Chapter 12 Summary

Chapter 12 Summary
Factorial Designs

Most research involves factorial designs, which include two or more independent variables in the same experiment in order to study their individual and their interactive effects on a dependent variable.

Factorial Designs

An interactive effect between two variables is an effect that is greater than simply summing the effects of the two variables. Rather than the effects simply adding together, the variables in interactions enhance each other.

In factorial designs, the independent variables are called factors. The notation for a factorial design shows how many independent variables there are and how many levels of each variable are included. For example, a 2 X 3 factorial design includes two independent variables, where there are two levels of the first and three levels of the second.

Theoretically, any number of factors and levels can be combined in a factorial design, but there are practical limits to the complexity. As the designs become more complex, the results become very difficult to interpret.

Main Effects and Interactions

Factorial studies are considerably more complicated than single-variable experiments. Because they are essentially several designs combined into one study, factorial experiments contain more than one hypothesis. For example, in a 2 X 2 factorial experiment there are three null hypotheses: (1) There is no difference between the levels of Factor A (no main effects for A), (2) there is no difference between the levels of Factor B (no main effects for B), and (3) there is no interaction. Further, because there are two or more independent variables to be manipulated, the potential threats to internal validity are more complex.

Possible Outcomes of Factorial Design

There are many possible outcomes for factorial studies. There may be main effects for one or more of the factors, but no interaction; there may be an interaction, but no main effects; or there may be both interactions and main effects; and finally, there may be no effects at all. Results can be displayed both by graphs and tables, as shown in Figure 12.4 in the textbook.

An Example: Children's Dark-Fears Study

Your textbook discusses a children's dark fears study; this study is a completely randomized, between-subjects, 2 X 2 factorial. We are interested in knowing if there is an A X B interaction, and whether there are main effects for Factors A and B. The appropriate statistical test for a factorial design is an analysis of variance.

When we interpret the results of an analysis of variance in which we have found both an interaction and a main effect, we always interpret the main effects in terms of the Interaction.

Analysis of Variance in Factorial Design

ANOVA calculations are usually carried out through the use of computer programs. The summary table for the children's dark-fears study is shown in Table 12.4 of the textbook. In this ANOVA, there is a significant A X B interaction (fear as measured by heart rate was highest when the darkness and fear images were presented together). There are also significant main effects for A and B. The main effects can be understood in light of the interaction. The primary conclusion from this hypothetical study is that neither feared images alone nor darkness alone appears to be sufficient stimuli for children's night fears, but the two together, darkness and fear images, is a sufficient condition for children's night fears.

Variations of Basic Factorial Design

Many factorial designs are either within-subjects factorials, in which each participant is tested under all conditions, or mixed designs, that blend different types of factors into a single study.

Within-Subjects (Repeated Measures) Factorial

The factorial designs discussed so far have all been between-subjects (randomized) designs. But with factorials, just as with single-variable designs, we may choose to employ a within-subjects design. The appropriate ANOVA for a within-subjects factorial is a repeated-measures ANOVA, which takes into account the correlated nature of the groups. Potential confounding of sequence effects must be controlled. When not precluded by serious sequence effects, the within-subjects (or repeated measures) factorial design has advantages over the between-subjects factorial. Therefore, within-subjects designs are often preferred for both single-variable and factorial experiments.

Mixed Designs

The textbook discusses two types of mixed designs: a factorial that includes a within-subjects factor and a between-subjects factor; and a factorial that includes a manipulated factor and a nonmanipulated factor.

Between-Subjects and Within-Subjects Variables. In the type of mixed design involving between-subjects and within-subjects variables, the critical issue is a statistical one. That is, the statistical formulas used in the analysis of variance will differ depending on which factors are within-subjects factors and which are between-subjects factors.
Manipulated and Nonmanipulated Variables. In the type of mixed design involving manipulated and nonmanipulated variables, the essential issue is one of interpretation of results. Manipulated factors are part of true experiments; therefore, causal inferences can safely be drawn. But this cannot be done with the nonmanipulated variables in the study. In a mixed design, interpreting main effects and interactions involving nonmanipulated factors must be done cautiously and with careful attention to possible confounding variables.
Mixed in Both Ways. It is also possible to develop a design that is mixed in both of the ways described (between- and within-subjects and manipulated and nonmanipulated factors are included in one study). Here both problems (the need to use the appropriate statistical procedure and the problems of interpretation of the nonmanipulated component) arise, and particular care is needed.

ANOVA: A Postscript

ANOVA is one of the most flexible and widely used statistical tools available for the evaluation of data. However complicated the designs become, the F-ratio is interpreted in the same way as in simpler designs. The problem is not in the computation of complex ANOVAs, but in the interpretation of the results, in which the researcher must visualize and interpret complex interactions.

Given the diversity of ANOVA procedures, it is not surprising that they have been extended into still other advanced designs such as analysis of covariance and multivariate analysis of variance.

Analysis of Covariance (ANCOVA)

In an analysis of covariance, the effects of an unimportant but powerful variable are removed statistically from the dependent measures before the groups are compared.

Multivariate Analysis of Variance (MANOVA)

In a multivariate analysis of variance, more than one dependent variable being analyzed at one time.

Ethical Principles

The dark fears study raises several ethical principles. Children are not legally able to give consent, so an adult responsible for their welfare (such as a parent) must provide assent. Nevertheless, the child must also agree (called assent to distinguish it from the legal term of consent). The fact that children would be made anxious by this study also raises critical issues of cost/benefit.