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BIOMETRICS 57, 62-73 March 2001 Covariate Measurement Error Adjustment for Matched Case-Control Studies Lisa M. McShane,'>* Douglas N. Midthune,2 Joanne F. D~rgan,~ 'National Cancer Institute, Biometric Research Branch, DCTD, Executive Plaza North, Room 739, 6130 Executive Boulevard, MSC 7434, Bethesda, Maryland 20892-7434, U.S.A. 2National Cancer Institute, Biometry Research Group, DCP, Executive Plaza North, Room 344, 6130 Executive Boulevard, Bethesda, Maryland 20892, U.S.A. 3F0x Chase Cancer Center, 7701 Burholme Avenue, Philadelphia, Pennsylvania 19111, U.S.A. 4Department of Mathematics, Statistics, and Computer Science, Bar Ilan University, Ramat Gan 52900, Israel 5Department of Statistics, Texas A&M University, College Station, Texas 77843-3143, U.S.A. * email: [email protected] Laurence S. and Raymond J. Carroll5 SUMMARY. We propose a conditional scores procedure for obtaining bias-corrected estimates of log odds ratios from matched case-control data in which one or more covariates are subject to measurement error. The approach involves conditioning on sufficient statistics for the unobservable true covariates that are treated as fixed unknown parameters. For the case of Gaussian nondifferential measurement error, we derive a set of unbiased score equations that can then be solved to estimate the log odds ratio parameters of interest. The procedure successfully removes the bias in naive estimates, and standard error estimates are obtained by resampling methods. We present an example of the procedure applied to data from a matched casecontrol study of prostate cancer and serum hormone levels, and we compare its performance to that of regression calibration procedures. KEY WORDS: Casecontrol study; Conditional logistic regression; Conditional scores; Hormones; Matched design; Measurement error; Prostate cancer. 1. Introduction There has been a proliferation of biorepositories holding serum or tissue specimens collected from subjects in large clinical tri- als or prospectively followed cohorts. Collected prediagnosis, these specimens can be used to examine relationships between risk of disease and serum and tissue biomarkers measured by laboratory assays. The nested casecontrol design, which in- volves matching on characteristics that might otherwise con- found exposure-disease relationships, is frequently used for such studies. Typically, one has only a single measurement of the biomarker per individual and it may be subject to measurement error arising from multiple sources. We envi- sion that each subject has a true underlying average measure for the biomarker of interest. The actual level on any occa- sion may vary from this average for numerous reasons. For example, for biomarkers measured in serum, biological vari- ation related to inherent patterns of secretion (e.g., diurnal rhythms) or changes in personal characteristics (e.g., diet) that are unmeasured or unknown to affect the biomarker of interest cause fluctuations in levels. Differences in specimen collection or handling may also cause fluctuations. We refer to the combined effects of random biological variation and specimen handling on biomarker levels as occasion-within- person variability. There is also laboratory assay variability, which may be subdivided into between-batch (i.e., assay) and within-batch variability. We consider the contributions of all of these sources of variability as measurement error with re- gard to an individual's true biomarker level. Measurement error in an explanatory exposure variable may result in attenuation of relative risk estimates and re- duced power for detecting exposure-disease relationships. For unmatched studies, a variety of measurement error correction methods have been proposed for logistic risk models (Ros- ner, Willett, and Spiegelman, 1989; Rosner, Spiegelman, and Willett, 1990, 1992; Carroll, Ruppert, and Stefanski, 1995, and references therein), but the matched design has received far less attention. Armstrong, Whittemore, and Howe (1989) propose a measurement error correction method assuming a normal discriminant analysis model. Their method assumes multivariate Gaussian covariates, but in that setting, it has the flexibility to handle differential measurement error. Forbes and Santner (1995) develop a correction method using a ret- rospective likelihood with a binary exposure variable. Their method assumes that the binary exposure variable is mea- 62Covariate Measurement Error Adjustment 63 sured without error and addresses situations in which continu- ous confounders may be measured with error. Prentice (1982) proposes a method for parameter estimation in Cox’s fail- ure time regression model when the covariates are measured with error. Noting the similarity between Cox’s partial like- lihood and the conditional logistic regression likelihood used for matched analyses, Prentice’s method is applicable here. But its implementation requires knowledge of the conditional distribution of the true covariates given their observed error- prone measurements or, at a minimum, sufficient information to compute the conditional expectation of the exponential terms in the likelihood. The conditional scores method we propose for matched studies is based on the prospective likelihood, and it allows for very general covariate distributions. In this approach, un- observable true covariates are treated as fixed unknown pa- rameters. One conditions on a sufficient statistic to remove them from the likelihood and produce a set of unbiased score equations to solve for log odds ratio parameter estimates. In Section 2, we review the matched casecontrol study de- sign and likelihood. A measurement error model appropriate for continuous biomarker variables measured by laboratory assay is presented in Section 3. In Section 4, the conditional scores measurement error adjustment procedure is developed under an assumption of Gaussian nondifferential measure- ment error. Regression calibration approaches are described in Section 5. In Section 6, we apply the procedures to example data from a study examining the relationship between serum hormone levels and risk of prostate cancer. Simulation studies are presented in Section 7. A discussion follows in Section 8. 2. Study Design The study


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