model was developed from the experimental data and statistically evaluated with validation data. Its average MAPE values of 1.47%, 4.81%, and 10.66% for training, testing, and validation, respectively, attest to its robustness. The excellent coefficient of determination values, R 2 , also justify the model’s robustness. Multi-objective optimization was also conducted to optimize material removal rate (MRR), resultant force, and vibration simultaneously. For sustainable and efficient UHT, optimal cutting velocity (158.8 m/min), feed (0.125 mm/rev), and depth of cut (0.074 mm) were proposed to generate optimal resultant force (224.8 N), MRR (2603.6 mm 3 /min), and vibration (0.03 m/s^2) simultaneously. These results can be beneficial in planning UHT processes for high-performance materials.

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1 Introduction

High-performance metals, viz. super alloys, hardened steels, titanium, and its alloys, are known for their high creep and corrosion resistances, low thermal conductivity, hot hardness, chemical inertness, and resistance to thermal shocks. These properties have made their demand for the fabrication of parts in the manufacturing industries, such as automotive, aeronautical, and marine, very high. In the same breath, the same properties have made their machinability difficult. For instance, their hardness has been reported to accelerate the tool wear process [ 1 ] and introduce workpiece burnouts, which affect the quality of the final product [ 2 ]. Cold work hardened AISI D2 steel is an essential hard metal when it comes to making extrusion dies, slitting cutters, wire dies, burnishing rolls, gauges, and master tools. Its machinability has been greatly impeded by its mechanical properties, and as a result, its application in part production could have been much higher. Therefore, characterizing its machinability is an area that needs thorough research.

Efforts by manufacturing stakeholders have seen the development of several machining approaches for hardened AISI D2 steel. For a long time, grinding has been the primary method of fabricating parts from hard metals [ 3 ]. However, its low material removal rate, high-power consumption, long set-up times, and inability to produce complex geometries rendered it unreliable for hard machining [ 4 ]. This gave way to better methods such as hard turning [ 5 ]. Hard turning refers to the single-point cutting of metals with Rockwell hardness (HRC) of 45 and above, primarily in the range 58–68 HRC [ 6 ] [ 7 ], and is reliable for near-net shape and finish machining. The deployment of ultra-precision machine tools, tools, machining conditions, and production of parts with tight dimensional and geometrical tolerances characterizes ultra-precision hard turning (UHT) [ 8 ]. UHT is known for using small feeds and depths of cut, enabling it to achieve form accuracy of less than 1 µm and surface roughness of less than 0.1 µm. These accuracy levels are superior to the traditional hard turning methods [ 9 ].

Tool design and fabrication for the UHT process has been one of the critical research areas in hard machining. Efforts have been made to fabricate strong, tough tools with high thermal shock resistance and chemically inert at high temperatures. Cermet tools [ 10 ], ceramic tools, coated carbide inserts [ 11 ], and cubic boron nitride (CBN) tools [ 12 ] are some of the tools that have found recognition for use in UHT. However, due to their strength, toughness, and hardness (which is only second to diamond), CBN tools and their variants (PCBN) are the best for hard turning. In addition, their high melting temperature of 2730 °C and high stability of up to around 2000 °C have made them the best choice for hard turning processes [ 13 ]. Works by [ 14 , 15 ] and [ 16 ] have shown the successful application of CBN tools in the hard turning of hardened steels.

The machinability of hardened AISI D2 can further be understood and improved through the prediction of machinability indices [ 17 ]. Force is a very significant machinability index when determining process stability. It has a strong correlation with almost all the other performance indices [ 18 ]. Force prediction, for a long period now, has been reported from analytical, numerical, and experimental (based on big data models) points of view. However, recently, industries have been rushing for intelligent machining to enhance productivity. Machine learning technologies, therefore, have been deployed to upheave production in line with the Industry 4.0 industrial revolution guidelines [ 19 ]. Machine learning (ML) models can be supervised, unsupervised, or reinforcement models. Supervised ML models such as Gaussian progressive relation (GPR), random forest (RF) regression, support vector machining (SVM), polynomial regression, gradient boosted trees (GBT), adaptive neuro-fuzzy inferencing system (ANFIS), decision trees (DT), and artificial neural network (ANN) have been the subject of interest in a bid to embrace intelligent machining. These ML models have been deployed largely for tool wear monitoring and machinability metrics forecasting both online and offline [ 20 ]. Process monitoring and prediction give prior knowledge of the process outcome, which can be informative when it comes to design and planning. It is through proper design and planning that the downtimes and production costs are reduced to improve production efficiency.

In addition to prediction, optimization of machinability indices is key in the planning and execution of sustainable and profitable production of AISI D2 steel products. Optimization can be objective as well as multi-objective [ 21 ]. Single objective optimization has been faulted for overlooking some of the process kinematics and the interdependence of machinability indices, hence not capturing the dynamic nature of processes. Multi-objective optimization finds a balance between selected machinability indices; hence, it can minimize vibration and acoustic emission while maximizing material removal rate, tool life, etc. Ant colony, teacher–learner-based optimization (TLBO), particle swarm optimization (PSO), grey relational method, multi-objective particle swarm optimization (MOPSO), firefly, reinforcement dung beetle algorithm, and genetic algorithm (GA) are some of the heuristics and metaheuristics optimization methods that have been reported this far [ 22 , 23 , 24 , 25 ].

Kumar et al. [ 26 ] studied the workability of heat-treated AISI D2 steel based on the cutting tool’s flank wear, product surface quality, and chip–tool interface temperature. They developed RSM and ANN models for forecasting the three machinability indices. Vahid et al. [ 27 ] developed an ACO (adaptive control optimization) system that relied on ANN and GP (geometrical programming) for prediction, and PSO for optimization when hard turning AISI D2 steel. According to Tabassum et al. [ 28 ], ANFIS prediction is superior to that of ANN and RSM when forecasting force from the hard turning of AISI 1045 (56 HRC) in both high-pressure coolant and dry cutting conditions. Non-dominated sorting genetic algorithm II (NSGA-II) was used by Meddour et al. [ 29 ] to find the optimal values of surface roughness and cutting forces simultaneously when hard turning AISI 4140 (60 HRC) using a mixed ceramics cutting tool. In addition, they developed RSM and ANN models for prediction. Similar work on ML modelling of AISI D2 processes was reported by Adizue et al. [ 30 ]. Further works on ML modelling have been reported by [ 31 , 32 , 33 , 34 ].

Parameter correlation studies by Patel and Gandhi [ 35 ], when hard turning AISI D2 steel using the CBN tool, showed that depth of cut was the most significant parameter in force generation. According to Rafighi et al. [ 17 ], variation of cutting edge radius had the most significant impact on cutting force when hard turning AISI D2 steel. They noted that the ceramic tool used led to a lower magnitude of forces than CBN inserts. According to [ 36 ], there is no correlation between the cutting velocity and cutting force when machining AISI D2 steel using the CBN tool.

Based on the literature review, there is a need for robust prognostic models and optimization methods to effectively characterize the UHT of AISI D2 steel using the CBN tool, based on the generated forces. Most of the reported works have been based on Taguchi’s design of experiments and, therefore, fail to investigate all the possible parameter combinations during the execution of the experiments, leading to incomprehensive and inaccurate conclusions. Thus, there is a need for more investigations based on full factorial experiments to exhaust all possible parameter combinations.

Due to the dynamic nature of the UHT process, no standard machine learning model or optimization method has been approved for force prediction when hard turning AISI D2 steel using the CBN tool. This is despite their prediction superiority over analytical and numerical methods [ 37 ]. The literature work on machine learning modelling and optimization of resultant force during UHT of AISI D2 steel is scarce. Hence, there is a need for either developing new, robust, and accurate models or evaluating and improving the existing ones for application during AISI D2 steel UHT using CBN tools.

Based on these gaps, this work analyses the kinematics of force components’ generation during UHT processes. In addition, it investigates the prediction of resultant forces using ANFIS during UHT of AISI D2 steel. The investigation is based on a full factorial design of experiments and an extra set of experiments for validation data for checking model robustness. Lastly, multi-objective particle swarm optimization (MOPSO) has been used to optimize the resultant force relative to material removal rate (MRR) and vibration to establish optimal parameters for sustainable and efficient production of parts made from cold work hardened AISI D2.

2 Materials and methods

2.1 design of experiments.

This work sought to characterize the UHT of cold work hardened AISI D2 steel through analysis, prediction, and optimization of force components based on full factorial experiments. Therefore, a theoretical analysis of the force components is conducted, and the resultant force is predicted through ANFIS modelling. The multi-objective optimization method is deployed to determine the optimal cutting parameters and resultant force.

UHT was conducted on a hollow, cold work hardened AISI D2 steel workpiece (62 HRC) of 12 mm internal and 60 mm external diameters and a length of 40 mm (Fig. 1 ). The internal hole provided an exit for the tool because as the diameter of the stock reduced, the machine tool’s rotational speed needed to be increased to maintain the constant cutting velocity. The hole, therefore, compensated for the limited rotational speed of the machine tool. AISI D2 steel was chosen because of its high-dimensional stability, good resistance to temper softening, and high compressive strength when hardened. These properties enhance its application in the development of extrusion dies and slitting cutters.

Workpiece of cold work hardened AISI D2 steel applied for the experiments

A single tip, uncoated, full-face layer CBN insert with 50% CBN grade, 2 µm grain size, and bound by a ceramic binder was chosen due to its exceptional performance in high-speed hard machining processes. The CBN insert used was SECO DCGW11T308S-01020-L1-B CBN010 and is specifically meant for finish hard turning operation due to its excellent wear resistance, good thermal shock resistance, chemical inertness when machining steel, hot hardness and strength, good thermal conductivity, and high homologous temperature. It had two cutting edges, a clearance angle of 7°, a cutting edge effective length of 3.5 mm, an insert corner radius of 0.8 mm, a thickness of 3.75 mm, a weight of 0.0004 kg, and an included angle of 55°. Figure 2 shows the CBN insert.

CBN insert used for the experiment

The machining process took place on a high-rigidity, ultra-precision CNC lathe (Hemburg-Mikroturn, 50 CNC) with a repetitive accuracy of ± 1 µm, maximum spindle speed of 6000 rpm, and a positional accuracy of 1 µm/150 mm. As shown in Fig. 3 , the workpiece was held on the machine with an ultra-precision three-jaw pneumatic clamp, and its concentricity was set using a dial gauge indicator. A full factorial design of experiments was deployed to explore all the possible parameter combinations. As shown in Table 1 , three cutting parameters (cutting velocity (m/min), feed (mm/rev), and depth of cut (mm)) with three levels were investigated. The cutting parameters’ levels were chosen relative to the manufacturer’s handbook, UHT lathe parameters, CBN tool geometry, and previous research work on hardened steel turning.

Experimental set-up

The three levels, therefore, meant that 27 runs of the experiment were conducted randomly. The 27 runs were carried out three times to ensure the accuracy of the final data, hence bringing the total number of experiments conducted to 81 runs. A further nine runs of experiments with unique parameter combinations were conducted to validate the models.

During the face hard turning process, three runs with different parameter combinations were carried on the surface of the workpiece cross-section. See Fig. 4 . When transiting from one run to the other, the spindle was rotated five times without any cutting action. During these non-cutting phases, the machine tool’s controller adjusted the cutting parameters relative to the next run in the used G code. After machining a set of three runs, the machined face was flattened before machining the next set. This was repeated until all the 90 runs had been completed.

Schematic representation of the face hard turning strategy

A three-axis Kistler dynamometer (Kistler, Type 9257A) was deployed to measure the signals of passive force ( F p ), cutting force ( F c ), and feed force ( F f ) components. The directions of the force components are shown in Fig. 5 . A transducer (Kistler, Type 9257) was applied for signal amplification. The force signals in Figs. 6 , 7 , and 8 show the individual signals for each run. These signals are separated by moments of change in parameter combination, as is shown in Fig. 8 . During these moments of change in parameter combination, the workpiece was rotated five times without any tool feed.

Force components

Cutting force ( F c ) signal

Feed force ( F f ) signal

Passive force ( F p ) signal

2.2 Adaptive neuro-fuzzy inferencing system (ANFIS)

This algorithm is a hybrid of two known machine learning algorithms, artificial neuro network (ANN) and fuzzy logic system (FLS). The hybrid utilizes the constituent algorithms’ strengths to establish a powerful and flexible model for complex nonlinear problems. Figure 9 shows the general ANFIS architecture. During modelling, the input layer receives the variables that are to be used for the prediction process. The membership functions (MF), for each of the variables, are then developed in the fuzzy layer. Equations ( 1 ) and ( 2 ) denote the nodal functions in the fuzzy layer.

ANFIS architecture

The membership functions are denoted by a j and b t . C 1, t represents the output from the t th node of the fuzzy layer. Nodes in the product layer generate the input parameters as per Eq. ( 3 ).

Normalization of the input parameters takes place in the third layer. The normalization function is denoted by Eq. ( 4 ), and it converts the input parameters to values between 0 and 1.

W t is determined by the firing strength of node t denoted by ω t . Equation ( 5 ) gives the output of the fuzzy system.

Whereby p t , q t , and r t are the consequent parameters whereas x and y represent the prediction variables. The summation process at the defuzzification layer is summarized by Eq. ( 6 ).

Training of the model involves the adjustment of the consequent parameters and MFs and is conducted by optimization algorithms such as backpropagation, least-square methods, and PSO [ 38 ] based on a cost function. The fuzzy output values, after solving Eq. ( 6 ), are clear, concise, numerical values that are suitable for data prediction and decision-making. ANFIS was chosen based on its excellent results when deployed by [ 39 , 40 ] and [ 41 ] in their works.

2.3 Multi-objective particle swarm optimization (MOPSO)

MOPSO is an improved version of the heuristic particle swarm optimization (PSO) proposed by Coello and Lechunga [ 42 ] for multi-objective problem optimization. The MOPSO process begins with the initialization of a population of particles. Each particle is identified by its position ( P j k ), velocity ( V j k ), and fitness. Each particle’s position is a potential solution for the problem at hand. To optimize the hard turning process, the particles are considered a combination of predictors P j = ( v c j , f j , a p j ) whereas the velocities are V j = ( v j1 , v j2 , v j3 ). The new position of the jth particle in k + 1 iteration, P j k+1 , is updated by the new velocity, V j k+1 , as shown by Eq. ( 7 ). Equation ( 8 ) shows the updating of the previous velocity V j k . The new position is incorporated in the objective function to determine the fitness value. This fitness value is compared to the previous values, and if it is superior, it is considered the personal best; otherwise, the previous fitness is considered the best. The best individual position in the population is considered the global best position.

Whereby P best k is the personal best position, P global k is the best overall position (solution), \(\omega\) is the inertia weight, c 1 represents the personal learning coefficient, and c 2 refers to the global learning coefficient, whereas r 1 and r 2 are random values in the range [0, 1].

As an extension to PSO, MOPSO deploys the concept of Pareto optimality. It involves the creation of external repositories for each particle at each iteration to store its non-dominated solutions. Another repository is created for the entire population for each iteration and one global repository for the entire population for the entire optimization process. A selection criterion is used to choose the best non-dominated solution, from the first two repositories. A leader or global solution is later chosen from the global repository. Figure 10 shows the process of executing MOPSO.

Flowchart of MOPSO algorithm

MOPSO was chosen for this work because of its fast convergence, great compatibility, high efficiency, and feasibility [ 43 ].

3 Results and discussion

The signal extraction process was performed in MATLAB R2022b software through coding for all 81 runs and an additional 9 for validation data. Numerical signal values were obtained from arbitrary ranges within the relevant run signal, as shown in Fig. 7 . The non-cutting ranges shown in Figs. 6 and 7 contain signals originating from machine excitation (due to tool acceleration towards the workpiece) and vibrations due to chuck rotation. These non-cutting signals were not considered during numerical extractions of signal values. The force components’ and vibrations’ values in Table 2 are the averages extracted from the signals of the three experimental replicates. Fr and MRR values were determined using Eqs. ( 9 ) and ( 10 ), respectively.

3.1 Force analysis

Generally, as shown in Table 2 , passive force had the largest magnitude. This is a distinctive characteristic of hard turning processes, which can be attributed to the exceptional circumstances of chip formation at hard turning (e.g. spring-back effect) [ 35 ]. The spring-back effect results from the elastic and plastic deformation phenomenon that the workpiece experiences during machining [ 1 ]. According to Rath et al. [ 44 ], chip formation in steel hard turning results from crack initiation on the surface of the workpiece. The crack is initiated ahead of the tool and propagates as the compressive stresses induced by the tool increase due to the feed force to form saw-tooth chips. In the wake of chip formation, the workpiece experiences some element of elastic recovery. The elastic recovery (spring-back) raises the magnitude of the passive force. According to Bartaya et al. [ 45 ], the high passive force can also be associated with the lower depth of cut relative to the CBN insert’s cutting edge radius and the negative rake angle of the tool.

Compared to conventional turning, the cutting forces magnitude in hard turning is smaller. This is probably due to residual thermal stresses induced by the high temperatures generated and the low feed rates (relative to cutting speed) associated with this process. The high temperatures soften the workpiece during a given machining pass. As a result, the subsequent passes (either close to the previous pass or covering some part of the previous pass due to small feeds and relatively high cutting speeds) tend to occur on softer surfaces. The low cutting forces can also result from the low depths of cut and feed deployed in hard turning processes [ 46 ].

Analysis of variance in Table 3 indicates that all three cutting parameters were significant in resultant force generation, as their p-values were below 0.05 confidence level. Feed has the highest contribution (69.74%), whereas cutting velocity (3.79%) is the least, and the interaction of feed and depth of cut has the strongest influence on resultant force generation. From Figs. 11 and 12 , an increment in the depth of cut leads to an increment in the resultant force. This can be attributed to an increase in cutting edge angle, which leads to an increment in the coefficient of friction, and subsequent increment in compressive stresses by the tool. Similarly, the feed directly correlates with the resultant force, as shown in Figs. 12 and 13 . This is probably due to the increase in the size of the theoretical chip area with an increase in feed. The changes in the chip area have a direct correlation with the coefficient of friction hence leading to the significant changes in the magnitude of generated force components. Cutting velocity and resultant force have an inverse correlation (Figs. 11 and 13 ). This can be attributed to increase in temperature with increasing cutting velocity at the primary shear zone. The increasing temperature softened the workpiece ahead of the tool, leading to low compressive stresses asserted by the tool, hence the generation of low resultant force [ 47 ].

Variation of resultant force with depth of cut and cutting velocity at a constant feed of 0.125 mm/rev

Variation of resultant force with depth of cut and feed at a constant cutting speed of 125 m/min

Variation of resultant force with cutting velocity and feed at a constant depth of cut of 0.06 mm

As shown in Table 4 , all the cutting parameters and their two-factor interactions were significant in determining material removal rates. All of them had p -values of 0.0. However, feed was the most significant parameter with a contribution of 60.10%. Depth of cut had the least influence at 8.45%, whereas velocity had a contribution of 21.63%. The interaction of velocity and feed had the strongest influence on MRR amongst the parameters’ interactions. It can be inferred from Figs. 14 , 15 , and 16 that all the cutting parameters had a positive correlation with MRR. Increasing cutting velocity could have led to temperature increment at the primary cutting zone, hence softening the workpiece, making it easier and quicker to machine. Likewise, the increment of feed increased the chip area and subsequently volume of the material to be machined per unit time. Similarly, increasing the depth of cut implied increased engagement between the workpiece and the cutting tool that translated to large volume of material to be machined.

Variation of MRR with cutting velocity and feed at a constant depth of cut of 0.06 mm

Variation of MRR with feed and depth of cut at a constant cutting velocity of 125 m/min

Variation of MRR with cutting velocity and depth of cut at a constant feed of 0.125 mm/rev

3.2 Tool vibration analysis

The correlation between vibration and the cutting parameters is quite stochastic, and no clear trend is observed in Figs. 17 , 18 , and 19 . This can be attributed to the complexity of the vibration signal. Apart from tool vibration, the accelerometer often records signals from other unwanted sources within the machining environment. These unwanted signals (noise) may be challenging to get rid of during signal processing. The ANOVA (Table 5 ), however, shows that feed and its interaction with cutting velocity are significant in determining tool vibration. Their contributions are 62.42% and 31.42%, respectively. At a confidence level of 0.05, feed has a p -value of 0.001 whereas its interaction with cutting velocity has a p-value of 0.02. The high influence of feed can be attributed to its direct correlation with the cutting force and the subsequent direct correlation between the cutting force and vibration. As the theoretical chip area increases with the increase in feed, the force generated becomes more due to the high compressive stresses asserted by the advancing tool. The significant impact of the interaction of feed and velocity on vibration can be attributed to possible tool wear. The variation of feed and cutting velocity determines the coefficients of friction and cutting temperature at the shear zone, and this influences tool wear progression. The tool vibration, therefore, will vary with the tool wear status. Figure 20 shows the vibration signals.

Variation of vibration with cutting velocity and feed at a constant depth of cut of 0.06 mm

Variation of vibration with cutting velocity and depth of cut at a constant feed of 0.125 mm/rev

Variation of vibration with depth of cut and feed at a constant velocity of 125 m/min

Vibration signal

3.3 ANFIS model development and evaluation

Development of the ANFIS model for resultant force prediction was conducted in MATLAB R2022b using data in Table 2 . The training and testing processes were conducted using data from 19 and 8 experimental runs, respectively. Data from nine experimental runs were later used for the validation of the model. The training process involved variation of the number of membership functions (MF) allocated to the inputs and output parameters, variation of MF type, variation of the type of training algorithms (backpropagation or hybrid (backpropagation and least-squares method)), as well as variation of the fuzzy inferencing system (FIS) (Sugeno or Mamdani). These variations gave different models whose suitability was gauged using the model performance parameters, mean absolute percentage error (MAPE), and coefficient of correlation, R. The adopted model comprised four Gaussian MF s for one input and two each for the other two inputs. Its output parameter was allocated a constant MF and was based on the Sugeno inferencing system. The hybrid algorithm trained the model using the weighted average of rules. Figure 21 shows the model’s structure.

ANFIS model structure

The results of the model performance analysis in Table 6 showed that the model developed was highly reliable for prediction. According to [ 48 ], the accuracy of a model is considered low if its MAPE is above 50%, satisfactory if it is between 20 and 50%, good if it is between 10 and 20%, and high if it is below 10%. Similarly, a coefficient of determination of 1 shows a strong positive correlation between the measured and predicted values whereas an R 2 value of − 1 indicates a strong negative correlation. On the other hand, 0 indicates a weak correlation. The MAPE of the model on the trained and test data was 1.47% and 4.81%, respectively, whereas the R 2 values were 0.9 (trained data) and 0.9 (test data). The high R 2 of 0.8 on validation data indicated a strong positive correlation between the measured and predicted resultant forces. This strong correlation was further confirmed by a MAPE of 10.66% on the validation data. The high MAPE and R 2 values (close to 1) on validation data indicated that the model was robust enough to predict resultant force during UHT of cold work hardened AISI D2 using the CBN tool. Figure 22 demonstrates the prediction ability of the developed model. Similar results are observed with the validation data as is shown in Fig. 23 .

Comparison of measured against ANFIS model predicted values

Comparison between validation and ANFIS-predicted data

3.4 MOPSO results

A relatively high MRR in UHT, relative to the optimal values of other process parameters, increases the production rate by lowering the lead-time on a single product. Therefore, maximization of MRR relative to other machinability metrics can upheave effective and efficient production [ 49 ]. Tool vibration is also a very important parameter in the determination of production quality and cost. Tool vibration correlates with the cutting force, surface roughness, tool wear, and MRR amongst other process parameters. Sahu et al. [ 50 ] demonstrated that surface roughness increased with an increase in tool vibration. Similarly, Guleria et al. [ 51 ] used a tool vibration signal processing technique to classify surface roughness. Tool vibrations introduce chatter marks on the product surface, hence lowering the surface’s quality. In addition to these, tool vibration is also a very critical cause of tool failures and breakages. Therefore, keeping tool vibration relatively minimal is reliable for the stability of the machining process and the acquisition of quality products. This work, therefore, opted to optimize UHT resultant force relative to MRR and vibration with the aim of maximizing MRR and minimizing both the resultant force and vibration.

MOPSO sought to find the best cutting parameter values that would give an optimal magnitude of the resultant force, MRR, and vibration. Coding of the MOPSO algorithm was conducted in MATLAB R2022b using Eqs. 11 , 12 , and 13 as the objective functions. These equations were developed by the response surface methodology approach in Minitab 21. The algorithm was best tuned according to the parameters in Table 7 .

Figure 24 shows the distribution of particles in the search space during the iteration processes of MOPSO. A series of red and black circular marks in the search space represent a series of Pareto fronts. The most optimal Pareto solution (leader) was selected from the many Pareto fronts. Table 8 shows the optimal values for the cutting parameters (cutting velocity, feed, and depth of cut) and response parameters.

Distribution of particles (possible solutions) in the search space

The large MRR value is desirable for enhancing the production rate of hardened AISI D2 steel components in an industrial setting. Since there is a direct correlation between vibration and the cutting forces [ 12 ], the low vibration value attained in this work shows that the acquired resultant force is the most optimal to warrant machine stability during UHT of AISI D2 steel within the investigated parameter range.

4 Conclusion

A comprehensive literature review on UHT of hardened AISI D2 steel is presented in this work. An in-depth analysis of the variation of force components during UHT of AISI D2 steel has been reported. ANFIS machine learning model for force prediction was developed and evaluated. The resultant force was later optimized relative to MRR and vibration using MOPSO. Consequently, the following conclusions were made:

The proposed ANFIS model was developed, and its performance evaluation was conducted using mean absolute percentage error (MAPE) and the coefficient of determination ( R 2 ). According to MAPE values of 10.66% and R 2 -values of 0.8 on validation data, the model’s prediction was highly satisfactory (Table 6 ). The model, therefore, can be incorporated into digital twin models for monitoring machining stability during the UHT process of AISI D2 metals.

From the optimization process, a feed of 0.125 mm/rev, a cutting velocity of 158.8 m/min, and a depth of cut of 0.074 mm will result in a stable UHT of hardened AISI D2 steel when using the CBN tool. The corresponding optimal resultant force, MRR, and vibration related to these cutting parameters were 224.8 N, 2603.6 mm 3 /min, and 0.03 m/s 2 (Table 8 ). These values are bound to improve the machining rate as well as lower the machining costs on a shop floor.

All the cutting parameters are significant in resultant force generation during UHT of AISI D2 steel, with feed being the most significant parameter with a contribution of 69.74%. Cutting velocity is a minor contributor (Table 3 ).

Passive force is the largest force component during UHT of hardened AISI D2 steel using the CBN tool (possibly due to the spring-back effect experienced during UHT). In contrast, the feed and cutting forces have low magnitudes (probably due to the high temperature generated at the primary shearing zone), see Table 2 . The knowledge of the magnitude of force components can be of help during tool design and tool choice for UHT processes.

Feed and cutting velocity are the most significant factors in the determination of MRR and vibration during UHT. Feed’s influence dominates that of cutting velocity.

The reported findings, therefore, offer invaluable insight into kinematics, forecasting, and optimization of UHT processes. The results can contribute immensely to the development of precision engineering in the machining of difficult-to-cut metals and other related materials.

4.1 Future directions

The authors recommend future studies to be carried out on composite materials with an interest in the chip removal mechanisms relative to the composite material’s structures. A digital twin model for monitoring machine stability, amongst other machinability indices, can be developed using the developed AI model.

Data availability

The dataset used in this work is available and can be issued by the corresponding author upon satisfactory request.

Code availability

The code used in this work is available and can be issued by the corresponding author upon satisfactory request.

Abbreviations

Ultra-precision hard turning

Mean average percentage error

Cubic boron nitride

Material removal rate

Multiple objective particle swarm optimization

Coefficient of determination

Adaptive neuro-fuzzy inferencing system

Resultant force

Tool vibration

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Acknowledgements

The authors acknowledge the support offered by the National Research, Development, and Information Office (NRDIO) through projects: Research on prime exploitation of the potential provided by industrial digitalization (ED_18-2018-0006) and transient deformation, thermal, and tribological processes at fine machining of metal surfaces of high hardness (OTKA-K-132430). In addition, the authors acknowledge the support of the National Laboratory of Artificial Intelligence, which NRDIO supports under the Ministry for Innovation and Technology. We acknowledge the BME administration for the permission to use the facilities at the Engineering workshop.

Open access funding provided by Budapest University of Technology and Economics. National Research Development and Information office (NRDIO), ED_18-2018-0006.

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Ogutu Isaya Elly, Ugonna Loveday Adizue, Amanuel Diriba Tura, Balázs Zsolt Farkas & M.Takács

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Research conceptualization and design were carried out by all the authors. Balázs Farkas configured the ultra-precision lathe and other data collection equipment. Ogutu Isaya Elly, Ugonna Loveday Adizue, and Amanuel Diriba Tura carried out material preparation, data collection, and analysis. Ogutu Isaya Elly prepared the very first draft of this paper. He later compiled several versions of the first draft, based on other authors’ comments and suggestions, to produce this final document. Márton Takács supervised the entire research. The final draft was later proofread and approved by all the authors.

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Elly, O.I., Adizue, U.L., Tura, A.D. et al. Analysis, modelling, and optimization of force in ultra-precision hard turning of cold work hardened steel using the CBN tool. J Braz. Soc. Mech. Sci. Eng. 46 , 624 (2024). https://doi.org/10.1007/s40430-024-05167-4

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2.1 Design of experiments. This work sought to characterize the UHT of cold work hardened AISI D2 steel through analysis, prediction, and optimization of force components based on full factorial experiments. Therefore, a theoretical analysis of the force components is conducted, and the resultant force is predicted through ANFIS modelling.

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