Some remarks on estimating a covariance structure model from a sample correlation matrix

A popular model in structural equation modeling involves a multivariate normal density with a structured covariance matrix that has been categorized according to a set of thresholds. In this setup one may estimate the covariance structure parameters from the sample tetrachoricl polychoric correlations but only if the covariance structure is scale invariant. Doing so when the covariance structure is not scale invariant results in estimating a more restricted covariance structure than the one intended. When the covariance structure is not scale invariant, then the model parameters must be estimated jointly from the sample thresholds and tetrachoricl polychoric correlations. In general, when fitting a covariance structure from a sample correlation matrix one should consider the population correlation structure under the null hypothesis. This is obtained by pre and post-multiplying the covariance structure by a diagonal matrix consisting of the inverse of the square root of the diagonal of the covariance structure under consideration. We provide computer algebra code for assessing whether a covariance structure is scale invariant and for assessing the identification of threshold and correlation structures.