5 Factorial Designs Research Methods in Psychology

Table Of Content

• Spacing of Factor Levels in the Unreplicated \(2^k\) Factorial Designs
• 3 - Unreplicated \(2^k\) Factorial Designs
• Selecting the Right Factors and Components in a Factorial Design: Design and Clinical Considerations
• Minitab Example for Centrifugal Contactor Analysis
• Lesson 5: Introduction to Factorial Designs
• Two-level factorial experiments

Once all desired changes have been made, click "OK" to perform the analysis. All of the plots will pop-up on the screen and a text file of the results will be generated in the session file. The names of each response can be changed by clicking on the column name and entering the desired name. In the figure, the area selected in black is where the responses will be inputted.

Spacing of Factor Levels in the Unreplicated \(2^k\) Factorial Designs

For a 2 level design, click the "2-level factorial (default generators)" radio button. Other designs such as Plackett-Burman or a General full factorial design can be chosen. For information about these designs, please refer to the "Help" menu. The following Yates algorithm table using the data for the null outcome was constructed.

3 - Unreplicated \(2^k\) Factorial Designs

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.

Selecting the Right Factors and Components in a Factorial Design: Design and Clinical Considerations

In addition, the efficiency of a factorial experiment depends in part on the extent to which higher order interactions are not found. If interactions are found, and inferential statistics must be used to unpackage such interactions, such simple effects tests would require examining the effects of ICs in only subgroups of the sample. In essence, if it is necessary to follow-up an interaction by identifying which particular subgroups differ from one another, some of the efficiency of the factorial design may be lost. However, it is important to note that interaction effects can be highly informative without simple effects tests (Baker et al., 2016; Box et al., 2005). For instance, some interactions may be due to the overall burden due to subjects receiving large numbers of components. This might result in subadditive or negative interactions in which interventions produce less benefit, or even produce net decreases in benefit, when they co-occur with another intervention(s).

The potential for statistical interaction, ie, when the effect of an intervention depends on the presence or absence of another intervention, is both a strength and a limitation. The strength is that the question of differential effects may actually be of scientific and clinical interest. What we want to do next is look at the residuals vs. variables A, B, C, D in a reduced model with just the main effects as none of the interactions seemed important.

2.2. Factorial Designs¶

Imagine, for example, that a researcher conducts an experiment on the effect of daily exercise on stress. The dependent variable, stress, is a construct that can be operationalized in different ways. For this reason, the researcher might have participants complete the paper-and-pencil Perceived Stress Scale and also measure their levels of the stress hormone cortisol. If the researcher finds that the different measures are affected by exercise in the same way, then he or she can be confident in the conclusion that exercise affects the more general construct of stress. Once the factorial effects have been computed, the natural question is whether they are large enough to be of statistical and scientific interest. Thus, if all factorial terms are included in the model, traditional regression-based inferences cannot be made because there is no estimate of residual error.

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Next, what we really want to look at is the factorial plots for these three factors, B, C and D and the interactions among these, BD and BC. All of the black dots are in fairly straight order except for perhaps the top two. If we look at these closer we can see that these are the BD and the BC terms, in addition to B, C, and D as our most important terms. Let's go back to Minitab and take out of our model the higher order interactions, (i.e. the 3-way and 4-way interactions), and produce this plot again (see below) just to see what we learn. The analysis of variance shows the individual effects and the coefficients, (which are half of the effects), along with the corresponding t-tests. Now we can see from these results that the A effect and C effect are highly significant.

Two-level factorial experiments

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). 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. 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. Time of day (day vs. night) is represented by different locations on the x-axis, and cell phone use (no vs. yes) is represented by different-colored bars.

Then these 15 linear combinations or contrasts are also normally distributed with some variance. If we assume that none of these effects are significant, the null hypothesis for all of the terms in the model, then we simply have 15 normal random variables, and we will do a normal random variable plot for these. We get a normal probability plot, not of the residuals, not of the original observations but of the effects. We have plotted these effects against what we would expect if they were normally distributed. Let's look at another example in order to reinforce your understanding of the notation for these types of designs. Below is a figure of the factors and levels as well as the table representing this experimental space.

Next, what we did at the end of the process is drop that factor entirely. If a particular factor in the screening experiment turns out to be not important either as a main effect or as part of any interaction we can remove it. This is the second strategy, and for instance in this example we took out factor B completely from the analysis. Even with just one observation per cell, by carefully looking at the results we can come to some understanding as to which factors are important.

Manipulation checks are usually done at the end of the procedure to be sure that the effect of the manipulation lasted throughout the entire procedure and to avoid calling unnecessary attention to the manipulation. Manipulation checks become especially important when the manipulation of the independent variable turns out to have no effect on the dependent variable. Imagine, for example, that you exposed participants to happy or sad movie music—intending to put them in happy or sad moods—but you found that this had no effect on the number of happy or sad childhood events they recalled. This could be because being in a happy or sad mood has no effect on memories for childhood events.

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.

The Big Five personality factors have been identified through factor analyses of people’s scores on a large number of more specific traits. For example, measures of warmth, gregariousness, activity level, and positive emotions tend to be highly correlated with each other and are interpreted as representing the construct of extraversion. As a final example, researchers Peter Rentfrow and Samuel Gosling asked more than 1,700 university students to rate how much they liked 14 different popular genres of music [RG03].

Laboratory experiments are conducted under controlled conditions, which allows for greater precision and accuracy. However, because laboratory conditions are not always representative of real-world conditions, the results of these experiments may not be generalizable to the population at large. In this design, the researcher manipulates one or more variables at different levels and uses a randomized block design to control for other variables.

Well, when we fit a full model it only has one observation per cell and there's no pure way to test for residuals. But when we fit a reduced model, now there is inherent replication and this pattern becomes apparent. We will set this up the same way in Minitab and this time Minitab will show the plot in three dimensions, two variables at a time. These can be very helpful to understand and present the relationship between several factors on the response.

Investigators may wish to adjust ICs to enhance their compatibility with other components. For instance, investigators might choose to reduce the burden of an IC by cutting sessions or contact times. This might reduce the meaning of the factor because it might make the IC unnecessarily ineffective or unrepresentative. The factorial design of experiment is described with examples in Video 1. 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).