Checking Assumptions in Latent Class Regression Models via a Markov Chain Monte Carlo Estimation Approach: An Application to Depression and Socio-Economic Status

Latent class regression models are useful tools for assessing associations between covariates and latent variables. However, evaluation of key model assumptions cannot be performed using methods from standard regression models due to the unobserved nature of latent outcome variables. This paper presents graphical diagnostic tools to evaluate whether or not latent class regression models adhere to standard assumptions of the model: conditional independence and non-differential measurement. An integral part of these methods is the use of a Markov Chain Monte Carlo estimation procedure. Unlike standard maximum likelihood implementations for latent class regression model estimation, the MCMC approach allows us to calculate posterior distributions and point estimates of any functions of parameters. It is this convenience that allows us to provide the diagnostic methods that we introduce. As a motivating example we present an analysis focusing on the association between depression and socioeconomic status, using data from the Epidemiologic Catchment Area study. We consider a latent class regression analysis investigating the association between depression and socioeconomic status measures, where the latent variable depression is regressed on education and income indicators, in addition to age, gender, and marital status variables. While the fitted latent class regression model yields interesting results, the model parameters are found to be invalid due to the violation of model assumptions. The violation of these assumptions is clearly identified by the presented diagnostic plots. These methods can be applied to standard latent class and latent class regression models, and the general principle can be extended to evaluate model assumptions in other types of models.

TRIO LOGIC REGRESSION - DETECTION OF SNP - SNP INTERACTIONS IN CASE-PARENT TRIOS

Statistical approaches to evaluate higher order SNP-SNP and SNP-environment interactions are critical in genetic association studies, as susceptibility to complex disease is likely to be related to the interaction of multiple SNPs and environmental factors. Logic regression (Kooperberg et al., 2001; Ruczinski et al., 2003) is one such approach, where interactions between SNPs and environmental variables are assessed in a regression framework, and interactions become part of the model search space. In this manuscript we extend the logic regression methodology, originally developed for cohort and case-control studies, for studies of trios with affected probands. Trio logic regression accounts for the linkage disequilibrium (LD) structure in the genotype data, and accommodates missing genotypes via haplotype-based imputation. We also derive an efficient algorithm to simulate case-parent trios where genetic risk is determined via epistatic interactions.

Computational Techniques for Spatial Logistic Regression with Large Datasets

In epidemiological work, outcomes are frequently non-normal, sample sizes may be large, and effects are often small. To relate health outcomes to geographic risk factors, fast and powerful methods for fitting spatial models, particularly for non-normal data, are required. We focus on binary outcomes, with the risk surface a smooth function of space. We compare penalized likelihood models, including the penalized quasi-likelihood (PQL) approach, and Bayesian models based on fit, speed, and ease of implementation. A Bayesian model using a spectral basis representation of the spatial surface provides the best tradeoff of sensitivity and specificity in simulations, detecting real spatial features while limiting overfitting and being more efficient computationally than other Bayesian approaches. One of the contributions of this work is further development of this underused representation. The spectral basis model outperforms the penalized likelihood methods, which are prone to overfitting, but is slower to fit and not as easily implemented. Conclusions based on a real dataset of cancer cases in Taiwan are similar albeit less conclusive with respect to comparing the approaches. The success of the spectral basis with binary data and similar results with count data suggest that it may be generally useful in spatial models and more complicated hierarchical models.

Inference on Survival Data with Covariate Measurement Error - An Imputation-based Approach

We propose a new method for fitting proportional hazards models with error-prone covariates. Regression coefficients are estimated by solving an estimating equation that is the average of the partial likelihood scores based on imputed true covariates. For the purpose of imputation, a linear spline model is assumed on the baseline hazard. We discuss consistency and asymptotic normality of the resulting estimators, and propose a stochastic approximation scheme to obtain the estimates. The algorithm is easy to implement, and reduces to the ordinary Cox partial likelihood approach when the measurement error has a degenerative distribution. Simulations indicate high efficiency and robustness. We consider the special case where error-prone replicates are available on the unobserved true covariates. As expected, increasing the number of replicate for the unobserved covariates increases efficiency and reduces bias. We illustrate the practical utility of the proposed method with an Eastern Cooperative Oncology Group clinical trial where a genetic marker, c-myc expression level, is subject to measurement error.

PENALIZED FUNCTIONAL REGRESSION

We develop fast fitting methods for generalized functional linear models. An undersmooth of the functional predictor is obtained by projecting on a large number of smooth eigenvectors and the coefficient function is estimated using penalized spline regression. Our method can be applied to many functional data designs including functions measured with and without error, sparsely or densely sampled. The methods also extend to the case of multiple functional predictors or functional predictors with a natural multilevel structure. Our approach can be implemented using standard mixed effects software and is computationally fast. Our methodology is motivated by a diffusion tensor imaging (DTI) study. The aim of this study is to analyze differences between various cerebral white matter tract property measurements of multiple sclerosis (MS) patients and controls. While the statistical developments proposed here were motivated by the DTI study, the methodology is designed and presented in generality and is applicable to many other areas of scientific research. An online appendix provides R implementations of all simulations.