Shrinkage Parameter Estimation in Penalized Logistic Regression Analysis of Case-Control Data

In genetic epidemiology, rare variant case-control studies aim to investigate the association between rare genetic variants and human diseases. Rare genetic variants lead to sparse covariates that are predominately zeros and this sparseness leads to estimators of log-OR parameters that are biased away from their null value of zero. Different penalized-likelihood methods have been developed to mitigate this sparse-data bias for case-control studies. In this research article, we study penalized logistic regression using a class of log-F priors indexed by a shrinkage parameter m to shrink the biased MLE towards zero. We propose a maximum marginal likelihood method for estimating m, with the marginal likelihood obtained by integrating the latent log-ORs out of the joint distribution of the parameters and observed data. We consider two approximate approaches to maximizing the marginal likelihood: (i) a Monte Carlo EM algorithm and (ii) a combination of a Laplace approximation and derivative-free optimization of the marginal likelihood. We evaluate the statistical properties of the estimator through simulation studies and apply the methods to the analysis of genetic data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI).

The Monotonic Polynomial Graded Response Model: Implementation and a Comparative Study

We present a monotonic polynomial graded response (GRMP) model that subsumes the unidimensional graded response model for ordered categorical responses and results in flexible category response functions. We suggest improvements in the parameterization of the polynomial underlying similar models, expand upon an underlying response variable derivation of the model, and in lieu of an overall discrimination parameter we propose an index to aid in interpreting the strength of relationship between the latent variable and underlying item responses. In applications, the GRMP is compared to two approaches: (a) a previously developed monotonic polynomial generalized partial credit (GPCMP) model; and (b) logistic and probit variants of the heteroscedastic graded response (HGR) model that we estimate using maximum marginal likelihood with the expectation–maximization algorithm. Results suggest that the GRMP can fit real data better than the GPCMP and the probit variant of the HGR, but is slightly outperformed by the logistic HGR. Two simulation studies compared the ability of the GRMP and logistic HGR to recover category response functions. While the GRMP showed some ability to recover HGR response functions and those based on kernel smoothing, the HGR was more specific in the types of response functions it could recover. In general, the GRMP and HGR make different assumptions regarding the underlying response variables, and can result in different category response function shapes.

Comparison of Uni- and Multidimensional Models Applied in Testlet-Based Tests

The bifactor model (BM) and the testlet response model (TRM) are the most common multidimensional models applied to testlet-based tests. The common procedure is to estimate these models using different estimation methods (see, e.g., DeMars, 2006 ). A possible consequence of this is that previous findings about the implications of fitting a wrong model to the data may be confounded with the estimation procedures they employed. With this in mind, the present study uses the same method (maximum marginal likelihood [MML] using dimensional reduction) to compare uni- and multidimensional strategies to testlet-based tests, and assess the performance of various relative fit indices. Data were simulated under three different models, namely BM, TRM, and the unidimensional model. Recovery of item parameters, reliability estimates, and selection rates of the relative fit indices were documented. The results were essentially consistent with those obtained through different methods ( DeMars, 2006 ), indicating that the effect of the estimation method is negligible. Regarding the fit indices, Akaike Information Criterion (AIC) showed the best selection rates, whereas Bayes Information Criterion (BIC) tended to select a model which is simpler than the true one. The work concludes with recommendations for practitioners and proposals for future research.