Correlation-adjusted standard errors and confidence intervals for within-subject designs: A simple multiplicative approach
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In within-subject designs, the multiple scores of a given participant are correlated. This correlation implies that the observed variance can be partitioned into between-subject variance and between-measure variance. The basic confidence interval about the mean does not separate these two sources and is therefore of little use in within-subject designs. Two solutions have been proposed, one (Loftus and Masson) requires the computation of the interaction terms including the subject and all within-subject factors, the other (Cousineau and Morey) requires a two-step transformation of the data. As shown, these two methods are nearly equivalent. Herein, I present a correlation-adjusted method which requires the mean correlation across all pairs of measurements. This solution is shown to be similar to the other two for data satisfying the compound symmetry assumption. It is found to be too liberal for data having homogeneous correlations and heterogeneous variances but a Welch correction for heterogeneous variances can be used. Finally, it is inadequate for data that do not satisfy the compound symmetry assumption but satisfy the sphericity assumption. A statistical test of compound symmetry is discussed.