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1.  Regression Calibration with Heteroscedastic Error Variance 
The problem of covariate measurement error with heteroscedastic measurement error variance is considered. Standard regression calibration assumes that the measurement error has a homoscedastic measurement error variance. An estimator is proposed to correct regression coefficients for covariate measurement error with heteroscedastic variance. Point and interval estimates are derived. Validation data containing the gold standard must be available. This estimator is a closed-form correction of the uncorrected primary regression coefficients, which may be of logistic or Cox proportional hazards model form, and is closely related to the version of regression calibration developed by Rosner et al. (1990). The primary regression model can include multiple covariates measured without error. The use of these estimators is illustrated in two data sets, one taken from occupational epidemiology (the ACE study) and one taken from nutritional epidemiology (the Nurses’ Health Study). In both cases, although there was evidence of moderate heteroscedasticity, there was little difference in estimation or inference using this new procedure compared to standard regression calibration. It is shown theoretically that unless the relative risk is large or measurement error severe, standard regression calibration approximations will typically be adequate, even with moderate heteroscedasticity in the measurement error model variance. In a detailed simulation study, standard regression calibration performed either as well as or better than the new estimator. When the disease is rare and the errors normally distributed, or when measurement error is moderate, standard regression calibration remains the method of choice.
PMCID: PMC3404553  PMID: 22848187
measurement error; logistic regression; heteroscedasticity; regression calibration
2.  The Comparison of Alternative Smoothing Methods for Fitting Non-Linear Exposure-Response Relationships with Cox Models in a Simulation Study* 
We examined the behavior of alternative smoothing methods for modeling environmental epidemiology data. Model fit can only be examined when the true exposure-response curve is known and so we used simulation studies to examine the performance of penalized splines (P-splines), restricted cubic splines (RCS), natural splines (NS), and fractional polynomials (FP). Survival data were generated under six plausible exposure-response scenarios with a right skewed exposure distribution, typical of environmental exposures. Cox models with each spline or FP were fit to simulated datasets. The best models, e.g. degrees of freedom, were selected using default criteria for each method. The root mean-square error (rMSE) and area difference were computed to assess model fit and bias (difference between the observed and true curves). The test for linearity was a measure of sensitivity and the test of the null was an assessment of statistical power. No one method performed best according to all four measures of performance, however, all methods performed reasonably well. The model fit was best for P-splines for almost all true positive scenarios, although fractional polynomials and RCS were least biased, on average.
PMCID: PMC2827890  PMID: 20231865

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