Two-Way ANOVA: SPSS
Assumptions Underlying Two-Way ANOVA
*Assumption 1*: The Dependent Variable Is Normally Distributed for Each of the Populations *Assumption 2*: The Population Variances of the Dependent Variable Are the Same for All Cells *Assumption 3*: The Cases Represent Random Samples from the Populations, and the Scores on the Dependent Variable Are Independent of Each Other
SPSS Tip
How would we conduct analyses if we had more than two factors? No problem! For example, with a three-way ANOVA, include all three factors in the Fixed Factor(s) box in the GLM-Univariate dialog box. In the Univariate: Options dialog box, include all three factors and their interactions in the Display Means for box.
Understanding Two-Way ANOVA
The initial tests conducted in a *two-way ANOVA* are the *omnibus or overall tests* of the main and interaction effects. These omnibus tests evaluate the following hypotheses: *First Main Effect*: Are the population means on the dependent variable the same among levels of the first factor averaging across levels of the second factor? *Second Main Effect*: Are the population means on the dependent variable the same among levels of the second factor averaging across levels of the first factor? *Interaction Effect*: Are the differences in the population means on the dependent variable among levels of the first factor the same across levels of the second factor? If one or more of the overall effects are significant, various follow-up tests can be conducted. The choice of which follow-up procedure to conduct depends on which effects are significant.
ANOVA: Assumption 3
The two-way ANOVA yields inaccurate p values if the independence assumption is violated.
ANOVA: Assumption 2
To the extent that this assumption is violated and the sample sizes differ for the cells, the p values from the overall two-way ANOVA are not trustworthy. In addition, the results of the follow-up tests that require equal variances should be mistrusted if the population variances differ.
Two-Way ANOVA
We can analyze data from different types of studies by using two-way ANOVA. --Experimental studies --Quasi-experimental studies --Field studies With a two-way analysis of variance (two-way ANOVA), each participant must have scores on three variables: two factors and a dependent variable. Each factor divides cases into two or more levels, while the dependent variable describes cases on a quantitative dimension. F tests are performed on the main effects for the two factors and the interaction between the two factors. Follow-up tests may be conducted to assess specific hypotheses if main effect tests, interaction tests, or both are significant.
The Research Question
Consequently, it might be useful to describe what questions the main and the interaction effects address. The questions are phrased in terms of differences among means, although they could be rewritten so that they address relationships between the factors and the dependent variable. *Method main effect*: Do the means on change in GPA differ among Method 1, Method 2, and control conditions? The means for the three method conditions are averaged across men and women students. *Gender main effect*: Do the means on change in GPA differ for men and women students? The means for men and women students are averaged across the three method conditions. *Method × Gender interaction effect*: Do the differences in the means on change in GPA among the three method conditions vary as a function of gender?
Explaining the /lmatrix Commands for Tetrad Contrasts
Customized hypotheses can be tested using the /lmatrix command. When using the /lmatrix command, we must define hypotheses by specifying coefficients for main and interaction effects (typically denoted as a, b, and ab), even though we typically conceptualize our research hypotheses in terms of means. Given this and other complexities associated with L matrices and the /lmatrix command, we offer a very restricted discussion of them. We refer the reader to Green, Marquis, Hershberger, Thompson, and McCollum (1999) for a general discussion of L matrices with GLM analyses. Fortunately it is relatively straightforward to test tetrad contrasts in a two-way ANOVA using L matrices. For these tetrad comparisons, the coefficients for the effects are the same as the coefficients for the means. Below we specify a six-step approach for defining the coefficients for the /lmatrix command. We illustrate the stepwise approach for the tetrad comparison evaluating whether the difference in population means between Method 1 and Method 2 for men minus the difference in population means between Method 1 and Method 2 for women is equal to zero.
Two-Way ANOVA Experimental Study Example
Ethel is interested in two methods of note-taking strategies and the effect of these methods on the overall GPAs of college freshmen. She believes that men would benefit most from Method 1, whereas women would benefit most from Method 2. After obtaining 30 men and 30 women volunteers in freshmen orientation, she randomly assigns 10 women and 10 men to Method 1, 10 women and 10 men to Method 2, and 10 women and 10 men to a control condition. During the first month of the spring semester, individuals in the two note-taking method groups receive daily instruction on the particular note-taking method to which they were assigned. The control group receives no note-taking instruction. Fall and spring GPAs for all participants are recorded. One factor for this study is note-taking method with three levels, and the second factor is gender with two levels. The design for this study may be described as a *3 × 2 ANOVA* (the number of levels of note-taking method by the number of levels of gender). Ethel's SPSS data file has 60 cases and three variables: a factor distinguishing among the three note-taking method groups, a second factor differentiating men from women, and a dependent variable, the students' spring semester GPA minus their fall semester GPA.
Simple Main Effect Tests Following a Significant Interaction
Simple main effect tests evaluate differences in population means among levels of one factor for each level of another factor. Conducting simple main effect tests involves two decisions. We must decide whether to examine the simple main effects for one of the factors or for both factors. Which simple main effects to explore should depend on the research hypotheses. For our example, we could explore two simple main effects: Method simple main effects involve differences in means among methods for men and differences in means among methods for women. Gender simple main effects involve differences in means between men and women for Method 1, differences in means between men and women for Method 2, and differences in means between men and women for the control group. We suspect that most researchers would be more interested in the method simple main effects. However, both could be of interest.
Two-Way ANOVA Field Study
Ted is interested in evaluating whether the gender of a client and the gender of a therapist affect the outcome of therapy. Specifically, he is interested in testing the gender-matching hypothesis that client-therapist pairs of the same gender produce the most positive outcomes. Ted has access to a large database and selects a sample of 40 client-therapist pairs so that there are an equal number of each type of pair: 10 pairs each of female client with female therapist, female client with male therapist, male client with female therapist, and male client with male therapist. All 40 clients are being treated for difficulty with coping with daily stresses. Each of the 40 therapists indicates that he or she has an eclectic approach to therapy. Finally, the length of therapy for any one pair is never fewer than five sessions. There are two factors in this study, gender of the client with two levels and gender of the therapist with two levels. The dependent variable is the amount of client improvement over the course of therapy. The design for this study may be described as a *2 × 2 ANOVA* (the number of levels of client gender by the number of levels of therapist gender). Ted's SPSS data file has 40 cases and three variables: a factor classifying the gender of the client, a second factor classifying the gender of the therapist, and a dependent variable assessing client improvement.
Effect Size Statistics for Two-Way ANOVA
The General Linear Model procedure computes an effect size index, labeled partial eta squared. It may be computed for a main or interaction source with the use of the following equation: Partial n^2 Main or Interactive Source= SS main or interaction Source/ SSmain or interaction Source + SS Error Partial η2 ranges in value from 0 to 1. A partial η2 is interpreted as the proportion of variance of the dependent variable that is related to a particular main or interaction source, excluding the other main and interaction sources. It is unclear what are small, medium, and large values for partial η2. What is a small versus a large η2 is dependent on the area of investigation. In all likelihood, the conventional cutoffs of .01, .06, and .14 for small, medium, and large η2 are too large for partial η2.
ANOVA: Assumption 1
The cells of the design (i.e., the combinations of levels of the two factors) define the different populations. For example, for a 3 × 2 ANOVA, there are six cells (3 × 2 = 6) and, consequently, the assumption requires the population distributions on the dependent variable to be normally distributed for all six cells. n many applications with a moderate or larger sample size, a two-way ANOVA may yield reasonably accurate p values even when the normality assumption is violated. In some applications with nonnormal populations, a sample size of 15 cases per group might be sufficiently large to yield fairly accurate p values. Larger sample sizes may be required to produce relatively valid p values if the population distributions are substantially nonnormal. In addition, the power of the ANOVA tests may be reduced considerably if the population distributions are nonnormal and, more specifically, thick-tailed or heavily skewed.
SPSS: Conducting Simple Main Effects Analyses
We cannot conduct simple main effects analyses by choosing options within dialog boxes. Instead, we must paste syntax created using dialog boxes and then make revisions to the estimated means statement (denoted as /EMMEANS). We present the required revisions for examining differences between men and women for each of the three methods as well as evaluating differences among the methods for each gender. Click Analyze, click General Linear Model, and then click Univariate. Conduct Steps 2 through 8 as described previously in the section "Conducting Tests of Main and Interaction Effects." Click Paste. The Syntax Editor should include the following statements: UNIANOVA gpaimpr BY gender method /METHOD = SSTYPE(3) /INTERCEPT = INCLUDE /EMMEANS = TableS(gender) /EMMEANS = TableS(method) /EMMEANS = TableS(gender*method) /PRINT = DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA = ALPHA(.05) /DESIGN = gender method gender*method. Copy the /EMMEANS = TableS(gender*method) statement so that it appears twice in the syntax: /EMMEANS = TableS(gender*method) /EMMEANS = TableS(gender*method) Revise these statements so that they appear as follows: /EMMEANS = TableS(gender*method) COMPARE(gender) ADJ(LSD) /EMMEANS = TableS(gender*method) COMPARE(method) ADJ(LSD) The first statement instructs SPSS to compare men and women for each method (i.e., gender simple main effects), while the second statement instructs SPSS to compare methods for each gender (i.e., method simple main effects). The syntax generated by conducting Steps 1 through 5 is shown in Figure 159. Highlight the syntax, click Run, and then click Selection.
Conducting Follow-up Analyses to a Significant Interaction
We may conduct two types of follow-up tests: simple main effects and interaction comparisons. We first discuss conducting tests of simple main effects.
What Error Term Should Be Used to Conduct the Simple Main Effects?
We must decide whether to use the same error term for each simple main effect analysis and assume homogeneity of error variances or to use different error terms for each simple main effect test and not assume homogeneity of error variances. If the error variances differ across simple main effects, it would be preferable to conduct the simple main effect tests using different error variances. Because of space limitations, we describe only how to conduct simple main effects assuming homogeneity of variances. We will describe two sets of steps. The first set of steps is for conducting simple main effect analyses, while second set of steps is for conducting pairwise comparisons of individual means if a simple main effect is significant.
