Analysis of Variance and Covariance with Repeated Measures

Analysis of Variance and Covariance with Repeated Measures 2V perfonns analysis of variance or covariance for a wide variety of fixedeffects and repeated-measures designs You can analyze models that have grouping factors, within factors, or both Grouping factors are also called between-groups or whole-plot factors Within-subjects factors are also called trial, split-plot, repeated measures, or simply within factors The grouping and the within factors must be crossed (not nested) in 2V Group sizes may be unequal for combinations of grouping factors, but each subject must have a response for every combination of within factors You can analyze both complete and incomplete fixed-effects factorial designs, including Latin square designs, incomplete block designs, and fractional factorial designs When there are within factors, 2V will perfonn an orthogonal decomposition of the within factors in addition to the overall analysis of variance The program can also perfonn contrasts over within factors Program 9D serves as a helpful adjunct to 2V; it can provide plots of group means and means of repeated measures, as well as repeated measures plots for individual subjects For a fixed-effects two-way analysis of variance with detailed data screening, see program 70 For mixed models with equal cell sizes, see program 8V; for unbalanced mixed models, see 3V For a multivariate as well as a univariate approach to repeated measures analysis of variance, see program 4V When some subjects in a repeated-measures design have data missing, see program SV SV also allows alternative structures for the withinsubjects covariance matrix See Volume for descriptions of 3V, 4V, SV, and 8V In this chapter, we present examples of a variety of designs for the analysis of variance and covariance with repeated and nonrepeated factors (For a general introduction to these subjects see Cox and Snell, 1981; Dunn and Clark, 1987; and Kirk, 1982.) One of the examples may be similar to your design If so, you may want to skip the discussion of the other examples We describe several examples in detail to explain the sums of squares and test statistics computed by 2V If your example has any grouping factors, we recommend that you read 521 2V ANOVA With Repeated Measures Example 2V.l, and if it has any within factors we recommend that you read the introduction to repeated measures preceding Example 2V.6 and 2V.7 The latter two examples both illustrate the same design with one grouping and one within factor; in 2V.7, results for an orthogonal decomposition are added Where to Find It Examples Fixed Effects Factorial Designs 522 2V.1 Analysis of variance with two grouping factors 524 2V.2 Analysis of variance with two grouping factors and two covariates 527 2V.3 Latin square design 531 2V.4 Incomplete block design 532 2V.5 Fractional factorial design 534 Repeated Measures Designs 535 2V.6 One grouping and one within factor 538 2V.7 Miniplots and breaking the within factor into orthogonal components 540 2V.8 One grouping and two within factors 542 2V.9 Contrast over a within factor 547 2V.10 One grouping and one within factor with a covariate constant across trials 549 2v.n One grouping and two within factors with a covariate changing across trials 551 Special Features • • • • • • • • • Miniplots of cell means 553 Box-Cox diagnostic plots 553 Split-plot designs 553 Covariates 554 Predicted values and residuals 555 Case weights 556 Using the FORM command to specify the design 556 Repeated measures for multiple dependent variables 557 Confidence intervals for trial cell means 557 2V Commands 558 Order of Instructions 562 Summary Table 563 Fixed Effects Factorial Designs (Grouping Variables Only) 2V performs an analysis of variance or covariance on fixed effects factorial designs with one or more grouping factors The grouping factors must be crossed and not nested You can obtain the correct sums of squares for nested factors by addition (see Dunn and Clark, 1987, for examples of this addition), but the desired F tests are not computed Analysis of variance is used to test null hypotheses about group means When there are two or more grouping factors, 2V tests null hypotheses about equality of main effects for each factor, and about interactions between factors