Using P-Splines to Estimate Nonlinear Covariate Effects in Latent Factor Models.

Abstract

Latent factor models are useful for summarizing information among multiple outcomes. In this thesis I apply semiparametric methods based on P-splines to latent factor models in order to estimate and test non-constant factor loadings, as well as estimate and test the nonlinear relationship between an observed continuous predictor and multiple observed outcomes that measure a latent factor.

In the first chapter, I develop a modeling strategy that estimates non-constant factor loadings as functions of multiple covariates. A highlight of my algorithm is the optimization of a type of generalized cross-validation criterion within each iteration of the EM algorithm for estimating the smoothing parameters of the splines. Through simulation studies I show the advantage of correctly estimating the non-constant factor loadings in reducing bias for the estimated factor score. I apply my model to studying the correlation among four highly correlated PM2.5 constituents.

In the second chapter I examine the use of likelihood ratio test (LRT) in assessing whether a factor loading is constant. In order to take into account the estimation of smoothing parameters in my testing procedure, I use maximum likelihood approach to smooth the P-splines, which treats the spline coefficients as random and I test the variance of the spline coefficients. importance sampling to compute the likelihood. I use a data-driven chi-square mixture approximation as the null LRT distribution. The method is applied to estimating the underlying lead exposure represented by four types of lead measurements on mothers from the ELEMENT study.

In the third chapter I use P-splines to estimate and test deviations of the latent factor mean from a linear trend. I also make the connection between my semiparametric latent factor model to a class of linear mixed models that estimate an overall exposure effect for multiple outcomes. My algorithm is based on standard linear mixed model and is implemented by adapting PROC MIXED from SAS into an iterative procedure. I apply my model to studying the lead exposure effect on children's behaviors as measured by the psychometric battery BASC-2.