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Local Likelihood Method for Estimating

Relative Risk Functions in Case-Control Studies

(Submitted to Journal of the Royal Statistical Society, B, 2004)

by

**K.F.
Cheng** and C.K.
Chu

**
Abstract**

We consider the analysis of case-control studies in which one exposure variable is continuous and another exposure variable is binary. Let the relative risk function describe the functional relation between the continuous exposure and disease risk. In this paper, we investigate the extension of the local likelihood method to the estimation of the unknown relative risk function and odds ratio parameter. Using the proposed initial estimates, the final estimates can be derived from solving two systems of estimating equations. One system of estimating equations is identical to that derived by Fan et al. (1995) under prospective sampling. Another system of estimating equations is identical to that used in the traditional linear logistic regression analysis of prospective data. The asymptotic properties of the estimators are presented in this paper so that the performance of different estimators can be compared and confidence intervals can be constructed. Two real data examples are given to illustrate the applications of our estimation procedures. Further, a simulation study is conducted to investigate the finite sample properties of the proposed estimators. From the results of our examples and simulation study, we find out that the local likelihood method is a useful nonparametric regression technique in estimating the unknown relative risk function and odds ratio parameter in case-control studies.

KEY WORDS:case-control; local likelihood; odds ratio; nonparametric regression; relative risk function.