  Applications of structural equation models (SEMs) are often restricted to
linear associations between variables. Maximum likelihood (ML) estimation in
non-linear models may be complex and require numerical integration.
Furthermore, ML inference is sensitive to distributional assumptions. In this
paper, we introduce a simple two-stage estimation technique for estimation of
non-linear associations between latent variables. Here both steps are based on
fitting linear SEMs: first a linear model is fitted to data on the latent
predictor and terms describing the non-linear effect are predicted by their
conditional means. In the second step, the predictions are included in a linear
model for the latent outcome variable. We show that this procedure is
consistent and identifies its asymptotic distribution. We also illustrate how
this framework easily allows the association between latent variables to be
modelled using restricted cubic splines and we develop a modified estimator
which is robust to non-normality of the latent predictor. In a simulation
study, we compare the proposed method to MLE and alternative two-stage
estimation techniques.
