Bootstrapping a Test Statistic for Vanishing Tetrads

Tetrads are differences in the product of pairs of covariances among four random variables. Typically, a structural equation model implies that some population tetrads will “vanish” (equal zero). Bollen and Ting proposed Confirmatory Tetrad Analysis (CTA) to test structural equation models based on their implied vanishing tetrads. The test statistic (Bollen) was derived from asymptotic theory. We analyze it in finite samples using Monte Carlo simulations. The smaller the sample size and the larger the number of vanishing tetrads, the greater the departure of the test statistic from its asymptotic distribution. We develop a bootstrapping procedure for computing the p-value of the CTA test statistic. The bootstrapping procedure generally is more accurate than using the chi-square distribution to compute the p-value of the test statistic in small to moderate sample sizes. Several empirical examples illustrate the bootstrap procedure.