The asymptotic distribution of a goodness of fit statistic for factorial invariance

Suppose that random factor models with k factors are assumed to hold for m, p-variate populations. A model for factorial invariance has been proposed wherein the covariance or correlation matrices can be written as [Sigma]i = LCiL' + [sigma]i2I, where Ci is the covariance matrix of factor variables and L is a common factor loading matrix, i = 1,..., m. Also a goodness of fit statistic has been proposed for this model. The asymptotic distribution of this statistic is shown to be that of a quadratic form in normal variables. An approximation to this distribution is given and thus a test for goodness of fit is derived. The problem of dimension is considered and a numerical example is given to illustrate the results.