Understanding and modeling genetic or nongenetic factors that influence susceptibility to complex traits has been the focus of many genetic studies. Large pedigrees with known complex structure may be advantageous in epidemiological studies since they can significantly increase the number of factors whose influence on the trait can be estimated. We propose a likelihood approach, developed in the context of generalized linear mixed models, for modeling dichotomous traits based on data from hundreds of individuals all of whom are potentially correlated through either a known pedigree or an estimated covariance matrix. Our approach is based on a hierarchical model where we first assess the probability of each individual having the trait and then formulate a likelihood assuming conditional independence of individuals. The advantage of our formulation is that it easily incorporates information from pertinent covariates as fixed effects and at the same time takes into account the correlation between individuals that share genetic background or other random effects. The high dimensionality of the integration involved in the likelihood prohibits exact computations. Instead, an automated Monte Carlo expectation maximization algorithm is employed for obtaining the maximum likelihood estimates of the model parameters. Through a simulation study we demonstrate that our method can provide reliable estimates of the model parameters when the sample size is close to 500. Implementation of our method to data from a pedigree of 491 Hutterites evaluated for Type 2 diabetes (T2D) reveal evidence of a strong genetic component to T2D risk, particularly for younger and leaner cases.