Semiparametric Regression Models for Repeated Events with Random Effects and Measurement Error Wenxin Jiang, Bruce W. Turnbull and Larry C. Clark Abstract Statistical methodology is presented for the regression analysis of multiple events in the presence of random effects and measurement error. Omitted covariates are modeled as random effects. Our approach to parameter estimation and significance testing is to start with a naive model of semi-parametric Poisson process regression, and then to adjust for random effects and any possible covariate measurement error. We illustrate the techniques with data from a randomized clinical trial for the prevention of recur- rent skin tumors. KEY WORDS: Consistency; Cox model; Estimating equations; Frailty; Measurement error; Omitted covariates; Point process; Poisson regression; Proportional intensities; Robust estimator; Selenium; Skin cancer; Specification analysis; Unobserved hetero- geneity; Validation data. 1. INTRODUCTION The research in this paper was motivated by some of the statistical discussions leading up to the publication of the somewhat controversial findings of the “Nutritional Prevention of Cancer” (NPC) trial — Clark et al. (1996). This trial, begun in 1983, studied the long-term safety and efficacy of a daily 200μg nutritional supplement of selenium (Se) for the prevention of cancer. This was a double-blind, placebo-controlled randomized clinical trial with 1312 patients accrued and followed for up to about ten years. A number of endpoints were considered, but here we shall concentrate on one of the two primary endpoints — namely squamous cell carcinoma (SCC) of the skin. The results for this endpoint are of particular interest because Clark et al. (1996) found a negative (but not statistically significant, P = 0.15) effect of selenium (Se) supplementation. This was opposite to previous 1 Wenxin Jiang is Assistant Professor, Department of Statistics, Northwestern University, Evanston, IL 60208. Bruce W. Turnbull is Professor, School of Operations Research and Department of Statistical Science, 227 Rhodes Hall, Cornell University, Ithaca, NY 14853. Larry C. Clark is Associate Professor, Arizona Cancer Center, University of Arizona, Tucson AZ 85724. The first author was supported in part by NSF grant DMS-9505799, the second author by NIH grant R01 CA 66218, and the third author by NIH grant R01 CA 49764. The authors are grateful to a referee, who made several very helpful suggestions. 1