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414 PART V: Analyzing and Reporting ResearchEffect Sizes for Two-Factor Designwith Independent GroupsA common measure of effect size for a complex design using ANOVA is etasquared ( 2 ), or proportion of variance accounted for, which was discussed earlierin the context of single-factor designs. In calculating eta squared, it is recommendedthat we focus only on the effect of interest (see Rosenthal & Rosnow,1991). Specifically, eta squared can be defined as 2 SS effect of interest___________________SS effect of interest + SS within(see Rosenthal & Rosnow, 1991, p. 352)Thus, eta squared may be obtained for each of the three effects in an A B design.As noted above (and see Rosenthal & Rosnow, 1991), when the sums of squaresfor the effects are not available, eta squared can be computed using the F ratio(and df ) for each effect of interest.ROLE OF CONFIDENCE INTERVALS IN THE ANALYSISOF COMPLEX DESIGNSThe analysis of a complex design can be aided by the construction of confidenceintervals for the means of interest. For example, each mean in a 2 3 designcan be bracketed with a confidence interval following the procedures outlinedin Chapter 11 and earlier in this chapter. Recall that the formula isUpper limit of 95% confidence interval: __X t .05 s __X Lower limit of 95% confidence interval: __X t .05 s __X When sample sizes are equal, the estimated standard error is defined ass __X _____ s pooledwhere n sample size for each group nBecause the square root of the MS error from the ANOVA Summary Table isequiv alent to s pooled , we can define the 95% confidence interval as95% CI _________ __X t .05 (MS error / n ) where t .05 is defined by the degrees of freedom associated with the MS error .Figure 12.3 shows the confidence intervals around the six means in thehypothetical experiment we introduced above. An examination of the CIs tellsus about the precision of our estimates. We want to examine the interval widthand the probable pattern of population means by looking to see if the intervalsaround the sample means overlap and, if so, to what degree they overlap. Recallthat a rule of thumb for interpreting confidence intervals suggests that ifthe intervals around means do not overlap, then the two means would likelybe statistically significant if tested using NHST (see Box 11.5 in Chapter 11).TWO-FACTOR ANALYSIS OF VARIANCE FOR A MIXED DESIGNThe two-factor analysis of variance for a mixed design is appropriate when oneindependent variable represents either the random groups or natural groupsdesign and the second independent variable represents the repeated measures