• • 0
• 14 An experimental study in which two or more categorical variables are simultaneously manipulated or observed in order to study their joint influence (interaction effect) and separate influences (main effects) on a separate dependent variable.

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.

1. The advantages and disadvantages of these two approaches are the same as those discussed in Chapter 5.
2. The between-subjects design is conceptually simpler, avoids order/carryover effects, and minimizes the time and effort of each participant.
3. 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).

1. But he or she 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).
2. 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.

#### What is meant by factorial design?

Definition – Factorial design is a type of research methodology that allows for the investigation of the main and interaction effects between two or more independent variables and on one or more outcome variable(s).

Overview – 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, 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 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. Figure 9.1 Factorial Design Table Representing a 2 × 2 Factorial Design 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. Figure 9.2 Factorial Design Table Representing a 2 × 2 × 2 Factorial Design 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.

1. In a factorial experiment, the decision to take the between-subjects or within-subjects approach must be made separately for each independent variable.
2. In a between-subjects factorial design, all of the independent variables are manipulated between subjects.
3. 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, 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.

Factorial Design The way in which a scientific experiment is set up is called a design. A Factorial Design is an experimental setup that consists of multiple factors and their separate and conjoined influence on the subject of interest in the experiment.

1. A factor is an in the experiment and a level is a subdivision of a factor.
2. Factors and levels are different conditions that the experimental subjects are exposed to.
3. A study or experiment is used to see if any of the conditions influence the subject and in what ways they are influential.
4. The benefit of a factorial design is that it allows the researchers to look at multiple levels at a time and how they influence the subjects in the study.

An example would be a researcher who wants to look at how recess length and amount of time being instructed outdoors influenced the grades of third graders. One factor would be recess length with two levels (long recess and short recess). The other factor in this study is outdoor instruction time with two levels (outdoor instruction and indoor only instruction).

This would be called a 2×2 factorial design because there are two factors that each have two levels which create four groups (long recess with outside instruction, short recess with outside instruction, long recess with inside only instruction, and short recess with inside only instruction). Four classes of third grader would be each placed in one of these four groups and receive different factors and levels.

Statistical tests then can be used to determine the differing effects that these factors and levels have on the students grades. : Factorial Design

## What is a real life example of a factorial design?

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 decide to look at how the amount of sleep and the amount of caffeine influence test performance. The researchers then decide 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. References Breckler, S.J., Olson, J.M., & Wiggins, E.C. (2006). Social Psychology Alive. Belmont, CA: Cengage Learning. Davis, S.F., & Buskist, W. (2008).21st Century Psychology: A Reference Handbook. Thousand Oaks, CA: SAGE Publications. Van der Merwe, L., & Viljoen, C.S.

(2000). Applied Elementary Statistics. Pearson. : What Is a Factorial Design? (Definition and Examples)

#### What is a 2 * 2 * 2 factorial design?

How to Analyze a 2×2 Factorial Design – Plotting the means is a visualize way to inspect the effects that the independent variables have on the dependent variable. However, we can also perform a two-way ANOVA to formally test whether or not the independent variables have a statistically significant relationship with the dependent variable. For example, the following code shows how to perform a two-way ANOVA for our hypothetical plant scenario in R: #make this example reproducible set. seed (0) df <- data. frame (sunlight = rep(c(' Low ', ' High '), each = 30 ), water = rep(c(' Daily ', ' Weekly '), each = 15, times = 2 ), growth = c(rnorm(15, 6, 2), rnorm(15, 7, 3), rnorm(15, 7, 2), rnorm(15, 10, 3))) #fit the two-way ANOVA model model <- aov(growth ~ sunlight * water, data = df) #view the model output summary(model) Df Sum Sq Mean Sq F value Pr(>F) sunlight 1 52.5 52.48 8.440 0.00525 ** water 1 31.6 31.59 5.081 0.02813 * sunlight:water 1 12.8 12.85 2.066 0.15620 Residuals 56 348.2 6.22 – Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1 Here’s how to interpret the output of the ANOVA:

The p-value associated with sunlight is,005, Since this is less than,05, this means sunlight exposure has a statistically significant effect on plant growth. The p-value associated with water is,028, Since this is less than,05, this means watering frequency also has a statistically significant effect on plant growth. The p-value for the interaction between sunlight and water is,156, Since this is not less than,05, this means there is no interaction effect between sunlight and water.

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## Is factorial design same as ANOVA?

Asked 7 years, 3 months ago Viewed 9k times \$\begingroup\$ I was wondering about the difference between ANOVA and factorial design ? I have applied the factorial design method for studying some models(in fact in is a model with three factors). I used Minitab,

In fact it was an application on design of experiments. Usually during my research ( which I am in the beginning ) I read in several places ( even on this website), asking about ANOVA? For me, till now, the idea is fully unclear. What is ANOVA ? and what is Factorial design ? how they differ ? When we apply ANOVA, and when we apply Factorial design ? What I know about ANOVA, Analysis of variance, is that it is may be What I read about the ANOVA decomposition of a function.

Any clarification is highly appreciated. Thanks. asked Jan 21, 2016 at 19:07 Nizar Nizar 787 1 gold badge 8 silver badges 21 bronze badges \$\endgroup\$ \$\begingroup\$ A factorial design is a type of experimental design, i.e. a plan how you create your data. An ANOVA is a type of statistical analysis that tests for the influence of variables or their interactions. \$\endgroup\$ 3 \$\begingroup\$ As far as I understand with DOE you design your experiments to get your data. In many cases the experiments are expensive and/or time consuming so you want to do only the minimum. DOE helps in that. With ANOVA you use the data you get from your experiments and figure out the mathematical formula for your dependent variables and factors. \$\endgroup\$

### What are two common reasons to use a factorial design?

Factorial designs permit the researcher to determine the effect of more than one independent variable on a dependent variable and to determine the possible interaction of multiple independent variables.

### What is the difference between simple and factorial design?

Main Effects and Interactions – In factorial designs, there are two kinds of results that are of interest: main effects and interaction effects (which are also just called “interactions”). A is the statistical relationship between one independent variable and a dependent variable—averaging across the levels of the other independent variable.

Thus there is one main effect to consider for each independent variable in the study. The top panel of Figure 8.3 shows a main effect of cell phone use because driving performance was better, on average, when participants were not using cell phones than when they were. The blue bars are, on average, higher than the red bars.

It also shows a main effect of time of day because driving performance was better during the day than during the night—both when participants were using cell phones and when they were not. Main effects are independent of each other in the sense that whether or not there is a main effect of one independent variable says nothing about whether or not there is a main effect of the other.

The bottom panel of Figure 8.3, for example, shows a clear main effect of psychotherapy length. The longer the psychotherapy, the better it worked. There is an effect (or just “interaction”) when the effect of one independent variable depends on the level of another. Although this might seem complicated, you already have an intuitive understanding of interactions.

It probably would not surprise you, for example, to hear that the effect of receiving psychotherapy is stronger among people who are highly motivated to change than among people who are not motivated to change. This is an interaction because the effect of one independent variable (whether or not one receives psychotherapy) depends on the level of another (motivation to change).

1. Schnall and her colleagues also demonstrated an interaction because the effect of whether the room was clean or messy on participants’ moral judgments depended on whether the participants were low or high in private body consciousness.
2. If they were high in private body consciousness, then those in the messy room made harsher judgments.

If they were low in private body consciousness, then whether the room was clean or messy did not matter. The effect of one independent variable can depend on the level of the other in several different ways. This is shown in Figure 8.4, In the top panel, independent variable “B” has an effect at level 1 of independent variable “A” but no effect at level 2 of independent variable “A.” (This is much like the study of Schnall and her colleagues where there was an effect of disgust for those high in private body consciousness but not for those low in private body consciousness.) In the middle panel, independent variable “B” has a stronger effect at level 1 of independent variable “A” than at level 2.

1. This is like the hypothetical driving example where there was a stronger effect of using a cell phone at night than during the day.
2. In the bottom panel, independent variable “B” again has an effect at both levels of independent variable “A,” but the effects are in opposite directions.
3. Figure 8.4 shows the strongest form of this kind of interaction, called a crossover interaction.

One example of a crossover interaction comes from a study by Kathy Gilliland on the effect of caffeine on the verbal test scores of introverts and extraverts (Gilliland, 1980). Introverts perform better than extraverts when they have not ingested any caffeine.

But extraverts perform better than introverts when they have ingested 4 mg of caffeine per kilogram of body weight. Figure 8.4 Bar Graphs Showing Three Types of Interactions. In the top panel, one independent variable has an effect at one level of the second independent variable but not at the other. In the middle panel, one independent variable has a stronger effect at one level of the second independent variable than at the other.

In the bottom panel, one independent variable has the opposite effect at one level of the second independent variable than at the other. Figure 8.5 shows examples of these same kinds of interactions when one of the independent variables is quantitative and the results are plotted in a line graph.

1. Note that in a crossover interaction, the two lines literally “cross over” each other.
2. Figure 8.5 Line Graphs Showing Three Types of Interactions.
3. In the top panel, one independent variable has an effect at one level of the second independent variable but not at the other.
4. In the middle panel, one independent variable has a stronger effect at one level of the second independent variable than at the other.

In the bottom panel, one independent variable has the opposite effect at one level of the second independent variable than at the other. In many studies, the primary research question is about an interaction. The study by Brown and her colleagues was inspired by the idea that people with hypochondriasis are especially attentive to any negative health-related information.

• Researchers often include multiple independent variables in their experiments. The most common approach is the factorial design, in which each level of one independent variable is combined with each level of the others to create all possible conditions.
• In a factorial design, the main effect of an independent variable is its overall effect averaged across all other independent variables. There is one main effect for each independent variable.
• There is an interaction between two independent variables when the effect of one depends on the level of the other. Some of the most interesting research questions and results in psychology are specifically about interactions.
1. Practice: Return to the five article titles presented at the beginning of this section. For each one, identify the independent variables and the dependent variable.
2. Practice: Create a factorial design table for an experiment on the effects of room temperature and noise level on performance on the MCAT. Be sure to indicate whether each independent variable will be manipulated between-subjects or within-subjects and explain why.
3. Practice: Sketch 8 different bar graphs to depict each of the following possible results in a 2 x 2 factorial experiment:
• No main effect of A; no main effect of B; no interaction
• Main effect of A; no main effect of B; no interaction
• No main effect of A; main effect of B; no interaction
• Main effect of A; main effect of B; no interaction
• Main effect of A; main effect of B; interaction
• Main effect of A; no main effect of B; interaction
• No main effect of A; main effect of B; interaction
• No main effect of A; no main effect of B; interaction

## Why do researchers use factorial research?

Abstract – Background: Quantitative research designs are broadly classified as being either experimental or quasi-experimental. Factorial designs are a form of experimental design and enable researchers to examine the main effects of two or more independent variables simultaneously.

They also enable researchers to detect interactions among variables. Aim: To present the features of factorial designs. Discussion: This article provides an overview of the factorial design in terms of its applications, design features and statistical analysis, as well as its advantages and disadvantages.

Conclusion: Factorial designs are highly efficient for simultaneously evaluating multiple interventions and present the opportunity to detect interactions amongst interventions. Such advantages have led researchers to advocate for the greater use of factorial designs in research when participants are scarce and difficult to recruit.

• Implications for practice: A factorial design is a cost-effective way to determine the effects of combinations of interventions in clinical research, but it poses challenges that need to be addressed in determining appropriate sample size and statistical analysis.
• Eywords: clinical trials; quantitative research; randomised controlled trials; research; research methods; study design.

©2021 RCN Publishing Company Ltd. All rights reserved. Not to be copied, transmitted or recorded in any way, in whole or part, without prior permission of the publishers.

## What are factorial designs pros and cons?

The Pros and Cons of Factorial Design – Factorial designs are extremely useful to psychologists and field scientists as a preliminary study, allowing them to judge whether there is a link between variables, whilst reducing the possibility of experimental error and confounding variables,

1. The factorial design, as well as simplifying the process and making research cheaper, allows many levels of analysis.
2. As well as highlighting the relationships between variables, it also allows the effects of manipulating a single variable to be isolated and analyzed singly.
3. The main disadvantage is the difficulty of experimenting with more than two factors, or many levels.

A factorial design has to be planned meticulously, as an error in one of the levels, or in the general operationalization, will jeopardize a great amount of work. Other than these slight detractions, a factorial design is a mainstay of many scientific disciplines, delivering great results in the field.

### What is factorial design and advantages?

The factorial designs have the advantage over the single factor design in that interaction of two or more variables can also be studied along with the main effect. In a single factor design the levels of only one factor is varied and the levels of other relevant variable are held constant.

#### What is main effect in factorial design?

Factorial Designs: Main Effects – The main effect in a factorial design is “the effect of one independent variable averaged over all levels of another independent variable” ( McBurney, 2004, p.289). Table 4 below shows hypothetical data for our 2 x 2 factorial design example.

TABLE 4. Hypothetical data from a 2 x 2 factorial design with drug dosage as Factor A and task description as Factor B; this shows a main effect for both Factor A and Factor B.

None (B 1 ) “Hard” (B 2 ) Row Mean
DRUG DOSAGE (A) 0 mg (A 1 )
 1 5 3 9

/td>

3 10 mg (A 2 ) 6 Column Mean 2 7

The number in each cell represents the mean number of errors on the memory task for subjects who experienced that condition. The row means are shown to the right of the table, and the column means are shown at the bottom of the table. There are two main effects to consider, one for Factor A and one for Factor B.

1. Regarding the main effect for Factor A, we ask: Did drug dosage affect performance on the memory task? To answer this question we compare the mean of A 1 over the two levels of Factor B (top row mean) with the mean of A 2 over the two levels of Factor B (bottom row mean).
2. We see that subjects on average made fewer errors when the pill did not contain the drug (mean = 3 errors) than when the pill did contain the drug (mean = 6 errors).

We conclude that there is a main effect for Factor A, that is, the drug enhanced errors on the memory task. Regarding the main effect for Factor B, we ask: Did leading subjects to believe that the task was difficult affect memory task performance? To answer this question we compare the mean of B 1 over the two levels of Factor A (left column mean) with the mean of B 2 over the two levels of Factor B (right column mean).

1. We see that subjects on average made fewer errors when they were told nothing about the task difficulty (mean = 2 errors) than when they were told that the task was “hard” (mean = 7 errors).
2. We conclude that there is a main effect for Factor B, that is, being told that the task was “hard” enhanced errors on the memory task.

Figure 2 below shows how these main effects for Factor A and Factor B would look on a graph. FIGURE 2. Graphs illustrating a main effect for Factor A (left panel) and Factor B (right panel). The two solid lines on the graph in the left panel show the effect of the drug (Factor A) for both levels of task description (Factor B). This graph illustrates mean number of errors as a function of drug dosage with task description as a parameter.

• Errors increase across the two levels of the drug (A 1 and A 2 ) both when subjects were told nothing about the task difficulty (line B 1 ) and when they are told that the task was “hard” (line B 2 ).
• The dashed red line shows the average increase (the two row means in Table 3) across the two levels of the drug; this is the main effect for Factor A.

The two solid lines on the graph in the right panel show the effect of the task description (Factor B) for both levels of the drug (Factor A). This graph illustrates mean number of errors as a function of task description with drug dosage as a parameter.

Errors increase across the two levels of task description (B 1 and B 2 ) when the pill does not contain the drug (line A 1 ) and when the pill does contain the drug (line A 2 ). The dashed blue line shows the average increase (the two column means in Table 2) across the two levels of task description; this is the main effect for Factor B.

What if there was no main effect for one or both of the factors? What might these data look like? Let’s consider the case in which there is a main effect for Factor A and no main effect for Factor B. Such results are shown in Table 5 and the data are plotted in Figure 3, both presented below.

TABLE 5. Hypothetical data from a 2 x 2 factorial design with drug dosage as Factor A and task description as Factor B; this shows a main effect for Factor A only.

None (B 1 ) “Hard” (B 2 ) Row Mean
DRUG DOSAGE (A) 0 mg (A 1 )
 5 1 5 9

/td>

3 10 mg (A 2 ) 7 Column Mean 5 5

FIGURE 3. Graphs illustrating a main effect for Factor A (left panel) and no main effect for Factor B (right panel). The row means in Table 5 show that subjects on average made fewer errors when the pill did not contain the drug (mean = 3 errors) than when the pill did contain the drug (mean = 7 errors).

Again, we conclude that there is a main effect for Factor A, that is, the drug enhanced errors on the memory task. The main effect for Factor A is illustrated by the non-flat, dashed red line in the left panel of Figure 3. Table 5 also reveals that subjects on average made the same number of errors (mean = 5 errors) when they were told nothing about the task difficulty and when they were told that the task was “hard.” We conclude that there is no main effect for Factor B, that is, being told that the task is “hard” does not increase errors on the memory task.

The flat, dashed blue line in the right panel of Figure 3 illustrates the absence of a main effect for Factor B.

## What is the most simple factorial design?

An experimental design in which the two or more levels of each independent variable or factor are observed in combination with the two or more levels of every other factor.

### What is a factorial design for dummies?

Factorial design involves having more than one independent variable, or factor, in a study. Factorial designs allow researchers to look at how multiple factors affect a dependent variable, both independently and together. Factorial design studies are named for the number of levels of the factors.

## What do you understand by 2×2 and 2×3 factorial design?

A factor is a discrete variable used to classify experimental units. For example, Gender might be a factor with two levels male and female and Diet might be a factor with three levels low, medium and high protein. The levels within each factor can be discrete, such as Drug A and Drug B, or they may be quantitative such as 0, 10, 20 and 30 mg/kg.

A factorial design is one involving two or more factors in a single experiment. Such designs are classified by the number of levels of each factor and the number of factors. So a 2×2 factorial will have two levels or two factors and a 2×3 factorial will have three factors each at two levels. Typically, there are many factors such as gender, genotype, diet, housing conditions, experimental protocols, social interactions and age which can influence the outcome of an experiment.

These often need to be investigated in order to determine the generality of a response. It may be important to know whether a response is only seen in, say, females but not males. One way to do this would be to do separate experiments in each sex. This OVAT or One Variable at A Time approach is, however, very wasteful of scientific resources.

A much better alternative is to include both sexes or more than one strain etc. in a single factorial experiment. Such designs can include several factors without using excessive numbers of experimental subjects. Factorial designs are efficient and provide extra information (the interactions between the factors), which can not be obtained when using single factor designs.

Split plot designs are considered at the end of this section. They are like a cross between a factorial and a randomised block design. They were derived from agricultural research in which is was sometimes impossible to irrigate, say, a small plot without affecting the adjacent plots. According to RA Fisher If the investigator confines his attention to any single factor we may infer either that he is the unfortunate victim of a doctrinaire theory as to how experimentation should proceed, or that the time, material or equipment at his disposal is too limited to allow him to give attention to more than one aspect of his problem.

Indeed in a wide class of cases (by using factorial designs) an experimental investigation, at the same time as it is made more comprehensive, may also be made more efficient if by more efficient we mean that more knowledge and a higher degree of precision are obtainable by the same number of observations.

(Fisher RA.1960. The design of experiments. New York: Hafner Publishing Company, Inc.248 p,) Unfortunately, although such designs are widely used, they are often incorrectly analysed. A survey found the following: Number of studies 513 Factorial designs 153 (30%) Correctly analysed 78 (50%) ( Niewenhuis et al., 2011, Nature Neurosci.14:1105) Examples Assuming that the animal is the experimental unit, the experiment on the right has two factors, the treatment (Control ve rsus Treated represented by the two columns) and the colour (White versus Green). This might represent the two sexes, or two strains or two diets or any other factor of possible interest. The aim is usually to see whether two factors are independent. This is a 2×2 factorial because there are two factors each at two levels.

• Using the Resource Equation method of sample size determination there are 16 animals and 4 groups, so E=16-4=12, which, though small, is satisfactory.
• Factorial designs are powerful because differences among the levels of each factor are determined by averaging across all other factors.
• This, if columns in the figure on the right represent Treated and Control the means are estimated by averaging across the two colours which might represent males and females.

This assumes that the males and females respond in the same way to the treatment, an assumption that is tested in the statistical analysis using a two-way analysis of variance with interaction, If the two sexes do not respond in the same way then this is known as an interaction and the differences will need to be looked at separately for each sex. This might be, for example, a Drug treatment with levels Control, Low high doses (columns) and Diet with three levels of a food additive represented by the three colours A 3x3x2 factorial is shown on the right. Here the three factors are Dose at three levels, Diet at three levels and Strain (stripes versus solid) at two levels. So it is a 3x3x2 factorial design. In this case there are 36 experimental units (animals) and 18 treatment groups so using the Resource Equation method of determining sample size, E=36-18 =18.

As E is between 10 and 20 it is probably an appropriate number of experimental units. Note that with factorial designs the concept of group size needs to be reconsidered. In this case each treatment and diet mean will be based on 12 subjects, averaged across the other factors. Strain means will be based on 18 animals averaged across both diets and treatments.

So although there are subgroups consisting of just two animals, the means are based on much larger numbers. A real example. In this study mice of two strains (BALB/c and C57BL) were dosed with a vehicle or with chloramphenicol at 2000mg/kg. This is a 2(strains) x 2(dose levels) factorial design. We want to know:

does treatment have an effect on RBC counts do strains differ in RBC counts do strains differ in their response to chloramphenicol (the interaction).

The treatment appears to reduce red blood cell (RBC) counts. There is no overlap between treated and control individuals. Also, C57BL seems to have lower counts than BALB/c. Whether or not there is an interaction can best be seen graphically This plot shows that BALB/c (triangles) mice have higher red blood counts than the C57BL (circles) both in the controls and in the treated group and the reduction due to the chloramphenicol was the same in both strains. So there is no interaction between strain and chloramphenicol in this case. In contrast, here are the results with two different strains (C3H and outbred CD-1). Chloramphenicol seems to reduce red blood cell counts and CD-1 seems to have higher counts than C3H. However, plotting the means (below) also shows that there is an interaction. Strain C3H (triangles) responded to chloraphenicol by a reduction in red blood cell counts, but in CD-1 (circles) there was no response. The data should be analysed by a two-way ANOVA with interaction to see whether the interaction effect is statistically significant, as shown in section 11.

Implications of strain x treatment interactions Strain by treatment interactions are almost universal. This means that results based on a single strain (or outbred stock) can not necessarily be generalised. It is often highly desirable to replicate over several strains using a factorial experimental design, particularly in toxicity testing where the aim is to prove a negative.

A good example is the response to bisphenol A (BPA) which is an endocrine disruptor in most strains and stocks of mice and rats, causing a range of developmental and other defects when administered at doses below the safe human exposure level. However in none of 13 studies were any effects observed when the CD:SD stock of rats was used. Split plot designs These are randomised block designs with a factorial treatment structure in which a main effect is confounded with blocks. They are worth knowing about because in some situations they may make efficient use of resources. Suppose the aim is to compare two or more treatments using a randomised block design.

For example, the experiment on the right has two animals in a cage, each receiving a different treatment. They could be two genotypes. Or it could be a within-animal experiment where the same animal is given two treatments sequentially or as a topical application to the skin on the left or right side.

Suppose it was decided that 12 cages would be sufficient (by the resource equation E=24-2=22, which is acceptable). Outcome measurements would be done on the animals, one cage at a time. This could be analysed as a randomised block design. But suppose half the cages had males and half females? In that case an estimate of sex differences for the outcome of interest could be obtained averaging across the two animals within a cage and a sex x treatment effect could also be estimated.

This would indicate whether the two sexes responded similarly to the treatments. Given the need to use both sexes and the need to group house rodents, this might be quite a useful design in some cases. A split-plot has two different experimental units (in this case animal (for comparing the treatments) and cage (for comparing the sexes) in one experiment.

Technically, it is a factorial design with a main effect confounded with blocks. The statistical analysis will be discussed in the statistical analysis pages.

### What is an example of a two factorial design of experiments?

When multiple factors can affect a system, allowing for interaction can increase sensitivity. When probing complex biological systems, multiple experimental factors may interact in producing effects on the response. For example, in studying the effects of two drugs that can be administered simultaneously, observing all the pairwise level combinations in a single experiment is more revealing than varying the levels of one drug at a fixed level of the other.

1. If we study the drugs independently we may miss biologically relevant insight about synergies or antisynergies and sacrifice sensitivity in detecting the drugs’ effects.
2. The simplest design that can illustrate these concepts is the 2 × 2 design, which has two factors (A and B), each with two levels ( a/A and b/B ).

Specific combinations of factors ( a/b, A/b, a/B, A/B ) are called treatments. When every combination of levels is observed, the design is said to be a complete factorial or completely crossed design. So this is a complete 2 × 2 factorial design with four treatments.

• Our previous discussion about experimental designs was limited to the study of a single factor for which the treatments are the factor levels.
• We used ANOVA 1 to determine whether a factor had an effect on the observed variable and followed up with pairwise t -tests 2 to isolate the significant effects of individual levels.

We now extend the ANOVA idea to factorial designs. Following the ANOVA analysis, pairwise t -tests can still be done, but often analysis focuses on a different set of comparisons: main effects and interactions. Figure 1 illustrates some possible outcomes in a 2 × 2 factorial experiment (values in Table 1 ).

Suppose that both factors correspond to drugs and the observed variable is liver glucose level. In Figure 1a, drugs A and B increase glucose levels by 1 unit. Because neither drug influences the effect of the other we say there is no interaction and that the effects are additive. In Figure 1b, the effect of A in the presence of B is larger than the sum of their effects when they are administered separately (3 vs.0.5 + 1).

When the effect of the levels of a factor depends on the levels of other factors, we say that there is an interaction between the factors. In this case, we need to be careful about defining the effects of each factor. Figure 1: When studying multiple factors, main and interaction effects can be observed, shown here for two factors (A, blue; B, red) with two levels each. ( a ) The main effect is the difference between τ values (light gray), which is the response for a given level of a factor averaged over the levels of other factors. ( b ) The interaction effect is the difference between effects of A at the different levels of B or vice versa (dark gray, Δ ).

1. C ) Interaction effects may mask main effects.
2. Table 1 Quantities used to determine main and interaction effects from data in Figure 1 The main effect of factor A is defined as the difference in the means of the two levels of A averaged over all the levels of B.
3. For Figure 1b, the average for level a is τ = (0 + 1)/2 = 0.5 and for level A is τ = (0.5 + 3)/2 = 1.75, giving a main effect of 1.75 − 0.5 = 1.25 ( Table 1 ).

Similarly, the main effect of B is 2 − 0.25 = 1.75. The interaction compares the differences in the mean of A at the two levels of B (2 − 0.5 = 1.5; in the Δ row) or, equivalently, the differences in the mean of B at the two levels of A (2.5 − 1 = 1.5).

Interaction plots are useful to evaluate effects when the number of factors is small (line plots, Fig 1b ). The x axis represents levels of one factor and lines correspond to levels of other factors. Parallel lines indicate no interaction. The more the lines diverge, or cross, the greater the interaction.

Figure 1c shows an interaction effect with no main effect. This can happen if one factor increases the response at one level of the other factor but decreases it at the other. Both factors have the same average value for each of their levels, τ = 0.5. However, the two factors do interact because the effect of one drug is different depending on the presence of the other.

There are various ways in which effects can combine; their clear and concise reporting is important. For a 2 × 2 design with two levels per factor, effects can be estimated directly from treatment means. In this case, effects should be summarized with their estimated value and a confidence interval (CI) and graphically reported as a plot of means with error bars 2,

Optionally, a two-sample t -test can be used to provide a P value for the null hypothesis that the two treatments have the same effect—a zero difference in their means. For example, with levels a/A and b/B we have four treatment means μ ab, μ Ab, μ aB and μ AB,

• The effect of A at level b is μ Ab − μ ab, which is estimated by substituting the observed sample means.
• The standard error of this estimate is s.e.
• = s √(1/ n Ab + 1/ n ab ), where s is the estimate of the population standard deviation, estimated by √MS E, where MS E is the residual mean square from the ANOVA, and n ij is the observed sample size for treatment A = i and B = j,

If the design is balanced, n Ab = n ab = n and s.e. = √ (2MS E / n ). The t -statistic is t = ( Ab − ab )/s.e. The CI can be constructed using Ab − ab ± t * × s.e., where t * is the critical value for the t -statistic at the desired α, Note, however, that the degrees of freedom (d.f.) are the error d.f. from the ANOVA, not 2( n − 1) as in the usual two-sample t -test 2, because the MS E rather than the sample variances is used in the s.e.

Computation. When there are more factors or more levels, the main effects and interactions are summarized over many comparisons as sums of squares (SS) and usually only the test statistic ( F -test), its d.f. and the P value are reported. If there are statistically significant interactions, pairwise comparisons of different levels of one factor for fixed levels of the other factors (sometimes called simple main effects) are often computed in the manner described above.

If the interactions are not significant, we typically compute differences between levels of one factor averaged over the levels of the other factor. Again, these are pairwise comparisons between means that are handled as just described, except that the sample sizes are also summed over the levels.

To illustrate the two-factor design analysis, we’ll use a simulated data set in which the effect of levels of the drug and diet were tested in two different designs, with 8 mice and 8 observations ( Fig.2a ). We’ll assume an experimental protocol in which a mouse liver tissue sample is tested for glucose levels using two-way ANOVA.

Our simulated simple effects are shown in Figure 1b —the increase in the response variable is 0.5 ( A/b ), 1 ( a/B ) and 3 ( A/B ). The two drugs are synergistic—A is 4× as potent in the presence of B, as can be seen by ( μ AB − μ aB )/( μ Ab − μ ab ) = Δ B /Δ b = 2/0.5 = 4 ( Table 1 ). ( a ) Two common two-factor designs with 8 measurements each. In the CR scenario, each mouse is randomly assigned a single treatment. Variability among mice can be mitigated by grouping mice by similar characteristics (e.g., litter or weight). The group becomes a block.

• Each block is subject to all treatments.
• B ) Partitioning of the total sum of squares (SS T ; CR, 16.9; RCB, 26.4) and P values for the CR and RCB designs in a,
• M represents the blocking factor.
• Vertical axis is relative to the SS T,
• The total d.f.
• In both cases = 7; all other d.f. = 1.
• We’ll use a completely randomized design with each of the 8 mice randomly assigned to one of the four treatments in a balanced fashion each providing a single liver sample ( Fig.2a ).

First, let’s test the effect of the two factors separately using one-way ANOVA, averaging over the values of the other factor. If we consider only A, the effects of B are considered part of the residual error and we do not detect any effect ( P = 0.48, Fig.2b ).

If we consider only B, we can detect an effect ( P = 0.04) because B has a larger main effect (2.0 − 0.25 = 1.75) than A (1.75 − 0.5 = 1.25). When we test for multiple factors, the ANOVA calculation partitions the total sum of squares, SS T, into components that correspond to A (SS A ), B (SS B ) and the residual (SS E ) ( Fig.2b ).

The additive two-factor model assumes that there is no interaction between A and B—the effect of a given level of A does not depend on a level of B. In this case, the interaction component is assumed to be part of the error. If this assumption is relaxed, we can partition the total variance into four components, now accounting for how the response of A varies with B.

• In our example, the SS A and SS B terms remain the same, but SS E is reduced by the amount of SS AB (4.6), to 2.0 from 6.6.
• The resulting reduction in MS E (0.5 vs.1.3) corresponds to the variance explained by the interaction between the two factors.
• When interaction is accounted for, the sensitivity of detecting an effect of A and B is increased because the F -ratio, which is inversely proportional to MS E, is larger.

To calculate the effect and interaction CIs, as described above, we start with the treatment means ab = 0.27, Ab = −0.39, aB = 0.86 and AB = 3.23, each calculated from two values. To calculate the main effects of A and B, we average over four measurements to find a = 0.57, A = 1.42, b = −0.06 and B = 2.05. The residual error MS E = 0.5 is used to calculate the s.e. of main effects: √ (2MS E / n ) = √ (2 × 0.5/4) = 0.5. The critical t -value at a = 0.05 and d.f. = 4 is 2.78, giving a 95% CI for the main effect of A to be 0.9 ± 1.4 ( F 1,4 = 2.9), where d.f. AB − aB ) − ( Ab − ab ) = 3.0 with s.e. = 1 and a CI of 3.0 ± 2.8 ( F 1,4 = 9.1). To improve the sensitivity of detecting an effect of A, we can mitigate biological variability in mice by using a randomized complete block approach 1 ( Fig.2a ). If the mice share some characteristic, such as litter or weight which contributes to response variability, we could control for some of the variation by assigning one complete replicate to each batch of similar mice.

The total number of observations will still be 8, and we will track the mouse batch across measurements and use the batch as a random blocking factor 2, Now, in addition to the effect of interaction, we can further reduce the MS E by the amount of variance explained by the block ( Fig.2b ). The sum-of-squares partitioning and P values for the blocking scenario are shown in Figure 2b,

In each case, the SS E value is proportionately lower than in the completely randomized design, which makes the tests more sensitive. Once we incorporate blocking and interaction, we are able to detect both main and interaction effects and account for nearly all of the variance due to sources other than measurement error (SS E = 0.8, MS E = 0.25).

• The interpretation of P = 0.01 for the blocking factor M is that the biological variation due to the blocking factor has a nonzero variance.
• Effects and CIs are calculated just as for the completely randomized design—although the means have two sources of variance (block effect and MS E ), their difference has only one (MS E ) because the block effect cancels.

With two factors, more complicated designs are also possible. For example, we might expose the whole mouse to a drug (factor A) in vivo and then expose two liver samples to different in vitro treatments (factor B). In this case, the two liver samples from the same mouse form a block that is nested in mouse.

• We might also consider factorial designs with more levels per factor or more factors.
• If the response to our two drugs depends on genotype, we might consider using three genotypes in a 2 × 2 × 3 factorial design with 12 treatments.
• This design allows for the possibility of interactions among pairs of factors and also among all three factors.

The smallest factorial design with k factors has two levels for each factor, leading to 2 k treatments. Another set of designs, called fractional factorial designs, used frequently in manufacturing, allows for a large number of factors with a smaller number of samples by using a carefully selected subset of treatments.

### What is the example of factorial method? A factorial is a mathematical operation that you write like this: n!, It represents the multiplication of all numbers between 1 and n. So if you were to have 3!, for example, you’d compute 3 x 2 x 1 (which = 6). Let’s see how it works with some more examples.

## What is factorial explanation examples?

The Factorial of a whole number ‘n’ is defined as the product of that number with every whole number less than or equal to ‘n’ till 1. For example, the factorial of 4 is 4 × 3 × 2 × 1, which is equal to 24. It is represented using the symbol ‘!’ So, 24 is the value of 4!.

The study of factorials is at the root of several topics in mathematics, such as number theory, algebra, geometry, probability, statistics, graph theory, discrete mathematics, etc. In the year 1677, Fabian Stedman, a British author, defined factorial as an equivalent of change ringing. Change ringing was a part of the musical performance where the musicians would ring multiple tuned bells.

In the year 1808, when a mathematician from France, Christian Kramp, came up with the symbol for factorial: n!. Thinking about how to calculate the factorial of a number? Let’s learn.

 1 What is Factorial? 2 n Factorial Formula 3 0 Factorial 4 Factorial of Hundred 5 Factorial of Negative Numbers 6 Use of Factorial 7 How to Calculate Factorial? 8 FAQs on Factorial