Covariance structure analysis uses chi 2 goodness-of-fit test statistics whose adequacy is not known. Scientific conclusions based on models may be distorted when researchers violate sample size, variate independence, and distributional assumptions. The behavior of 6 test statistics is evaluated with a Monte Carlo confirmatory factor analysis study. The tests performed dramatically differently under 7 distributional conditions at 6 sample sizes. Two normal-theory tests worked well under some conditions but completely broke down under other conditions. A test that permits homogeneous nonzero kurtoses performed variably. A test that permits heterogeneous marginal kurtoses performed better. A distribution-free test performed spectacularly badly in all conditions at all but the largest sample sizes. The Satorra-Bentler scaled test statistic performed best overall.

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a b s t r a c tIn this paper we propose a test for testing the equality of the mean vectors of two groups with unequal covariance matrices based on N 1 and N 2 independently distributed p-dimensional observation vectors. It will be assumed that N 1 observation vectors from the first group are normally distributed with mean vector µ 1 and covariance matrix Σ 1 .Similarly, the N 2 observation vectors from the second group are normally distributed with mean vector µ 2 and covariance matrix Σ 2 . We propose a test for testing the hypothesis that µ 1 = µ 2 . This test is invariant under the group of p × p nonsingular diagonal matrices. The asymptotic distribution is obtained as (N 1 , N 2 , p) → ∞ and N 1 /(N 1 + N 2 ) → k ∈ (0, 1) but N 1 /p and N 2 /p may go to zero or infinity. It is compared with a recently proposed noninvariant test. It is shown that the proposed test performs the best.