A method of testing for serial correlation in univariate repeated-measures analyses of variance

Abstract

Many behavioral experiments utilize a two-factor split-plot design having one between-subjects grouping factor and one within-subjects repeated-measures factor. Subjects are assigned at random to the various levels of the grouping factor, but all subjects are exposed to all levels of the repeated-measures factor. Very often, the repeated-measures factor represents levels of a temporal variable (e.g., laboratory sessions or experimental trials). There are two assumptions concerning the variance-covariance matrices that must be met for a valid univariate test of the trials effect or the Trials by Groups interaction effect. First, the variance-covariance matrices for a set of orthonormal variables corresponding to the effect of the repeated factor must be equal across groups. The Box M test may be used to test this assumption (Huynh & Mandeville, 1979). Second, the common covariance matrix must be spherical. An approximate test for this assumption appears in Huynh and Feldt (1970).1 When these assumptions are not met, the use of a multivariate test or reducing the degrees of freedom on repeated-measures factors (Geisser & Greenhouse, 1958) to yield a conservative but more robust univariate F test are recommended. The purpose of this discussion is to comment briefly on a particular covariance matrix structure that is not spherical and to introduce a method with which to further explore the covariance matrix structure when violations of the sphericity assumption occur. When the repeated-measures factor is time, observations made closer together tend to be more highly correlated than those made further apart. One such pattern is the serial correlation pattern, and it can be a common cause of sphericity assumption violations in behavioral research. We recently examined the effects of the serial correlation pattern on conventional alpha levels (Hearne, Clark, & Hatch, in press) and found that in extreme cases of the pattern (correlation of trial levels one step apart =.8) actual alpha levels were more than three times the nominal level. Thus, certain types of covariance matrix structure that violate the sphericity