{'fraction_non_alphanumeric': 0.02293379489398529, 'fraction_numerical': 0.004759844223279966, 'mean_word_length': 4.974160206718346, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Measurement error correction for logistic regression models with an "alloyed gold standard".

Recently, some authors have questioned the validity of methods which correct relative risk estimates for measurement error and misclassification when the "gold standard" used to obtain information about the measurement error process is itself imperfect. When such an "alloyed" gold standard is used to validate the usual exposure measurement, the bias in the "regression calibration" (Rosner et al., Stat Med 1989; 8:1051-69) measurement-error correction factor for relative risks estimated from logistic regression models is derived. This quantity is a function of the correlations of the "alloyed" gold standard (X) and the usual exposure assessment method (Z) with the truth, of the ratio of the variances of X and Z, and of the correlation between the errors in the "alloyed" gold standard and the errors in the usual exposure assessment method. In this paper, it is proven that if the errors between Z and X are uncorrelated, the regression calibration method has no bias even when the gold standard is "alloyed." When a third method of exposure assessment is available and it is reasonable to assume that the errors in this method are uncorrelated with the errors in the other two exposure assessment methods, point and interval estimates of the correlation between the errors in X and Z are derived. These methods are illustrated here with data on the measurement of physical activity, vitamins A and E, and poly- and monounsaturated fat. In addition, when a third exposure assessment method is available, a modification of standard regression calibration is derived which can be used to calculate point and interval estimates of relative risk that are corrected for measurement error in both X and Z. This new method is illustrated here with data from the Health Professionals Follow-up Study, a study investigating the associations between physical activity and colon cancer incidence and between vitamin E intake and coronary heart disease. It is shown that in these examples, correlations of the errors in X and Z tended to be small. Even when moderate, estimates of relative risk corrected for error in both X and Z were not very different from the estimates which assumed that X was a true gold standard.