Relating factor models for longitudinal data to quasi-simplex and NARMA models

In this article we show the one-factor model can be rewritten as a quasi-simplex model. Using this result along with addition theorems from time series analysis, we describe a common general model, the nonstationary autoregressive moving average (NARMA) model, that includes as a special case, any latent variable model with continuous indicators and continuous latent variables. As an example, we show the NARMA representations of the linear growth curve model and the growth curve model with estimated basis vector coefficients. In certain instances rewriting competing models may help the investigator to compare different models. Here we compare the "hybrid" behavior genetics model of Eaves and Hewitt to the quasi-simplex model of Boomsma and Molenaar and show that both have equivalent NARMA representations which differ only in order.