ORIGINAL RESEARCH Imposing Nonlinear Constraints When Estimating Genetic and Cultural Transmission Under Assortative Mating: A Simulation Study Using Mx and BUGS Ste ´phanie M. van den Berg Received: 7 April 2008 / Accepted: 29 October 2008 Ó Springer Science+Business Media, LLC 2008 Abstract Modeling both genetic and cultural transmis- sion in parent-offspring data in the presence of phenotypic assortment requires the imposition of nonlinear con- straints. This article reports a simulation study that determined how well the structural equation modeling software package Mx and the Bayesian-oriented BUGS software package can handle such nonlinear constraints under various conditions. Results generally showed good and comparable results for Mx and BUGS, although BUGS was much slower than Mx. However, since BUGS uses Markov-chain Monte Carlo estimation it could be used for parent-offspring models with non-normal data and/or item-response theory models. Keywords Bayesian  MCMC  Nonlinear constraints  Gene-environment correlation  Phenotypic assortment Introduction For many traits, a correlation exists between the pheno- typic values of partners that have offspring together (assortative mating). For example, the spouse correlation for IQ is about 0.3 (van Leeuwen et al. 2008). In addition, for quite a few phenotypes it is not unreasonable to assume that the phenotypic value in the parents has a direct influence on the phenotypes in the offspring, an influence that cannot be explained by genetic transmis- sion. For example, high IQ parents may be able to provide a healthier or more stimulating environment for their children than low IQ parents, which might have an indirect effect on the children’s IQ test performance. Such an influence is usually termed cultural transmission, as opposed to genetic transmission. Based on earlier work (Cavalli-Sforza and Feldman 1973; Cloninger et al. 1979), Fulker (1982) developed a model for genetic and cultural transmission for data on twins and their parents that accounted for assortative mating and gene–environment correlation due to the combined action of genetic and cultural transmission. The model is presented in Fig. 1. It assumes that the corre- lation between parents is the result of phenotypic assortment, that is, that assortment takes place on the basis of phenotypic values. Such phenotypic assortment induces correlations between the antecedents of the phe- notype, the genotypic and environmental values. This in turn results in increased genetic variance in the offspring due to increased homozygosity. In Fulker’s model, cul- tural transmission is modeled as effects of parental phenotypic values on the environment of the offspring (indexed by parameter z). For traits that are both heritable and show cultural transmission, a correlation is induced between genetic and environmental effects on the phenotype in the offspring. This correlation arises because the phenotype of the parent is partly genetically determined and his/her genes are also transmitted to the offspring. When it can be assumed that assortment and cultural transmission have been fairly constant over several generations, parameters will have reached equilibrium values. The total phenotypic variance will then be constant over generations so that (with stan- dardized phenotypic scores) Edited by Stacey Cherny. S. M. van den Berg (&) Department of Research Methodology, Measurement and Data Analysis (OMD), Faculty of Behavioral Sciences, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands e-mail: stephanie.vandenberg@utwente.nl 123 Behav Genet DOI 10.1007/s10519-008-9239-7