The general diagnostic model (GDM) utilizes located latent classes for modeling a multidimensional proficiency variable. In this paper, the GDM is extended by employing a log-linear model for multiple populations that assumes constraints on parameters across multiple groups. This constrained model is compared to log-linear models that assume separate sets of parameters to fit the distribution of latent variables in each group of a multiple-group model. Estimation of these constrained log-linear models using iterative weighted least squares (IWLS) methods is outlined and an application to NAEP data exemplifies the differences between constrained and unconstrained models in the presence of larger numbers of group-specific proficiency distributions. The use of log-linear models for the latent skill space distributions using constraints across populations allows for efficient computations in models that include many proficiency distributions.