ON THE ROBUSTNESS OF TOTAL INDIRECT EFFECTS ESTIMATED IN THE JORESKOG-KEESLING-WILEY COVARIANCE STRUCTURE MODEL.

Abstract

In structural equation models, researchers often examine two types of causal effects: direct and indirect effects. Direct effects involve variables that "directly" influence other variables, whereas indirect effects are transmitted via intervening variables. While researchers have paid considerable attention to the distribution of sample direct effects, the distribution of sample indirect effects has only recently been considered. Using the (delta) method (Rao, 1973), Sobel (1982) derived the asymptotic distribution for estimators of indirect effects in recursive systems. Sobel (1986) then derived the asymptotic distribution for estimators of total indirect effects in the Joreskog covariance structure model (Joreskog, 1977). This study examined the applicability of the large sample theory described by Sobel (1986) in small samples. Monte Carlo methods were used to evaluate the behavior of estimated total indirect effects in sample sizes of 50, 100, 200, 400, and 800. Two models were used in the analysis. Model 1 was a nonrecursive model with latent variables, feedback, and functional constraints among the effects (Duncan, Haller, & Portes, 1968; Sobel, 1986). Model 2 was a recursive model with observable variables (Duncan, Featherman, & Duncan, 1972). In addition, variations in these models were studied by randomly increasing and decreasing model parameters. The principal findings of the study suggest certain guidelines for researchers who use Sobel's procedures to evaluate total indirect effects in structural equation models. In order for the behavior of the estimates to approximate the asymptotic properties, sample sizes of 400 or more are indicated for nonrecursive systems similar to Model 1, and for recursive systems such as Model 2, sample sizes of 200 or more are suggested. At these sample sizes, researchers can expect sample indirect effects to be accurate point estimators, and confidence intervals for the effects to behave as theory predicts. A caveat to the above guidelines is that, when the total indirect effects are "small" in magnitude, relative to the scale of the model, convergence to the asymptotic properties appears to be very slow. Under these conditions, sampling distributions for the "smaller" valued estimates were positively skewed. This caused estimates to be significantly different from true values, and confidence intervals to behave contrary to theoretical expectations.