Use of covariates in randomized controlled trials.

A recent discussion in this journal illustrates some recurrent misunderstandings about the role of covariates in randomized controlled trials (RCTs). This letter aims at clarifying this role and at pointing out a pitfall in SPSS repeated measures ANOVA. We hope that our commentary will contribute to a further improvement in the use of advanced statistics in neuropsychology. Anstey et al. (2006) reported the effects of cataract surgery on neuropsychological test performance in an RCT. They used repeated measures ANOVA to test for group by time interaction, adjusting for two covariates: age and baseline visual acuity. Gilmore (2007) disputed the correctness of their analyses, pointing out a significant group difference in baseline visual acuity and suggesting that a betweensubject covariate cannot adjust within-subject effects or interactions involving within-subject effects. Anstey et al. (2007) replied that their model included a covariate by time interaction, thereby adjusting the group by time interaction of interest for baseline visual acuity. While the analysis by Anstey et al. (2007) is by and large correct, all three publications mentioned above show some statistical misconceptions that we briefly discuss in this commentary. First of all, in an RCT baseline group differences are caused by chance fluctuation and will not be significant, except because of type I errors or if dropouts are excluded from the test on baseline differences. The only significant baseline difference in Anstey et al. (2006), that in visual acuity, had a p-value of .03, but this was one out of eight tests for baseline differences (see their Table 1). So the risk of a type I error caused by multiple testing was considerable. This baseline difference would not be significant after Bonferroni correction, that is, using a 5 0.0508 (the alternative explanation of selective dropout is discussed at the end of this commentary). Relatedly, if treatment assignment is random and there is no selective dropout, then covariates are not needed to adjust for baseline differences; instead, they serve to increase the power of the treatment effect test by reducing unexplained outcome variance. So age and baseline visual acuity did not have to be included into the analysis by Anstey et al. (2007), but their inclusion may have increased the test power. So let us first look at model (1) of Anstey et al. (2007), which does not contain covariates. This is a repeated measures ANOVA with group, time, and group by time effects. Their test of group by time interaction is equivalent to ANOVA of the group difference with respect to change from baseline (posttest minus pretest). The alternative method is ANCOVA of the group difference at posttest, with the pretest as covariate. The difference between both methods can be clarified with the following ANCOVA equation: