Using SAS Software to Check Assumptions for Analysis of Covariance, Including Repeated Measures Designs

Transcription

1 Using SAS Software to Check Assumptions for Analysis of Covariance, Including Repeated Measures Designs Richard P. Steiner, The University of Akron, Akron, OH N. Margaret Wineman, The University of Akron, Akron, OH ABSTRACT Analysis of covariance (ANCOVA) is a powerful statistical tool for adjusting an analysis to acoount for the effects of concomitant variables. The technique may be applied to completely randomized designs (CRO) as well as repeated measures designs. In addition to normality assumptions, ANCOVA depends on assumptions about variances and slopes. Although not usually provided directly, most statistical software has the capability of testing the equal slopes assumption in a CRO ANCOVA; however, the slope assumptions of repeated measures ANCOVA are more complex and more difficult to check. The SYSLIN procedure in SAS/ETS software is an elegant tool to assist with checking the slope assumptions of repeated measures ANCOVA. The assumptions of ANCOVA are discussed and SAS programs for checking these assumptions are presented. A method for comparing groups when the equal slopes assumption is violated is also presented. INTRODUCTION The analysis of covariance (ANCOVA) is a powerful statistical tool widely used today. In studies where a dependent variable is compared between two or more treatment groups, ANCOVA may be used to statistically control for the effects of other variables which are related to the dependent variable, but are not of primary interest themselves. Such variables are referred to as concomitant variables or covariates. The basic notion of ANCOVA is to adjust the analysis so that comparisons of the dependent variable among treatment groups will be "fair" and not clouded by covariate differences among the groups. The statistical mechanism by which the dependent variable is adjusted for the covariates is linear regression. For simplicity the following discussion will pertain to the problem with a single covariate; however the results extend to multiple covariates. The discussion of repeated measures ANCOVA will pertain to a design with one withinsubjects (repeated) factor and one between-subjects (grouping) factor. In addition to the usual normality assumption of ANOVA, the assumptions upon which ANOOVA are based may be classified broadly as assumptions about variances and assumptions about slopes. ASSUMPTIONS ABOUT VARIANCES The first -- and for completely randomized designs (ORO), the only -- variance assumption is that the conditional variance of the dependent variable, given the covariate, is the same in each treatment group. This is analogous to the homogeneity of variance assumption encountered in ANOVA.. In repeated measures ANOOVA this first assumption is that the covariance structure among the dependent variables representing the within-subjects factors is the same in each group. Specifically, the covariance matrices for the treatment groups are assumed to be equal. This means the groups are assumed to share a common covariance matrix. This could be termed homogeneity of variances and covariances. A second variance assumption applies to repeated measures ANOOVA. This assumption, called compound symmetry, deals with the form of the common covariance matrix. It is assumed that the variances of the dependent variables are equal. It is also assumed that the covariances among the dependent variables also are equal. However, It is not required that the variances equal the covariances. This assumption is somewhat more restrictive than necessary, but it is statistically sufficient. Two procedures, the Greenhouse-Geisser (1959) and Huynh-Feldt (1976) adjustments, are available to correct the analysis n the com mon covariance matrix does not have the required form. Both adjustments produce conservative tests; the lormer is more conservative than the latter. A conservative test has less power to reject the null hypothesis when it is lalse than would normally be available at the specnied level of signilicance. ASSUMPTIONS ABOUT SLOPES For ORO ANOOVA, the dependent variable is regressed on the covariate separately in each treatment group. These slope~ are then pooled, and this pooled slope is used to adjust the analysis 01 the group effect. The assumption implied by the pooling is that the separate treatment group slopes are equal. Two types of regression models are used in repeated measures ANCOVA. The lirst type is called betweensubjects regressions by Winer (1971). In these regressions the mean dependent variable score lor each subject is regressed on the mean covariate score lor each subject. These regressions are performed separately lor each 01 the treatment groups. The slope coefficients from these regressions are then pooled, and this pooled slope is used to adjust the analysis 01 the between-subjects (group) effect. The assumption implied by the pooling is that the between-subjects slopes are equal. 1045

2 Regressions of a second type are used to adjust the analysis of the whhin-subjects effects and are called within-subjects regressions. Adjustment of the withinsubjects effects are necessary only if the covariate varies from level-to-ievel of the repeated fector. Here dependent variable scores are regressed on their corresponding covariate scores for each level the whhin-subjects factor. The slope coefficients of these regressions are pooled forming a pooled slope for the whhin-subjects effects. Again, the assumption here is that the slopes that are combined to form this pooled slope are equal. In addhion to the equal slopes assumptions stated above, each regression in the ANCOVA is subject to the usual. regression assumptions: IinearHy of the response and homogeneny of variance about the regression line. (That is, the condhional variance of the dependent variable given the covariate (X) does not depend on X.) If the regression assumptions are violated in any of the regressions the. qualny of the ANCOVA may be eroded. When adjusted cell means are reported, statistical software uses an 'overall' pooled slope, which is derived from all the pooled slopes described above. This adjustment assumes that all the regressions in the ANCOVA have equal slopes. CHECKS FOR VARIANCE ASSUMPTIONS SAS programs using DATA step programming, PROC SUMMARY, and SAS/IML" software were constructed to perform tests for equalny of variances and equality of covariance matrices. The macro % VARTEST performs Bartlett's test and Levene's test for homogeneity of variance across the groups specified on the GROUPS parameter. Levene's test is computed using absolute deviations from the group medians, as suggested by Conover et al. (1981). The macro is invoked as follows: %VARTEST (DATA = SASdatasetname, VAR = varlist, GROUPS = (varlis~ [varl var2"... varn]) The VAR parameter lists the variables to be tested. A set of tests is performed for each variable in varlist. The GROUPS parameter defines the groups over which the homogeneity of variance tests are performed. For variable names separated by spaces, or in an implied list (e.g. Xl X5), separate tests are perlormed for the groups defined by the values of each variable. When two or more variable names are connected by asterisks ('), one set of tests is performed comparing the groups defined by all combinations of the values of the connected variables. Thus, tests for homogenehy of variance acmss the cells of a factorial design can be tested readily. Residuals from the ANCOVA model can be saved and tested for homogeneity of variance across groups. Two versions of Box's (1950) test for homogenehy of covariance matrices are carried out by the macro %MVARTEST. A chi-square approximation is used when sample sizes are large (> 20) and the number of variables and the number of groups are both less than 6. Otherwise an F-approximation is used. The details of these tests are provided by TImm (1975). The O/OMVARTEST macro has the same parameters as the %VARTEST macro, except that the VAR parameter defines the variables used to form the covariance matrices. PROC GLM with the PRINTE option on the REPEATED statement furnishes a test for compound symmetry, labeled "Test for Sphericity". (See SASISTA ~ User's Guide, Vol. 2, Version 6, 4th ed., p. 954) The following SAS statements specify a repeated measures ANOVA with one grouping factor (A) and one repeated factor (B). each whh three levels: PROC GLM; CLASS A; MODEL Yl Y2 Y3 = A; REPEATED/ PRINTE; The output generated by these statements includes a section displaying the test for sphericity like the following: Test for Spl':ericity: Mau:::hl.y 1 s Criterion = Chisquare lipproxination = with 2 df Prob > Chlsquare = Ppplied to ~ Cca"PJlJelit.s: Test for Spmrici.ty: Mauchly's Criterion = Chisquare ~ion = with 2 df Prob > Chlsquare = The test "Applied to Orthogonal Components' is the pertinent test. The null hypothesis is that the spherichy condition is satisfied. In the above example, we would reject the null hypothesis at the 0.05 level of significance (P = ) and conclude that the Greenhouse-Geisser (G-G) or Huynh-Feldt (H-F) adjustments should be used. P-values for both G-G adjusted and H-F adjusted analyses are provided as part of PROC GLM output. CHECKS FOR SLOPE ASSUMPTIONS The test for equal slopes in CRD ANCOVA can be carried out in a very straightforward manner using the TEST statement in PROC REG or PROC GLM. For a design wnh one grouping factor (A) wtth three levels and one covariate (X), the following model is fn: In this model A, = 1 for each observation at level 1 of factor A; A, = 0 otherwise. Similarly, A2 = 1 for each observation at level 2 of factor A; A2 = 0 otherwise. For three groups, two such indicator variables are needed. In general, for k groups, k - 1 indicator variables are required. The variables A, X and A2X are interaction terms that can be created in a SAS DATA step using the statements AlX = Al*X; A2X = A2*X; The regression model above allows for different slopes for the covariate in each group. The differences in slope of group 1 and group 2 from the reference group (group 3) are given by ~4 and lis, respectively. The null hypothesis of equal slopes across all groups is 1046

3 The SAS statements below test this null hypothesis. PROC REG; MODEL Y = Al A2 X AlX A2X; TEST AlX=O, A2X=O; In repeated measures ANCOVA, the test for equality of the between-subjects slopes is carried out in a manner similar to that just described for a CRD ANCOVA. The difference is that the dependent variable in the regression is the mean (for each subject) of the dependent variables that define the levels of the repeated factor. Similarly, the covariate in the regression is the mean of the variables representing the values of the covariate at the different levels of the repeated factor. The following DATA step statements create the necessary variables for a design w~h a grouping factor (A) w~h 3 levels, a repeated factor (B) wtth 2 levels, and a single covariate: YEAR = ~(Yl,Y2); XBAR = ~(Xl,X2); AlXBAR = Al *XBAR; A2XBAR = A2*XBAR; A I and A2 are as defined earlier. Yl and Y2 are the dependent variables representing the levels of the repeated factor; XI is the covariate at level 1 of Band X2 is the covariate at level 2 of B. The computation of XBAR is necessary only H the values of the covariate differ across the levels of factor B. If the covariate is fixed, the values of XI and X2 will be the same for each subject, and can be included as a single variable, X. In this case X is used in place of XBAR. The SAS statements to perform the test of equal~y of the between-subjects slopes are PROC REG; MODEL YEAR = Al A2 XBAR A1XBAR A2XBAR; TEST AlXBAR=O, A2XBAR=O; Testing for equality of wtthin-subjects slopes is more difficult, since the regressions involved are not independent. The SYSLIN procedure in SASIETS software is an elegant tool to assist with checking the slope assumptions of repeated measures ANCOVA. These regressions are done as seemingly unrelated regressions using the SUR option. STEST statements are used to compare the relevant slopes. The following statements illustrate the technique: PROC SYSLIN SUR; Bl: MODEL Yl = Xl Al A2; B2: MODEL Y2 = X2 Al A2; STEST B2.X2 - Bl.Xl; Again, YI and Y2 are the dependent variables representing the levels of the wtthin-subjects factor; XI is the covariate applied to YI and X2 is the covariate applied to Y2. The above procedure tests whether the slope of the regression of the dependent variable on the covariate is the same across the levels of the repeated factor, B. The test assumes, however, that within each level of B the regressions have the same slope at each level of A. To test for equal slopes across all cells of the A x B design, the following statements may be used: PRoe SYSLIN SUR; Bl: MODEL Yl = Xl Al A2 AlXl A2Xl; B2: MODEL Y2 = X2 Al A2 A1X2 A2X2; STEST B2.X2-Bl.Xl, Bl.AlXl, Bl.A2Xl, B2.A1X2, B2.A2X2; The variables A1XI. A2XI, etc. are interaction variables as described in earlier analyses. For example. A 1 X1 = AI'XI. COMPARISON THE EQUAL VIOLATED OF GROUP MEANS WHEN SLOPES ASSUMPTION IS When the covariate slopes are equal. the group means of the dependent variable may vary wtth the level of the covariate, but the differences among the adjusted group means remain constant regardless of the level of the covariate. However, when the equal slopes assumption is violated, the implication is that differences among the group means of the dependent variable may vary depending upon the level of the covariate. Thus, separate comparisons of group means at different levels of the covariate are necessary for understanding how the groups differ. SAS software was used to implement a multiple comparisons procedure for comparing group means at different covariate levels when covariate slopes are not equal. A full model is fit, allowing different covariate slopes, and simultaneous confidence intervals for the dependent variable mean are constructed at different levels of the covariate. The levels chosen for the macro described below were the quartiles of the covariate and, optionally, the extremes. A macro, o/ollnecomp, was written to conduct this type of analysis for a CRD ANCOVA with one covariate. The macro is invoked as follows: O/OUNECOMP (DATA = SASdatasetname, DEP = depvar, IND = covariate, GRP = groupingvar, TYPE = [BONFERRONI] [WORKING-HOTELLlNG] [11, CLEV = #, EXTREMES = [YES] [NO]) The DEP, IND, and GRP parameters define the dependent variable, covariate, and grouping variable, respectively, for the analysis. The TYPE parameter allows the user to specify the type of confidence interval used. Bonferroni and Working-Hotelling confidence intervals are simultaneous; "T" requests individual Student-t confidence intervals, which are not adjusted for multiple comparisons. The first letter of each TYPE keyword is sufficient to specify it. CLEV specifies the level of confidence desired (as a percent). CLEV = 95 is the default. The EXTREMES parameter controls whether or. not comparisons are made at the extremes of the covariate. EXTREMES = YES includes the extremes, EXTREMES = NO does not. The default is YES. The program performs the following steps: I. The quartiles and extremes (if requested) are obtained using PROC UNIVARIATE and output in a data set. 2. This data set is then TRANSPOSEd and the values of the group indicator variables are propagated across 1047

4 the covariate values using SET statements. The resuhing data set has one case for each combination of group indicator values and covariate values output from Step 1. The variables in the data set are the group indicators and the covariate. 3. Next the input data set, &DATA, is reduced to complete cases, and the data set created in Step 2 is APPENDed to it, with the dependent variable (&DEP) set to missing for cases coming from the Step 2 data set. 4. Interaction variables are created and PROC REG is used to fit a model analogous to Equation (1). An OUTPUT statement is used to compute predicted values and their standard errors (STOP) for each group at the quartiles (and extremes, if specified) of the covariate. 5. Finally, the data set created in Step 4 is read into PROC IMl, where the confidence limits are computed and printed along with the group indicators, quartiles (and extremes, n specnied), and predicted values of the dependent variable. Values of the covariate for which the confidence intervals overlap for all groups indicate no group differences in the mean of the dependent variable at that covariate value. Values of the covariate for which the confidence intervals do not overlap for some groups indicate that these groups have different dependent variable means at that covariate value. In a recent study, Wineman at al. (in press) compared coping behaviors between people with a spinal cord injury (SCI) and people with multiple sclerosis (MS). As part of the analysis, an analysis of covariance was used to compare the use of emotion focused coping (EMOTION) between the two groups, adjusting for the covariate lengthdx, the length of time (in months) since diagnosis with either MS or SCI. The test for equal slopes between the groups suggested that the slopes were not equal. The %linecomp macro was used to compare the groups. The macro invocation was %linecomp(data = Q, DEP = EMOTION, IND = lengthdx, GRP = SAMPLE, 1YPE = B, CLEV = 95, EXTREMES = YES). The output is given below: The quartiles for lengthdx were 5, 10, and 17 months. The minimum was 1 month; the maximum was 67 months. At each covariate value two confidence intervals are given, one for each SAMPLE. The confidence intervals within these pairs are compared to test for differences in the mean of EMOTION between the groups. Notice that each pair of the confidence intervals overlaps, suggesting that there is no evidence to suggest that the mean of EMOTION differs between the groups, when adjusting for LENGTHDX. While the quartiles and extremes of the covariate were selected here as "representative" locations to examine differences in group means of the dependent variable, other or additional values of the covariate could be used as well. Rogosa (1980) discusses methods based on comparing the groups "in the middle part of the data". This middle covariate value is the point at which the vertical distance between the (nonparallel) covariate regressions is the same as the difference in adjusted means computed in an ordinary ANCOVA on the data. Another technique for comparing group means in the presence of nonparallel covariate regressions is known as the Johnson Neyman technique (Rogosa, 1980; Huitema, 1980). The technique is similar to the one presented in this paper, but the objective of the Johnson Neyman technique is to find a region of significance consisting of the set of all covariate values for which the confidence limits do not overlap. A difficulty with the Johnson Neyman region of signnicance is that it may be outside the observed range of the covariate. A second potential problem with simultaneous inference versions of the Johnson-Neyman approach is that confidence bands for the entire regression lines are required. Working-Hotelling bands are typically used. By selecting only a few points for comparisons, within the observed range of the covariate, and using the Bonferroni method, it may be possible to increase power while maintaining the error protection afforded by a simuitaneous procedure. Huitema (1980) presents some other alternatives to traditional analysis of covariance including nonlinear covariance analysis and rank analysis of covariance. ACKNOWLEDGMENT This research was funded in part by the National Institute for Nursing Research of the National Institutes of HeaHh (R01 NR02662). PRlNT SI'M'lE llnilllllx Ul'ER ill ill UPPER B OS REFERENCES Box, G. E. P. (1950) Problems in the analysis of growth and linear curves. Biometrics, 6: Conover, W. J., Johnson, M. E., and M. M. Johnson. (1981) A comparative study for tests of homogeneity of variances, with applications to the outer continental sheh bidding data. Technometrics, 23: Greenhouse, S. H. and S. Geisser. (1959) On methods in the analysis of profile data. Psychometrika, 24: Huitema, B. E. (1980) The Analysis of Covariance and Alternatives. New York: John Wiley & Sons. 1048

5 REFERENCES (Continued) Huynh, H. and L S. Feldt (1976) Estimation of the Box correction for degrees of freedom from sample data in randomized block and spin-plot designs. Journal of Educational Statistics, 1: Rogosa, D. (1980) Comparing nonparallel regression lines. Psychological Bunetin, 88: SAS Instnute Inc. (1990) SAS/STA T User's Guide, Version 6, Fourth Edition. Cary, NC: SAS Instnute Inc. Timm, N. H. (1975) Multivariate Analysis with Applications in Education and Psychology. Monterey, CA: Brooks/Cole Publishing Co. Wineman, N. 1.1., Durand, E. J., and R. P. Steiner (in press) A comparative analysis of coping behaviors in individuals wnh muttiple sclerosis or a spinal cord injury. Research in Nursing and Health. Winer, B. J. (1971) Statistical Principles in Experimental Design, 2nd ed. New York: McGraw-Hili Book Co. SAS, SASIETS, SASIIML, and SAS/STAT are registered trademarks or trademarks of SAS Instnute Inc. in the USA and other countries. indicates USA registration. 1049