Since the seminal paper of Kenny and Judd (1984) several methods have been proposed for dealing with non-linear latent variable models. In all these methods more information from the data than just means and covariances is required. In this paper we also use more than just first- and second-order moments; however, we restrict ourselves to a selection of third-order moments. The key issue in this paper is a procedure for selection of the third-order moments for estimating the parameters and testing the goodness-of-fit of a model. The procedure we propose is based on the power of the test associated to the different choices of third-order moments, where the power is defined as the probability that a model without nonlinear factors is rejected by the goodness-of-fit model test when there are in fact nonlinear factors. The main conclusion of this paper is that evaluation of power for selection of third-order moments can easily be done by multivariate analysis of third- order moments, a moment test, without reference to a structural equation model. A consequence of this result is that in practice the selection of third-order moments is conceptually and computationally simple. Examples will illustrate our method.