Mauchly Test

The Sphericity Assumption

The sphericity assumption is the assumption that the difference scores of paired levels of the repeated measures factor have equal population variance. As with the other ANOVA assumptions of normality and homogeneneity of variance, it is important to note that the sphericity assumption refers to population parameters rather than sample statistics. Also worth noting is that the sphericity assumption by definition is always met for designs with only two levels of a repeated measures factor. One need not conduct a Mauchly test on such data, and a test conducted automatically by statistical software will not output p values.

Sphericity is a more general form of compound symmetry, the condition of equal population covariance (among paired levels) and equal population variance (among levels). Whereas compound symmetry is a sufficient but not necessary precondition for conducting valid repeated measures F tests (assuming normality), sphericity is both a sufficient and necessary precondition. Historically, statisticians and social scientists often failed to recognize these distinctions between compound symmetry and sphericity, leading to frequent confusion over the definitions of both, as well as the true statistical assumptions that underlie repeated measures ANOVA. In fact, Mauchly's definition of sphericity is what is now considered compound symmetry, although the Mauchly test nevertheless assesses what is now considered sphericity.

Implementation and Computation

Like any null hypothesis significance test, the Mauchly test assesses the probability of obtaining a value for the test statistic as extreme as that observed given the null hypothesis. In this instance, the null hypothesis is that of sphericity, and the test statistic is Mauchly's W. Mathematically, the null hypothesis of sphericity (and alternative hypothesis of nonsphericity) can be written in terms of difference scores:

(for all k(k − 1)/2 unique difference scores created from k levels of repeated variable y) or in terms of matrix algebra:

where C is any (k − 1) × (k orthonormal coefficient matrix associated with the hypothesized repeated measure effect; C′ is the transpose of C; Σ is the k × k population covariance matrix; λ is a positive, scalar number; and I is the (k − 1) × {k − 1) identity matrix. Mauchly's test statistic, W, [Page 777]can be expressed concisely only in terms of matrix algebra:

where S is the k × k sample covariance matrix.

One can rely on either an approximate or exact sampling distribution to determine the probability value of an obtained W value. Because of the cumbersome computations required to determine exact p values and the precision of the chi-square approximation, even statistical software packages (e.g., SPSS, an IBM company, formerly called PASW® Statistics) typically rely on the latter. The chi-square approximation is based on the statistic – (n − 1)dW with degrees of freedom (df) = k(k − 1)/2 − 1,

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