Page 48 to the denominator. Therefore the ratio should yield approximately 1.00 in the absence of an effect and a value that is significantly greater than 1.00 if there is an effect. What follows is appropriate when both variables yield fixed effects. Accordingly, the following are the long range E(MS)s: The appropriate ratio for testing the effect of gender on ratings of importance is F = MS /MS = 1,392.4/8.4 = 165.762. With 1 and 8 df this indicates B swg that the average ratings were higher for females. Note that the df associated with factor B is low, resulting in a much less powerful test than for factor A and interaction. Likewise, the appropriate ratio for testing the effect of the explanations on rated importance is F = MS /MS = 445.4/8.817 = A A×swg 50.516. With 3 and 24 df this indicates that the various attributions had differential average effects on ratings. Finally, the ratio for testing the interaction effects of group membership and attribution is F = MS /MS = 52.067/8.817 = 5.905, which, with 3 and 24 df, is A×B A×swg significant and indicates that the effect of attribution depends upon gender. Such would be the conclusion without considering assumptions underlying the tests. The group factor was treated as a fixed-effect variable. However, organismic variables sometimes are considered as random-effect variables (Edwards, 1985). If factor B is considered as a random sample of females and males, then and the effect of the repeated measures factor would be evaluated with MS as AB error. Then F = 445.4/52.067 = 8.554, and with 3 and 3 df this factor would not be considered significant. Assumptions Underlying the Mixed Design The assumptions underlying the two-factor study with repeated measures on one factor are a blending of those of an independent- Page 47 TABLE 5.2 Summary of Analysis of Variance for Two Factors with Repeated Measures on One Factor Source SS df MS F Between-Subjects 1,459.6 9 165.762 Groups (B) 1,392.4 1 1,392.4 SwithinG 67.2 8 8.4 Within-Subjects 1,704 30 50.516 Tratment 1,336.2 3 445.4 5.905 Group × Treatment 156.2 3 52.067 Treatment × SwithinG 211.6 24 8.817 Multivariate Analyses of Interaction Effects Greenhouse-Geisser Epsilon = 0.39125 Huynh-Feldt Epsilon = 0.47549 Lower-Bound Epsilon = 0.33333 Hypothetical Significance Test Name Value Exact F df Error df of F 0.95486 42.30710 3.000 6.000 .000 Pillais 42.30710 3.000 6.000 .000 Hotellings 21.15355 0.04514 42.30710 3.000 6.000 .000 Wilks 0.95486 Roys SS . The degrees of freedom are determined in the usual way. Thus, df = K A×swg B – 1, df = J – 1, and df = (J – 1) (K – 1). The df warrants some A A×B swg consideration. The SS represents the squared differences between each swg person's mean and the group mean. Because there are n = 5 means per group jk and the sum of the deviations of the means from the group mean must equal zero, there is 1 df lost per group, or df = K(n – 1) = 2(5 – 1) = 8. Likewise, because swg jk SS represents the difference between a person's score and mean, the sum of all ws J differences must equal zero. Thus 1 df is lost from each set of J deviations, one per subject, or df = N(J – 1) = 10(4 – 1) = 30. Finally, df is the same as ws A×swg that for any interaction, or df = (J – 1)[K(n – 1)] = (4 – 1)[2(5 – 1)] = 24. A×swg jk The rationale for the F ratios is based on expected mean squares of each component of total variability of the ratings. As shown earlier, the numerator has the added effect component, but is otherwise identical Page 49 groups and a single-factor repeated measures design—that is, within-group variability is the same for each of the groups and scores are normally distributed and independent among groups. Should variances be heterogeneous, Looney and Stanley (1989) recommend a more robust test. Brown and Forsythe (1974) describe one by Welch and a modified F* whose denominator and df are adjusted. The usual repeated measures assumptions apply to the scores at each level of the group factor (B). In the present example, this means that the population variance-covariance matrices for males and for females are equal and their pooled matrix has a sphericity pattern. Huynh (1978) referred to these two assumptions as multisample sphericity. If the assumptions are tenable, the treatment and interaction F tests are valid, using pooled MS as the A×swg error term. If either assumption is violated, sphericity is untenable. Then, the test for interaction is more vulnerable to a type I error, especially with unequal ns per group (Huynh & Feldt, 1980). Available tests to evaluate the assumptions usually are not recommended. Sphericity almost always is violated and several approaches to handling such data have been investigated. The approaches vary in robustness (i.e., protection against the type I error and power rate). These include the conservative Greenhouse-Geisser correction, the adjustment, the e * adjustment, and multivariate analysis. (Huynh, in 1978, proposed an improved approximate test and found it to be slightly more robust than the e* adjustment, but recommended the latter because it requires fewer calculations.) Robustness varies with degree of nonsphericity, normality of the distribution of scores, sample size, and number of levels of the repeated measures factor relative to sample size. With slight departures from sphericity (e ³.75), an adjustment by e * of degrees of freedom for the repeated measures factor and interaction component is most robust when sample size exceeds the number of levels of the repeated measures factor (Rogan et al., 1979) and in the reverse situation with only two levels of the group factor (Maxwell and Arvey, 1982). With greater departures from sphericity (e < .75), the adjustment has been found to be more robust, especially with more than two groups and moderate sample size (Maxwell & Arvey, 1982), as has multivariate analysis (Rogan et al., 1979), provided that scores are normally distributed. For the present set of data = .39125. Adjusted df for the effect of attribution are .39125(3) = 1.17 and .39125(24) = 9.39. This effect still is significant (F 7.21). However, interaction is no longer significant. (025) » Page 46 interaction effects. These components and their members are shown with the breakdown for the score of 35 achieved by the third female: If the total deviation and each of the members are squared, and the squared deviations summed for all scores of the 10 persons, these sums lead to the empirical model that will guide the analysis: SS = SS + SS + SS + SS T B swg A×B A×swg Computation of Sums of Squares The rated importance of each explanation for each of the 10 individuals is in Table 5.1. Once again, calculations are most likely to be performed by computer. Just remember that each component of SS is based on the T estimations of each member, weighted by the number of scores contributing to that member. Calculation of Mean Squares and F Ratios At this point a summary table is very useful. Table 5.2 (top) is more elaborate than the usual summary table in order to illustrate that SS = SS + SS ; T bs ws SS = SS + SS ; and SS = SS + SS + bs B swg ws A A×B Page 50 Multivariate Analyses Evidence suggests inconsistency in the extent to which univariate tests or multivariate tests are most suitable when multisample sphericity is violated in an exploratory study. Some investigators recommend use of one of the adjustments on dfs (e.g., Huynh, 1978), others recommend multivariate analysis, especially with large Ns (e.g., McCall & Applebaum, 1973; O'Brien & Kaiser, 1985), whereas others recommend either approach, depending upon parameters of the study (e.g., Maxwell & Arvey, 1982; Rogan et al., 1979). As a reasonable compromise, Looney and Stanley (1989) recommended conducting both analyses. Accordingly, data of this study also were analyzed by MANOVA with a = .025. Although Hotelling's T2 is recommended with two groups and the Pillai-Bartlett test (Pillai, 1955) recommended for more than two groups, the SPSS-X program conducts both. The results agreed with those of the univariate analysis of main effects. Despite nonsphericity, the effect of attribution was large enough to be evident in the univariate test. Additionally, the multivariate analysis of the interaction effect of gender and attribution was significant. Results are shown in Table 5.2. Analyses of Means Differences The choice of the most appropriate procedure to follow in evaluating the effects of an independent variable depends on a number of factors, including the extent to which the sphericity assumption has been met. When sphericity is violated, even small departures from it produce large biases in the F ratios associated with contrasts between means. Compensation for the bias is a function of whether contrasts were planned or are post hoc. A priori tests should be performed with separate, rather than pooled, error terms for testing the effects of the repeated measures variable and partial interactions or simple main effects (both to be explained shortly). These would be accompanied by a reduction in degrees of freedom from K(n – 1)(J – 1) to jk K(n – 1) (Boik, 1981). jk The general procedures for post hoc testing are more involved. Tests on Main Effects Consider the situation in which a main effect is significant and the variable has more than two levels. The group