wiki factorial experiments

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:

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

Experiment II: Mean Scores

Experiment III: Mean Scores

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.

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Factorial Designs

41 setting up a factorial experiment, learning objectives.

• Explain why researchers often include multiple independent variables in their studies.
• Define factorial design, and use a factorial design table to represent and interpret simple factorial designs.

Just as it is common for studies in psychology to include multiple levels of a single independent variable (placebo, new drug, old drug), it is also common for them to include multiple independent variables. Schnall and her colleagues studied the effect of both disgust and private body consciousness in the same study. Researchers’ inclusion of multiple independent variables in one experiment is further illustrated by the following actual titles from various professional journals:

• The Effects of Temporal Delay and Orientation on Haptic Object Recognition
• Opening Closed Minds: The Combined Effects of Intergroup Contact and Need for Closure on Prejudice
• Effects of Expectancies and Coping on Pain-Induced Intentions to Smoke
• The Effect of Age and Divided Attention on Spontaneous Recognition
• The Effects of Reduced Food Size and Package Size on the Consumption Behavior of Restrained and Unrestrained Eaters

Just as including multiple levels of a single independent variable allows one to answer more sophisticated research questions, so too does including multiple independent variables in the same experiment. For example, instead of conducting one study on the effect of disgust on moral judgment and another on the effect of private body consciousness on moral judgment, Schnall and colleagues were able to conduct one study that addressed both questions. But including multiple independent variables also allows the researcher to answer questions about whether the effect of one independent variable depends on the level of another. This is referred to as an interaction between the independent variables. Schnall and her colleagues, for example, observed an interaction between disgust and private body consciousness because the effect of disgust depended on whether participants were high or low in private body consciousness. As we will see, interactions are often among the most interesting results in psychological research.

By far the most common approach to including multiple independent variables (which are often called factors) in an experiment is the factorial design. In a  factorial design , each level of one independent variable is combined with each level of the others to produce all possible combinations. Each combination, then, becomes a condition in the experiment. Imagine, for example, an experiment on the effect of cell phone use (yes vs. no) and time of day (day vs. night) on driving ability. This is shown in the  factorial design table  in Figure 9.1. The columns of the table represent cell phone use, and the rows represent time of day. The four cells of the table represent the four possible combinations or conditions: using a cell phone during the day, not using a cell phone during the day, using a cell phone at night, and not using a cell phone at night. This particular design is referred to as a 2 × 2 (read “two-by-two”) factorial design because it combines two variables, each of which has two levels.

If one of the independent variables had a third level (e.g., using a handheld cell phone, using a hands-free cell phone, and not using a cell phone), then it would be a 3 × 2 factorial design, and there would be six distinct conditions. Notice that the number of possible conditions is the product of the numbers of levels. A 2 × 2 factorial design has four conditions, a 3 × 2 factorial design has six conditions, a 4 × 5 factorial design would have 20 conditions, and so on. Also notice that each number in the notation represents one factor, one independent variable. So by looking at how many numbers are in the notation, you can determine how many independent variables there are in the experiment. 2 x 2, 3 x 3, and 2 x 3 designs all have two numbers in the notation and therefore all have two independent variables. The numerical value of each of the numbers represents the number of levels of each independent variable. A 2 means that the independent variable has two levels, a 3 means that the independent variable has three levels, a 4 means it has four levels, etc. To illustrate a 3 x 3 design has two independent variables, each with three levels, while a 2 x 2 x 2 design has three independent variables, each with two levels.

In principle, factorial designs can include any number of independent variables with any number of levels. For example, an experiment could include the type of psychotherapy (cognitive vs. behavioral), the length of the psychotherapy (2 weeks vs. 2 months), and the sex of the psychotherapist (female vs. male). This would be a 2 × 2 × 2 factorial design and would have eight conditions. Figure 9.2 shows one way to represent this design. In practice, it is unusual for there to be more than three independent variables with more than two or three levels each. This is for at least two reasons: For one, the number of conditions can quickly become unmanageable. For example, adding a fourth independent variable with three levels (e.g., therapist experience: low vs. medium vs. high) to the current example would make it a 2 × 2 × 2 × 3 factorial design with 24 distinct conditions. Second, the number of participants required to populate all of these conditions (while maintaining a reasonable ability to detect a real underlying effect) can render the design unfeasible (for more information, see the discussion about the importance of adequate statistical power in Chapter 13). As a result, in the remainder of this section, we will focus on designs with two independent variables. The general principles discussed here extend in a straightforward way to more complex factorial designs.

Assigning Participants to Conditions

Recall that in a simple between-subjects design, each participant is tested in only one condition. In a simple within-subjects design, each participant is tested in all conditions. In a factorial experiment, the decision to take the between-subjects or within-subjects approach must be made separately for each independent variable. In a  between-subjects factorial design , all of the independent variables are manipulated between subjects. For example, all participants could be tested either while using a cell phone  or  while not using a cell phone and either during the day  or  during the night. This would mean that each participant would be tested in one and only one condition. In a within-subjects factorial design, all of the independent variables are manipulated within subjects. All participants could be tested both while using a cell phone and  while not using a cell phone and both during the day  and  during the night. This would mean that each participant would need to be tested in all four conditions. The advantages and disadvantages of these two approaches are the same as those discussed in Chapter 5. The between-subjects design is conceptually simpler, avoids order/carryover effects, and minimizes the time and effort of each participant. The within-subjects design is more efficient for the researcher and controls extraneous participant variables.

Since factorial designs have more than one independent variable, it is also possible to manipulate one independent variable between subjects and another within subjects. This is called a  mixed factorial design . For example, a researcher might choose to treat cell phone use as a within-subjects factor by testing the same participants both while using a cell phone and while not using a cell phone (while counterbalancing the order of these two conditions). But they might choose to treat time of day as a between-subjects factor by testing each participant either during the day or during the night (perhaps because this only requires them to come in for testing once). Thus each participant in this mixed design would be tested in two of the four conditions.

Regardless of whether the design is between subjects, within subjects, or mixed, the actual assignment of participants to conditions or orders of conditions is typically done randomly.

Non-Manipulated Independent Variables

In many factorial designs, one of the independent variables is a non-manipulated independent variable . The researcher measures it but does not manipulate it. The study by Schnall and colleagues is a good example. One independent variable was disgust, which the researchers manipulated by testing participants in a clean room or a messy room. The other was private body consciousness, a participant variable which the researchers simply measured. Another example is a study by Halle Brown and colleagues in which participants were exposed to several words that they were later asked to recall (Brown, Kosslyn, Delamater, Fama, & Barsky, 1999) [1] . The manipulated independent variable was the type of word. Some were negative health-related words (e.g.,  tumor, coronary ), and others were not health related (e.g.,  election, geometry ). The non-manipulated independent variable was whether participants were high or low in hypochondriasis (excessive concern with ordinary bodily symptoms). The result of this study was that the participants high in hypochondriasis were better than those low in hypochondriasis at recalling the health-related words, but they were no better at recalling the non-health-related words.

Such studies are extremely common, and there are several points worth making about them. First, non-manipulated independent variables are usually participant variables (private body consciousness, hypochondriasis, self-esteem, gender, and so on), and as such, they are by definition between-subjects factors. For example, people are either low in hypochondriasis or high in hypochondriasis; they cannot be tested in both of these conditions. Second, such studies are generally considered to be experiments as long as at least one independent variable is manipulated, regardless of how many non-manipulated independent variables are included. Third, it is important to remember that causal conclusions can only be drawn about the manipulated independent variable. For example, Schnall and her colleagues were justified in concluding that disgust affected the harshness of their participants’ moral judgments because they manipulated that variable and randomly assigned participants to the clean or messy room. But they would not have been justified in concluding that participants’ private body consciousness affected the harshness of their participants’ moral judgments because they did not manipulate that variable. It could be, for example, that having a strict moral code and a heightened awareness of one’s body are both caused by some third variable (e.g., neuroticism). Thus it is important to be aware of which variables in a study are manipulated and which are not.

Non-Experimental Studies With Factorial Designs

Thus far we have seen that factorial experiments can include manipulated independent variables or a combination of manipulated and non-manipulated independent variables. But factorial designs can also include  only non-manipulated independent variables, in which case they are no longer experiments but are instead non-experimental in nature. Consider a hypothetical study in which a researcher simply measures both the moods and the self-esteem of several participants—categorizing them as having either a positive or negative mood and as being either high or low in self-esteem—along with their willingness to have unprotected sexual intercourse. This can be conceptualized as a 2 × 2 factorial design with mood (positive vs. negative) and self-esteem (high vs. low) as non-manipulated between-subjects factors. Willingness to have unprotected sex is the dependent variable.

Again, because neither independent variable in this example was manipulated, it is a non-experimental study rather than an experiment. (The similar study by MacDonald and Martineau [2002] [2]  was an experiment because they manipulated their participants’ moods.) This is important because, as always, one must be cautious about inferring causality from non-experimental studies because of the directionality and third-variable problems. For example, an effect of participants’ moods on their willingness to have unprotected sex might be caused by any other variable that happens to be correlated with their moods.

• Brown, H. D., Kosslyn, S. M., Delamater, B., Fama, A., & Barsky, A. J. (1999). Perceptual and memory biases for health-related information in hypochondriacal individuals. Journal of Psychosomatic Research, 47 , 67–78. ↵
• MacDonald, T. K., & Martineau, A. M. (2002). Self-esteem, mood, and intentions to use condoms: When does low self-esteem lead to risky health behaviors? Journal of Experimental Social Psychology, 38 , 299–306. ↵

Experiments that include more than one independent variable in which each level of one independent variable is combined with each level of the others to produce all possible combinations.

Shows how each level of one independent variable is combined with each level of the others to produce all possible combinations in a factorial design.

All of the independent variables are manipulated between subjects.

A design which manipulates one independent variable between subjects and another within subjects.

An independent variable that is measured but is non-manipulated.

Research Methods in Psychology Copyright © 2019 by Rajiv S. Jhangiani, I-Chant A. Chiang, Carrie Cuttler, & Dana C. Leighton is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

Share This Book

• Create account
• Contributions
• Discussion for this IP address

Recipes for the Design of Experiments/Chapter 1: One Factor, Two Level Experiments

1.1 One Factor, Two Level Experiments (Shamus W, Alexis Z)

In experimental design, the number of factors and levels dictate how effects are calculated and which statistical inference tests are used. A one factor two level experiment studies the effect of only one independent variable on the response of a dependent variable. Only two levels of the factor are studied in this experiment.

A main effect is the effect that the change in level of a factor has on the response. In a one factor, two level experiment, the main effect is the difference in the average of the response variable caused by the change of the factor from one level to the other level. An interaction effect is when the difference in response variable for one factor's levels is dependent on the levels of the other factor. In one factor experiments, there are no interaction effects.

To be able to make inferences about a population, samples of the population are taken. These samples should be randomly selected and randomly assigned to a treatment, and the experimental runs should also be randomized. It is also assumed that the data are from a normal distribution. A normal distribution is when the probability distribution of a sample can be defined by the equation in the figure on the right:

For one factor, two level experiments, a t-test is used to indicate whether the difference between the two sample response averages is due to a difference in the effect of the two levels, or randomization. T-tests are commonly used when sample size is small.  The formula for a t-Test test statistic is provided on the right:

To determine whether we can reject the null hypothesis the test statistic needs to be compared to the critical t-value. The degrees of freedom and significance level are used to calculate the critical value. The critical t-value and test statistic can both be plotted on a distribution of test statistics. On this distribution, the “upper tail” region of the curve is region under the curve that is greater than the critical t-value. For the Null Hypothesis, we assume the variance of each sample (μ) is equal, μ1 is equal to μ2. If the test statistic is lower than the critical t-value, we accept the null hypothesis.

1.2 Main Effects (Liang Z, Joonhyuk B)

In this section, we would like to discuss about the Main Effects in experimental design. Especially, when our treatments have a factorial structure, we could utilize a factorial analysis of the data. The main concepts of this factorial analysis are main effects and interactions. When we design an experiment to analyze the problems we meet, we cannot avoid thinking about the effect relationships. For example, when economists study the difference of average hourly earnings (AHE) among people, they need to consider whether it is their education level, gender, or other effects causing the difference. In pharmaceutical science, scientists need to consider whether a particular medicine works for people. Furthermore, in a tennis racket experiment discussed during the class, we could change all factors of string tension, racket mass, balance of racket, and hardness of ball at the same time in order to check power (main effects). However, we cannot guarantee research results without interaction effects between factors. we are not sure which combination of factors has the greatest power because the paper does not consider interaction effects between factors. In this chapter, we start from the beginning – One Factor, Two Level Experiments. We will see what main effects are and how we test them.

1.2.1. Concept of Main Effect

A Main Effect is the effect of an independent variable on a dependent variable across the levels without considering other effects. In other words, Main effects are differences in means over levels of one factor collapsed over levels of the other factor. For example, the main effect of Method is simply the difference between the means of final exam score for the two levels of Method, ignoring or collapsing over experience. In One Factor, Two Level Experiments, we consider only one factor, and two levels. For instance, in AHE example above, if we want test whether studying in college has effect on AHE difference, we have one factor – College Degree, and two levels (Bachelor = 1, High School or Below =0). If we study latter chapters, we may also consider other factors such as gender, district difference, etc. We may not conclude them in this chapter.

1.2.2. Experiment Design

Right now, you may have a question. How can we design the experiment and get the result close to the truth? You may have note that other effects may bias our test result if we do not deal with them properly. Here, we can use complete randomized design (one-factor). In a completely randomized design , there is only on factor, and subjects are randomly assigned to treatments. Here we do not need to consider the effect of individual difference, and it is a one-way experimental treatment, such as the effect of fertilizer on wheat production, etc. We can use t-test if the factor has 2 levels, and F-test if the factor has 3 levels. This method will be discussed in detail in later section in this chapter. In our AHE example above, we may randomly pick people who go to the college and who does not, and then analyze their salary difference. In pharmaceutical science example, we may pick people and separate them into two groups randomly, one with medicine and another with placebo. For another example, let’s look at a research in which 10-year-olds and 17-year-olds are given IQ tests. These students will be selected completely at random, without regard to their actual test scores, to see if teacher expectations alone have an impact on student performance. We include age as another factor to see if teacher expectations have a different effect depending on the age of the student. This would be a 2 (teacher expectations: high or average) x 2 (age of student: 10 years or 17 years) factorial design.

1.3 Interaction Effects (Andreas V, Prasanna D)

Interaction describes the failure of one factor to generate the same influence on response at varying levels of another factor. This is a major issue with one-factor-at-a-time (OFAT) approach. The interaction effect between factors can be either substantial or virtually non-existent, and it is important to determine which factors influence each other at varying levels of severity. Factorial experiments are designed to negate the interaction effects in a scientific experiment. An example of high interaction effect could be racket weight vs. power, and an example of a very weak or no interaction effect may be the type of hat worn by a golfer and the type of driver used.

Let us understand the significance of interaction effects and their computation by the means of an example. Let us say we have a group of 100 students which have been randomly divided into four groups of 25 students each as follows:

• Group 1 studies during the day and gives a math test, obtaining a mean score of 13.
• Group 2 studies during the day and gives a social science test, obtaining a mean score of 9.
• Group 3 studies at night and gives a math test, obtaining a mean score of 9.
• Group 4 studies at night and gives a social science test, obtaining a mean score of 11.

As can be seen here we have two factors: 'Study Time' and 'Subject'. 'Study Time' has two levels: 'Day' and 'Night', and 'Subject' has two levels: 'Math' and 'Social Science'. The mean effect for both 'Study Time' and 'Subject' is -1. The interaction effect can be computed as follows:

• Take the mean of (Night, Math) and (Day, Social Science), which is equal to 9.
• Take the mean of (Night, Social Science) and (Day, Math), which is equal to 12.
• Take the difference of (2) from (1), which is equal to 9 - 12 = -3.

Thus, in this example, the interaction effect is much greater than either of the two main effects.

1.4 Simple Two-way Comparative Experiments (Trilce E, Bjarke H)

Simple two-way comparative experiments evaluate the effects of two different treatments on subsets of a population. The subjects of the experiments are then grouped into pairs, based on some blocking variable. By assuming that the two groups are probabilistically equivalent, we can use random assignment to select the individuals or samples that receive the treatment within each pair. Consider the example of a study conducted by a Portland cement mortar [ 1 ] . The engineer in charge of the study has created two populations of 10 samples each, with one set of experiments receiving the treatment. In this case, he added a polymer latex emulsion to determine if this impacts the curing time and tension strength of the mortar.

The factor is the mortar formulation and the two levels are: "modified" and "unmodified". The observations are shown in the following box plot.

Link to box plot: http://imgur.com/AMTyEQl

In order to evaluate the treatment effect, the statistical technique of hypothesis testing allows the comparison of two formulations to be made on objective terms. By using hypothesis testing, such as either t-test, one way Analysis of Variance (ANOVA), as well as confidence interval procedures, we can compare the two treatment means to determine whether the populations differ due to the effect of the treatment of due to random chance.

1.5 The t-test and the 1-way Analysis of Variance (ANOVA)(TC)

A t-test is an inferential statistic which is used to “draw conclusions about the properties of populations from related properties of the sample.” (Dunn, 10/16, Introduction to the Design of Experiments) The paired t-test is the gold standard procedure for testing null hypotheses. The formula for a one sample t-test can be found in the following link http://imgur.com/gallery/miVeT A t-test is generally utilized while determining if two datasets are significantly different from one another.

ANOVA, which stands for Analysis of Variance, is Fisher’s statistical method of analysis for factorial experiments. It takes into account all possible combinations of factors and levels, each in solitary experimental runs. ANOVA can be used to aid in the determination of main effects. One-way ANOVA is a specialized model for computing main effects of factors on response variables. In other words, “with ANOVA, an inference procedure is included to assess the likelihood that there is a model relationship between the factor(s) and response variable that is something other than randomization.” (Dunn, 10/16, Introduction to the Design of Experiments)

1.6 Sample Recipes

This is the collection of R recipes for the analysis of one factor, two level experiments. Each of these recipes are structured in the Setting, Design, Analysis paradigm and the Design portion is structured in four parts: Exploratory Analysis, Testing, Estimation and Model Adequacy Checking.

The Setting, Design, Analysis Paradigm

The Project Outline

• Exploratory Analysis
• Estimation (of Parameters)
• Model Adequacy Checking

Sample Recipes

-> These data are a collection of various measures, including variables such as elevation, temperature (surface and air), ozone, air pressure, and cloud cover. For this section, a t-test was conducted to explore whether a statistically significant difference existed between the two temperature variables. The H0 was that no difference existed between the mean temperature values of both factors, whereas the HA was that a statistically significant difference existed between the two factors. Based on the results of the test, the H0 was rejected W.R.T. an alpha value of .05. However, after model adequacy checking was performed, the results were invalidated due to the fact that the t-test assumes normality, and the Shapiro-Wilks normality test confirmed that the data are not normal. Further steps to "coerce" the data to be normally distributed include performing data transformations, but this action has not yet been completed. http://rpubs.com/manzat/28671

-> The following is an analysis of a data set which includes a collection of wind speed, barometric pressure, longitude, latitude and time points for a large number of storms recorded by NASA. In my analysis I decided to examine if there was a significant statistical difference in the wind speed and pressure readings between hurricanes and extratropical storms. This analysis was performed through the use of a t-test and a QQ plot as a check for model adequacy. http://rpubs.com/adamato/28910

-> Using fuel economy data from the EPA collected from 1985-2015, a one-factor, two-level experiment is performed to see if the “make” of a vehicle has a statistically significant effect on the fuel economy of that vehicle. The two “makes” that are considered in this analysis include ‘Toyota’ and ‘Audi’. Additionally, this analysis separately considers two different metrics for measuring fuel economy, including both highway fuel economy [in mpg] (“hwy”) and city fuel economy [in mpg] (“cty”). Upon performing this analysis, it was determined that the fact that these two vehicles are classified by different “makes” likely does appear to have an effect on the average city fuel economy that is achieved when driving either “make” of vehicle. However, the fact that these two vehicles are classified by different “makes” does not appear to have an effect on the average highway fuel economy that is achieved when driving either “make” of vehicle. - Brendan Howell http://rpubs.com/howelb/29127

-> The following analysis is conduced based on the EPA fuel economy data from 1985 to 2015. To explore the effect of vehicle engine power (number of cylinders) on fuel economy, a one factor (number of cylinders), two level (cyl = 4 and cyl = 6) experiment was carried out and a linear regression model is estimated to quantify the marginal effect. The result shows that the vehicle fuel economy decreases with the number of cylinders in the vehicle. http://rpubs.com/serena049/doehw1

-> The html file below utilizes fuel economy data that has been collect by the EPA from 1985 to 2015. Specific data is analyzed to conduct a one-factor, two-level experiment to determine if the variation in the make of a vehicle is responsible for the variation in highway gas mileage. The factor, "make," is grouped into two levels, "Toyota" and "Honda," with a response variable of "hwy" (the highway mileage). An unpaired, two-sample t-test, with a null hypothesis that the highway mileage means of both vehicle makes are equal, will be conducted. http://rpubs.com/maxwinkelman/28916

This analysis utilizes ‘nycflights13’ data set containing all information about flights that departed from NYC (JFK, EWR and LGA) in 2013. There are 336,776 flights in total. In order to study the underlying causes of flight delay an experiment was conducted which can be termed as ‘one factor- two level’ experiment. Here ‘time-delay’ is the factor and ‘departure time-delay’ and ‘arrival time-delay’ are two levels. We conduct a t-test to test the null hypothesis , ‘mean of the departure time delay is equal to the mean of the arrival time delay’. This is essentially to try and test whether there are other factors that contribute to the delay of an aircraft apart from the delay in its take-off time. For this purpose we take a sub-set of the data (only UA flights departing from JFK). Our t-test experiment leads to the rejection of the null hypothesis. However, our exploratory data analysis as well as diagnostic check leads us to the conclusion that the data is not normally distributed. Therefore the assumption based on which we conduct the t-test is flawed. This leads us to the conclusion that our t-test is not valid for our current data set. http://rpubs.com/Uzma_1004/28917

The following analysis is based on the data set called 'nasaweather' collected from National Hurricane Center. It takes the classification of different types of storms and their respective air pressure measurement to perform a t-test. As you will see in the analysis, the results show that we reject the null hypothesis that the means of Tropical Storms does not equal the means of Hurricanes. http://rpubs.com/hsiac/28942

The following analysis investigates if electric/hybrid cars are actually more fuel efficient than cars with more traditional gasoline engines. This analysis looked at both highway and city driving where the data was derived from an EPA fuel economy dataset from 1985 to 2015. Individual analyses of highway and city driving were conducted in a one-factor, two-level manner using t-tests. Cars with electric/hybrid engines were determined to get significantly better gas mileage than cars with traditional gasoline engines on the highway as well as around the city. http://rpubs.com/JohnMariani/28953

->The following link utilizes the R data "storms" from the "nasaweather" package. "Storms" includes 4 observations a day for every storm named from 1995 to 2000. The storm name, year, month, day, hour, latitude, longitude, pressure, wind speed, and type of storm were recorded for each observation. This analysis will focus on only two of the observations a day (hour 0 and 12) and one factor, two level testing to examine if the variation in wind could be explained by the hour of each day. Exploritory data analysis, a t-test, and normality testing is performed. There is also a section at the end that covers contingencies if the assumptions are broken. http://rpubs.com/svoboa/28937

The following analysis is based on the 'nycflights13' dataset which contains information of flights from NYC, including JFK, EWR and LGA. Also plane information and weather condition are recorded in this dataset. It is known to all that several airlines are usually not on time, and this analysis is in order to test effect of airlines on time delay and help passengers choose the best one. We take test for Delta Airline as an example, considering it as a 'one factor - two level' model, where 'departure delay' and 'arrival delay' are two levels. To block possible cofounding factors we only select flights from JFK. T-test shows there is no relationship between two factors, however, following tests reveal that data is not normally distributed, therefore it is not correct to conduct t-test in this case. This analysis is the first version, it is still needed to be improved in the future. http://rpubs.com/chenh16/29271

-> The following is an experiment designed to investigate the relationship between year and storm wind speed from the 'nasaweather' dataset. The experiment will look into the normality of the data and also attempt to discover whether or not there is a significant difference between of the means via two-sample t-test. http://rpubs.com/macchm/30638

->This experiment is testing to see if there are differences in the Mean Surface Temperature from Clear Sky Composite and Mean Near-Surface Air Temperature in the month of January. The Mean Surface Temperature from Clear Sky Composite is the monthly mean temperature based on the energy being emitted from the Earth’s surface under clear sky conditions in K. The Mean Near-Surface Air Temperature is the monthly mean temperature of the air near the surface of the Earth in K. http://rpubs.com/hsiac/30643 -by Cheryl Tran

-> The following link is an experiment that utilizes flight data from NYC, in particular date of flight, departure and arrival times, departure and arrival times, departure and arrival locations and some information about the plane. The experiment looks to examine the effects of the origin on the departure delay time. The assumption being that the a flight's delay is due to the problems in the departure airport. The study firsts looks to explore the data with summary statistics, box plots and histograms. The data is analyzed using t-tests and the normality of the data is also checked using QQ plots and the Shapiro-Wilk test. Lastly it addresses possible contingencies in the experiment. http://rpubs.com/Tothk2/31630 -By Kevin Toth

-> The following analysis examined flight data from planes leaving from three of New York City's major airports. The data includes factors such as the date, the time of arrival/departure, the arrival/departure delays, the airline carrier, the origin and destination, along with multiple others. In particular, this experiment analyzed delays, and how they are related to the origin and departure city, along with the specific airline carrier responsible for the flights. A one factor, two level t-test was completed for two different airline carriers and two different origin locations. Unfortunately, after completing a Shapiro-Wilks test for normality, it was found that the data is nonparametric, and would therefore need additional testing using nonparametric methods. http://rpubs.com/braunj6/31855

-> The following analysis of a one factor two level experiment uses a t-test to test the difference in means of departure delays across two of NYC's airports. http://rpubs.com/konraz/39536

• ↑ Montgomery, Design & Analysis of Experiment ECE, 8th Edition

• Book:Recipes for the Design of Experiments

Factorial experiment

In statistics , a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors. A full factorial design may also be called a fully crossed design . Such an experiment allows the investigator to study the effect of each factor on the response variable , as well as the effects of interactions between factors on the response variable.

Advantages and disadvantages of factorial experiments

Example of advantages of factorial experiments, contrasts, main effects and interactions, implementation, analysis example, explanatory footnotes, external links.

For the vast majority of factorial experiments, each factor has only two levels. For example, with two factors each taking two levels, a factorial experiment would have four treatment combinations in total, and is usually called a 2×2 factorial design . In such a design, the interaction between the variables is often the most important. This applies even to scenarios where a main effect and an interaction are present.

If the number of combinations in a full factorial design is too high to be logistically feasible, a fractional factorial design may be done, in which some of the possible combinations (usually at least half) are omitted.

Other terms for "treatment combinations" are often used, such as runs (of an experiment), points (viewing the combinations as vertices of a graph , and cells (arising as intersections of rows and columns).

Factorial designs were used in the 19th century by John Bennet Lawes and Joseph Henry Gilbert of the Rothamsted Experimental Station . [1]

Ronald Fisher argued in 1926 that "complex" designs (such as factorial designs) were more efficient than studying one factor at a time. [2] Fisher wrote,

"No aphorism is more frequently repeated in connection with field trials, than that we must ask Nature few questions, or, ideally, one question, at a time. The writer is convinced that this view is wholly mistaken. Nature, he suggests, will best respond to a logical and carefully thought out questionnaire; indeed, if we ask her a single question, she will often refuse to answer until some other topic has been discussed."

A factorial design allows the effect of several factors and even interactions between them to be determined with the same number of trials as are necessary to determine any one of the effects by itself with the same degree of accuracy.

Frank Yates made significant contributions, particularly in the analysis of designs, by the Yates analysis .

The term "factorial" may not have been used in print before 1935, when Fisher used it in his book The Design of Experiments . [3]

Many people examine the effect of only a single factor or variable. Compared to such one-factor-at-a-time (OFAT) experiments, factorial experiments offer several advantages [4] [5]

• Factorial designs are more efficient than OFAT experiments. They provide more information at similar or lower cost. They can find optimal conditions faster than OFAT experiments.
• When the effect of one factor is different for different levels of another factor, it cannot be detected by an OFAT experiment design. Factorial designs are required to detect such interactions . Use of OFAT when interactions are present can lead to serious misunderstanding of how the response changes with the factors.
• Factorial designs allow the effects of a factor to be estimated at several levels of the other factors, yielding conclusions that are valid over a range of experimental conditions.

The main disadvantage of the full factorial design is its sample size requirement, which grows exponentially with the number of factors or inputs considered. [6] Alternative strategies with improved computational efficiency include fractional factorial designs , Latin hypercube sampling , and quasi-random sampling techniques .

In his book, Improving Almost Anything: Ideas and Essays , statistician George Box gives many examples of the benefits of factorial experiments. Here is one. [7] Engineers at the bearing manufacturer SKF wanted to know if changing to a less expensive "cage" design would affect bearing life. The engineers asked Christer Hellstrand, a statistician, for help in designing the experiment. [8]

Box reports the following. "The results were assessed by an accelerated life test. … The runs were expensive because they needed to be made on an actual production line and the experimenters were planning to make four runs with the standard cage and four with the modified cage. Christer asked if there were other factors they would like to test. They said there were, but that making added runs would exceed their budget. Christer showed them how they could test two additional factors "for free" – without increasing the number of runs and without reducing the accuracy of their estimate of the cage effect. In this arrangement, called a 2×2×2 factorial design, each of the three factors would be run at two levels and all the eight possible combinations included. The various combinations can conveniently be shown as the vertices of a cube ... " "In each case, the standard condition is indicated by a minus sign and the modified condition by a plus sign. The factors changed were heat treatment, outer ring osculation, and cage design. The numbers show the relative lengths of lives of the bearings. If you look at [the cube plot], you can see that the choice of cage design did not make a lot of difference. … But, if you average the pairs of numbers for cage design, you get the [table below], which shows what the two other factors did. … It led to the extraordinary discovery that, in this particular application, the life of a bearing can be increased fivefold if the two factor(s) outer ring osculation and inner ring heat treatments are increased together."

"Remembering that bearings like this one have been made for decades, it is at first surprising that it could take so long to discover so important an improvement. A likely explanation is that, because most engineers have, until recently, employed only one factor at a time experimentation, interaction effects have been missed."

The simplest factorial experiment contains two levels for each of two factors. Suppose an engineer wishes to study the total power used by each of two different motors, A and B, running at each of two different speeds, 2000 or 3000 RPM. The factorial experiment would consist of four experimental units: motor A at 2000 RPM, motor B at 2000 RPM, motor A at 3000 RPM, and motor B at 3000 RPM. Each combination of a single level selected from every factor is present once.

This experiment is an example of a 2 2 (or 2×2) factorial experiment, so named because it considers two levels (the base) for each of two factors (the power or superscript), or #levels #factors , producing 2 2 =4 factorial points.

Designs can involve many independent variables. As a further example, the effects of three input variables can be evaluated in eight experimental conditions shown as the corners of a cube.

This can be conducted with or without replication, depending on its intended purpose and available resources. It will provide the effects of the three independent variables on the dependent variable and possible interactions.

Factorial experiments are described by two things: the number of factors, and the number of levels of each factor. For example, a 2×3 factorial experiment has two factors, the first at 2 levels and the second at 3 levels. Such an experiment has 2×3=6 treatment combinations or cells. Similarly, a 2×2×3 experiment has three factors, two at 2 levels and one at 3, for a total of 12 treatment combinations. If every factor has s levels (a so-called fixed-level or symmetric design), the experiment is typically denoted by s k , where k is the number of factors. Thus a 2 5 experiment has 5 factors, each at 2 levels. Experiments that are not fixed-level are said to be mixed-level or asymmetric .

There are various traditions to denote the levels of each factor. If a factor already has natural units, then those are used. For example, a shrimp aquaculture experiment [9] might have factors temperature at 25°C and 35°C, density at 80 or 160 shrimp/40 liters, and salinity at 10%, 25% and 40%. In many cases, though, the factor levels are simply categories, and the coding of levels is somewhat arbitrary. For example, the levels of an 6-level factor might simply be denoted 1, 2, ..., 6.

Treatment combinations are denoted by ordered pairs or, more generally, ordered tuples . In the aquaculture experiment, the ordered triple (25, 80, 10) represents the treatment combination having the lowest level of each factor. In a general 2×3 experiment the ordered pair (2, 1) would indicate the cell in which factor A is at level 2 and factor B at level 1. The parentheses are often dropped, as shown in the accompanying table.

To denote factor levels in 2 k experiments, three particular systems appear in the literature:

• The values 1 and 0;
• the values 1 and −1, often simply abbreviated by + and −;
• A lower-case letter with the exponent 0 or 1.

If these values represent "low" and "high" settings of a treatment, then it is natural to have 1 represent "high", whether using 0 and 1 or −1 and 1. This is illustrated in the accompanying table for a 2×2 experiment. If the factor levels are simply categories, the correspondence might be different; for example, it is natural to represent "control" and "experimental" conditions by coding "control" as 0 if using 0 and 1, and as 1 if using 1 and −1. [note 1] An example of the latter is given below . That example illustrates another use of the coding +1 and −1.

For other fixed-level ( s k ) experiments, the values 0, 1, ..., s −1 are often used to denote factor levels. These are the values of the integers modulo s when s is prime. [note 2]

The expected response to a given treatment combination is called a cell mean , [12] usually denoted using the Greek letter μ. (The term cell is borrowed from its use in tables of data .) This notation is illustrated here for the 2 × 3 experiment.

A contrast in cell means is a linear combination of cell means in which the coefficients sum to 0. Contrasts are of interest in themselves, and are the building blocks by which main effects and interactions are defined.

In the 2 × 3 experiment illustrated here, the expression

is a contrast that compares the mean responses of the treatment combinations 11 and 12. (The coefficients here are 1 and –1.) The contrast

Interaction in a factorial experiment is the lack of additivity between factors, and is also expressed by contrasts. In the 2 × 3 experiment, the contrasts

belong to the A × B interaction ; interaction is absent (additivity is present ) if these expressions equal 0. [13] [14] Additivity may be viewed as a kind of parallelism between factors, as illustrated in the Analysis section below .

Since it is the coefficients of these contrasts that carry the essential information, they are often displayed as column vectors . For the example above, such a table might look like this: [15]

The columns of such a table are called contrast vectors : their components add up to 0. Each effect is determined by both the pattern of components in its columns and the number of columns .

The patterns of components of these columns reflect the general definitions given by Bose : [16]

• A contrast vector belongs to the main effect of a particular factor if the values of its components depend only on the level of that factor.
• A contrast vector belongs to the interaction of two factors , say A and B , if (i) the values of its components depend only on the levels of A and B , and (ii) it is orthogonal (perpendicular) to the contrast vectors representing the main effects of A and B . [note 3]

Similar definitions hold for interactions of more than two factors. In the 2 × 3 example, for instance, the pattern of the A column follows the pattern of the levels of factor A , indicated by the first component of each cell.

The number of columns needed to specify each effect is the degrees of freedom for the effect, [note 4] and is an essential quantity in the analysis of variance . The formula is as follows: [18] [19]

• A main effect for a factor with s levels has s −1 degrees of freedom.
• The interaction of two factors with s 1 and s 2 levels, respectively, has ( s 1 −1)( s 2 −1) degrees of freedom.

The formula for more than two factors follows this pattern. In the 2 × 3 example above, the degrees of freedom for the two main effects and the interaction — the number of columns for each — are 1, 2 and 2, respectively.

In the tables in the following examples, the entries in the "cell" column are treatment combinations: The first component of each combination is the level of factor A , the second for factor B , and the third (in the 2 × 2 × 2 example) the level of factor C . The entries in each of the other columns sum to 0, so that each column is a contrast vector.

A 3 × 3 experiment: Here we expect 3-1 = 2 degrees of freedom each for the main effects of factors A and B , and (3-1)(3-1) = 4 degrees of freedom for the A × B interaction. This accounts for the number of columns for each effect in the accompanying table.

The two contrast vectors for A depend only on the level of factor A . This can be seen by noting that the pattern of entries in each A column is the same as the pattern of the first component of "cell". (If necessary, sorting the table on A will show this.) Thus these two vectors belong to the main effect of A . Similarly, the two contrast vectors for B depend only on the level of factor B , namely the second component of "cell", so they belong to the main effect of B .

The last four column vectors belong to the A × B interaction, as their entries depend on the values of both factors, and as all four columns are orthogonal to the columns for A and B . The latter can be verified by taking dot products .

A 2 × 2 × 2 experiment: This will have 1 degree of freedom for every main effect and interaction. For example, a two-factor interaction will have (2-1)(2-1) = 1 degree of freedom. Thus just a single column is needed to specify each of the seven effects.

The columns for A , B and C represent the corresponding main effects, as the entries in each column depend only on the level of the corresponding factor. For example, the entries in the B column follow the same pattern as the middle component of "cell", as can be seen by sorting on B .

The columns for AB , AC and BC represent the corresponding two-factor interactions. For example, (i) the entries in the BC column depend on the second and third ( B and C ) components of cell , and are independent of the first ( A ) component, as can be seen by sorting on BC ; and (ii) the BC column is orthogonal to columns B and C , as can be verified by computing dot products.

Finally, the ABC column represents the three-factor interaction: its entries depend on the levels of all three factors, and it is orthogonal to the other six contrast vectors.

Combined and read row-by-row, columns A, B, C give an alternate notation, mentioned above, for the treatment combinations (cells) in this experiment: cell 000 corresponds to +++, 001 to ++−, etc.

In columns A through ABC , the number 1 may be replaced by any constant, because the resulting columns will still be contrast vectors. For example, it is common to use the number 1/4 in 2 × 2 × 2 experiments [note 5] to define each main effect or interaction, and to declare, for example, that the contrast

is "the" main effect of factor A , a numerical quantity that can be estimated. [20]

For more than two factors, a 2 k factorial experiment can usually be recursively designed from a 2 k −1 factorial experiment by replicating the 2 k −1 experiment, assigning the first replicate to the first (or low) level of the new factor, and the second replicate to the second (or high) level. This framework can be generalized to, e.g. , designing three replicates for three level factors, etc .

A factorial experiment allows for estimation of experimental error in two ways. The experiment can be replicated , or the sparsity-of-effects principle can often be exploited. Replication is more common for small experiments and is a very reliable way of assessing experimental error. When the number of factors is large (typically more than about 5 factors, but this does vary by application), replication of the design can become operationally difficult. In these cases, it is common to only run a single replicate of the design, and to assume that factor interactions of more than a certain order (say, between three or more factors) are negligible. Under this assumption, estimates of such high order interactions are estimates of an exact zero, thus really an estimate of experimental error.

When there are many factors, many experimental runs will be necessary, even without replication. For example, experimenting with 10 factors at two levels each produces 2 10 =1024 combinations. At some point this becomes infeasible due to high cost or insufficient resources. In this case, fractional factorial designs may be used.

As with any statistical experiment, the experimental runs in a factorial experiment should be randomized to reduce the impact that bias could have on the experimental results. In practice, this can be a large operational challenge.

Factorial experiments can be used when there are more than two levels of each factor. However, the number of experimental runs required for three-level (or more) factorial designs will be considerably greater than for their two-level counterparts. Factorial designs are therefore less attractive if a researcher wishes to consider more than two levels.

A factorial experiment can be analyzed using ANOVA or regression analysis . [21] To compute the main effect of a factor "A" in a 2-level experiment, subtract the average response of all experimental runs for which A was at its low (or first) level from the average response of all experimental runs for which A was at its high (or second) level.

Other useful exploratory analysis tools for factorial experiments include main effects plots, interaction plots , Pareto plots , and a normal probability plot of the estimated effects.

When the factors are continuous, two-level factorial designs assume that the effects are linear . If a quadratic effect is expected for a factor, a more complicated experiment should be used, such as a central composite design . Optimization of factors that could have quadratic effects is the primary goal of response surface methodology .

Montgomery [4] gives the following example of analysis of a factorial experiment:.

An engineer would like to increase the filtration rate (output) of a process to produce a chemical, and to reduce the amount of formaldehyde used in the process. Previous attempts to reduce the formaldehyde have lowered the filtration rate. The current filtration rate is 75 gallons per hour. Four factors are considered: temperature (A), pressure (B), formaldehyde concentration (C), and stirring rate (D). Each of the four factors will be tested at two levels.

Onwards, the minus (−) and plus (+) signs will indicate whether the factor is run at a low or high level, respectively.

The non-parallel lines in the A:C interaction plot indicate that the effect of factor A depends on the level of factor C. A similar results holds for the A:D interaction. The graphs indicate that factor B has little effect on filtration rate. The analysis of variance (ANOVA) including all 4 factors and all possible interaction terms between them yields the coefficient estimates shown in the table below.

Because there are 16 observations and 16 coefficients (intercept, main effects, and interactions), p-values cannot be calculated for this model. The coefficient values and the graphs suggest that the important factors are A, C, and D, and the interaction terms A:C and A:D.

The coefficients for A, C, and D are all positive in the ANOVA, which would suggest running the process with all three variables set to the high value. However, the main effect of each variable is the average over the levels of the other variables. The A:C interaction plot above shows that the effect of factor A depends on the level of factor C, and vice versa. Factor A (temperature) has very little effect on filtration rate when factor C is at the + level. But Factor A has a large effect on filtration rate when factor C (formaldehyde) is at the − level. The combination of A at the + level and C at the − level gives the highest filtration rate. This observation indicates how one-factor-at-a-time analyses can miss important interactions. Only by varying both factors A and C at the same time could the engineer discover that the effect of factor A depends on the level of factor C.

The best filtration rate is seen when A and D are at the high level, and C is at the low level. This result also satisfies the objective of reducing formaldehyde (factor C). Because B does not appear to be important, it can be dropped from the model. Performing the ANOVA using factors A, C, and D, and the interaction terms A:C and A:D, gives the result shown in the following table, in which all the terms are significant (p-value < 0.05).

• Combinatorial design
• Design of experiments
• Orthogonal array
• Plackett–Burman design
• Taguchi methods
• Welch's t-test
• ↑ This choice gives the correspondence 01 ←→ +−, the opposite of that given in the table. There are also algebraic reasons for doing this. [10] The choice of coding via + and − is not important "as long as the labeling is consistent." [11]
• ↑ This choice of factor levels facilitates the use of algebra to handle certain issues of experimental design. If s is a power of a prime, the levels may be denoted by the elements of the finite field GF(s) for the same reason.
• ↑ Orthogonality is determined by computing the dot product of vectors.
• ↑ The degrees of freedom for an effect is actually the dimension of a vector space , namely the space of all contrast vectors belonging to that effect. [17]
• ↑ And 1/2 k-1 in 2 k experiments.
• ↑ Fisher, Ronald (1926). "The Arrangement of Field Experiments" (PDF) . Journal of the Ministry of Agriculture of Great Britain . 33 . London, England: Ministry of Agriculture and Fisheries: 503–513.
• ↑ "Earliest Known Uses of Some of the Words of Mathematics (F)" . jeff560.tripod.com .
• 1 2 Montgomery, Douglas C. (2013). Design and Analysis of Experiments (8th   ed.). Hoboken, New Jersey: Wiley . ISBN   978-1-119-32093-7 .
• ↑ Oehlert, Gary (2000). A First Course in Design and Analysis of Experiments (Revised   ed.). New York City: W. H. Freeman and Company . ISBN   978-0-7167-3510-6 .
• ↑ Tong, C. (2006). "Refinement strategies for stratified sampling methods". Reliability Engineering & System Safety . 91 (10–11): 1257–1265. doi : 10.1016/j.ress.2005.11.027 .
• ↑ George E.P., Box (2006). Improving Almost Anything: Ideas and Essays (Revised   ed.). Hoboken, New Jersey: Wiley . ASIN   B01FKSM9VY .
• ↑ Hellstrand, C.; Oosterhoorn, A. D.; Sherwin, D. J.; Gerson, M. (24 February 1989). "The Necessity of Modern Quality Improvement and Some Experience with its Implementation in the Manufacture of Rolling Bearings [and Discussion]". Philosophical Transactions of the Royal Society . 327 (1596): 529–537. doi : 10.1098/rsta.1989.0008 . S2CID   122252479 .
• ↑ Kuehl (2000 , pp.   200–205)
• ↑ Cheng (2019 , Remark 8.1)
• ↑ Box, Hunter & Hunter (1978 , p.   307)
• ↑ Hocking (1985 , p.   73) . Hocking and others use the term "population mean" for expected value.
• ↑ Graybill (1976 , p.   559-560)
• ↑ Beder (2022 , pp.   29–30)
• ↑ Beder (2022 , Example 5.21)
• ↑ Bose (1947 , pp.   110–111)
• ↑ Cheng (2019 , p.   77)
• ↑ Kuehl (2000 , p.   202)
• ↑ Cheng (2019 , p.   78)
• ↑ Box, Hunter & Hunter (2005 , p.   180)
• ↑ Cohen, J (1968). "Multiple regression as a general data-analytic system". Psychological Bulletin . 70 (6): 426–443. CiteSeerX   10.1.1.476.6180 . doi : 10.1037/h0026714 .

Related Research Articles

Analysis of variance ( ANOVA ) is a collection of statistical models and their associated estimation procedures used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t -test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.

In statistics, an effect size is a value measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the value of a parameter for a hypothetical population, or to the equation that operationalizes how statistics or parameters lead to the effect size value. Examples of effect sizes include the correlation between two variables, the regression coefficient in a regression, the mean difference, or the risk of a particular event happening. Effect sizes are a complement tool for statistical hypothesis testing, and play an important role in power analyses to assess the sample size required for new experiments. Effect size are fundamental in meta-analyses which aim at provide the combined effect size based on data from multiple studies. The cluster of data-analysis methods concerning effect sizes is referred to as estimation statistics.

In statistics, an interaction may arise when considering the relationship among three or more variables, and describes a situation in which the effect of one causal variable on an outcome depends on the state of a second causal variable. Although commonly thought of in terms of causal relationships, the concept of an interaction can also describe non-causal associations. Interactions are often considered in the context of regression analyses or factorial experiments.

The one-factor-at-a-time method, also known as one-variable-at-a-time , OFAT , OF@T , OFaaT , OVAT , OV@T , OVaaT , or monothetic analysis is a method of designing experiments involving the testing of factors, or causes, one at a time instead of multiple factors simultaneously.

In the statistical theory of the design of experiments, blocking is the arranging of experimental units that are similar to one another in groups (blocks) based on one or more variables. These variables are chosen carefully to minimize the impact of their variability on the observed outcomes. There are different ways that blocking can be implemented, resulting in different confounding effects. However, the different methods share the same purpose: to control variability introduced by specific factors that could influence the outcome of an experiment. The roots of blocking originated from the statistician, Ronald Fisher, following his development of ANOVA.

In statistics, a central composite design is an experimental design, useful in response surface methodology, for building a second order (quadratic) model for the response variable without needing to use a complete three-level factorial experiment.

In statistics, fractional factorial designs are experimental designs consisting of a carefully chosen subset (fraction) of the experimental runs of a full factorial design. The subset is chosen so as to exploit the sparsity-of-effects principle to expose information about the most important features of the problem studied, while using a fraction of the effort of a full factorial design in terms of experimental runs and resources. In other words, it makes use of the fact that many experiments in full factorial design are often redundant , giving little or no new information about the system.

In the design of experiments and analysis of variance, a main effect is the effect of an independent variable on a dependent variable averaged across the levels of any other independent variables. The term is frequently used in the context of factorial designs and regression models to distinguish main effects from interaction effects.

Plackett–Burman designs are experimental designs presented in 1946 by Robin L. Plackett and J. P. Burman while working in the British Ministry of Supply. Their goal was to find experimental designs for investigating the dependence of some measured quantity on a number of independent variables (factors), each taking L levels, in such a way as to minimize the variance of the estimates of these dependencies using a limited number of experiments. Interactions between the factors were considered negligible. The solution to this problem is to find an experimental design where each combination of levels for any pair of factors appears the same number of times , throughout all the experimental runs. A complete factorial design would satisfy this criterion, but the idea was to find smaller designs.

In engineering, science, and statistics, replication is the process of repeating a study or experiment under the same or similar conditions to support the original claim, which crucial to confirm the accuracy of results as well as for identifying and correcting the flaws in the original experiment. ASTM, in standard E1847, defines replication as "... the repetition of the set of all the treatment combinations to be compared in an experiment. Each of the repetitions is called a replicate ."

In statistics, Box–Behnken designs are experimental designs for response surface methodology, devised by George E. P. Box and Donald Behnken in 1960, to achieve the following goals:

In computational biology and bioinformatics, analysis of variance – simultaneous component analysis is a method that partitions variation and enables interpretation of these partitions by SCA, a method that is similar to principal components analysis (PCA). Analysis of variance ( ANOVA ) is a collection of statistical models and their associated estimation procedures used to analyze differences. Statistical coupling analysis (SCA) is a technique used in bioinformatics to measure covariation between pairs of amino acids in a protein multiple sequence alignment (MSA).

In statistics, one-way analysis of variance is a technique to compare whether two or more samples' means are significantly different. This analysis of variance technique requires a numeric response variable "Y" and a single explanatory variable "X", hence "one-way".

In statistics, a Yates analysis is an approach to analyzing data obtained from a designed experiment, where a factorial design has been used. Full- and fractional-factorial designs are common in designed experiments for engineering and scientific applications. In these designs, each factor is assigned two levels, typically called the low and high levels, and referred to as "-" and "+". For computational purposes, the factors are scaled so that the low level is assigned a value of -1 and the high level is assigned a value of +1.

In mathematics, an orthogonal array is a "table" (array) whose entries come from a fixed finite set of symbols, arranged in such a way that there is an integer t so that for every selection of t columns of the table, all ordered t -tuples of the symbols, formed by taking the entries in each row restricted to these columns, appear the same number of times. The number t is called the strength of the orthogonal array. Here are two examples:

In statistics, restricted randomization occurs in the design of experiments and in particular in the context of randomized experiments and randomized controlled trials. Restricted randomization allows intuitively poor allocations of treatments to experimental units to be avoided, while retaining the theoretical benefits of randomization. For example, in a clinical trial of a new proposed treatment of obesity compared to a control, an experimenter would want to avoid outcomes of the randomization in which the new treatment was allocated only to the heaviest patients.

In the design of experiments, completely randomized designs are for studying the effects of one primary factor without the need to take other nuisance variables into account. This article describes completely randomized designs that have one primary factor. The experiment compares the values of a response variable based on the different levels of that primary factor. For completely randomized designs, the levels of the primary factor are randomly assigned to the experimental units.

A glossary of terms used in experimental research.

Software that is used for designing factorial experiments plays an important role in scientific experiments and represents a route to the implementation of design of experiments procedures that derive from statistical and combinatorial theory. In principle, easy-to-use design of experiments (DOE) software should be available to all experimenters to foster use of DOE.

A robust parameter design , introduced by Genichi Taguchi, is an experimental design used to exploit the interaction between control and uncontrollable noise variables by robustification—finding the settings of the control factors that minimize response variation from uncontrollable factors. Control variables are variables of which the experimenter has full control. Noise variables lie on the other side of the spectrum. While these variables may be easily controlled in an experimental setting, outside of the experimental world they are very hard, if not impossible, to control. Robust parameter designs use a naming convention similar to that of FFDs. A 2 (m1+m2)-(p1-p2) is a 2-level design where m1 is the number of control factors, m2 is the number of noise factors, p1 is the level of fractionation for control factors, and p2 is the level of fractionation for noise factors.

• Beder, Jay H. (2022). Linear Models and Design . Cham, Switzerland: Springer . doi : 10.1007/978-3-031-08176-7 . ISBN   978-3-031-08175-0 . S2CID   253542415 .
• Bose , R. C. (1947). "Mathematical theory of the symmetrical factorial design". Sankhya . 8 : 107–166.
• Box, G. E. ; Hunter, W. G.; Hunter, J. S. (1978). Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building . Wiley. ISBN   978-0-471-09315-2 .
• Box, G. E. ; Hunter, W. G.; Hunter, J. S. (2005). Statistics for Experimenters: Design, Innovation, and Discovery (2nd   ed.). Wiley. ISBN   978-0-471-71813-0 .
• Cheng, Ching-Shui (2019). Theory of Factorial Design: Single- and Multi-Stratum Experiments . Boca Raton, Florida: CRC Press. ISBN   978-0-367-37898-1 .
• Dean, Angela; Voss, Daniel; Draguljić, Danel (2017). Design and Analysis of Experiments (2nd   ed.). Cham, Switzerland: Springer . ISBN   978-3-319-52250-0 .
• Graybill, Franklin A. (1976). Fundamental Concepts in the Design of Experiments (3rd   ed.). New York: Holt, Rinehart and Winston. ISBN   0-03-061706-5 .
• Hicks, Charles R. (1982). Theory and Application of the Linear Model . Pacific Grove, CA: Wadsworth & Brooks/Cole. ISBN   0-87872-108-8 .
• Hocking, Ronald R. (1985). The Analysis of Linear Models . Pacific Grove, CA: Brooks/Cole . ISBN   978-0534036188 .
• Kuehl, Robert O. (2000). Design of Experiments: Statistical Principles of Research Design and Analysis (2nd   ed.). Pacific Grove,CA: Brooks/Cole. ISBN   978-0534368340 .
• Wu, C. F. Jeff; Hamada, Michael S. (30 March 2021). Experiments: Planning, Analysis, and Optimization . John Wiley & Sons. ISBN   978-1-119-47010-6 .
• Factorial Designs (California State University, Fresno)
• GOV.UK Factorial randomised controlled trials (Public Health England)

• Index of dispersion
• Contingency table
• Frequency distribution
• Grouped data
• Partial correlation
• Pearson product-moment correlation
• Kendall's τ
• Spearman's ρ
• Scatter plot
• Control chart
• Correlogram
• Forest plot
• Q–Q plot
• Radar chart
• Stem-and-leaf display
• Violin plot

• Credible interval
• Bayes factor
• Maximum posterior estimator

Category : Definitions/Factorial Experiments

This category contains definitions related to Factorial Experiments . Related results can be found in Category:Factorial Experiments .

A factorial experiment is an experiment structured so that several different types of treatment can be compared, at different qualitative or quantitative levels.

The design will allow the experimenter to see whether the effects of each factor is additive , or whether they interact .

Results are typically analysed by partitioning the between-treatments sum of squares in an ANOVA exercise into:

Designs may involve any number of factors and any number of levels of each factor .

In a randomized block design , every factor -level combination appears once in each block .

Subcategories

This category has only the following subcategory.

• Definitions/Confounding ‎ (1 P)

Pages in category "Definitions/Factorial Experiments"

The following 8 pages are in this category, out of 8 total.

• Definition:Confounding
• Definition:Factor of Experiment
• Definition:Factorial Experiment
• Definition:Factorial Experiment/Factor
• Definition:Factorial Experiment/Interaction
• Definition:Factorial Experiment/Main Effect
• Definition:Interaction of Factors
• Definition:Main Effect of Factors
• Definitions/Statistics

Navigation menu