Commentary: Correlated errors and energy adjustment—where are the data?

In this issue of the International Journal of Epidemiology (IJE), Day et al.1 conduct a sensitivity analysis of the effects of correlated errors of mis-measured covariates in linear regression models, when the data to permit direct adjustment for the effects of correlated errors are not available. They show yet again what has been documented many times before, but not cited by Day et al.,2–6 that measurement error in more than one model covariate in which both the underlying true values and the errors may be correlated leads to a conflation of the well-known effects of confounding with the well-known effects of measurement error. When the correlations between the underlying variables, the error correlations, and the measurement errors are large, these results can be misleading and unpredictable. It is not entirely clear what is new here, although the detailed example is certainly useful to further emphasize this point to those who didn’t get it the first few times around. There are many instances in nutritional epidemiology when it is of interest to evaluate the independent effects of several nutrients on a health endpoint. The sensitivity analysis approach given by Day et al. is of limited use, since as they and others before them have shown, the regression coefficients relating the nutrients to the health endpoint could take on virtually any value. With the standard study design, it is impossible to validly estimate or even put bounds of any informative width on the regression coefficients of interest. So how can we remove bias in multiple regression coefficients for mis-measured nutrients which both are correlated and may have correlated error? Are there study designs and then methods of data analysis available to correct for these biases? Of course! Day et al.’s revisitation of the problem without reviewing the proposed solutions does not highlight the potential for progress in nutritional epidemiology. Very briefly, Carroll & Stefanski,7 Rosner et al.,8,9 Spiegelman et al.,10,11 and Xie et al.12 presented the regression calibration method for main study/validation study designs, which permit approximately unbiased point and interval estimates of effect to be estimated from linear, Cox, and logistic regression models when the usual methods of exposure assessment can be validated against an unbiased, perhaps ‘alloyed’ gold standard in a sub-sample, possibly outside of the original main study. The method is inherently multivariate and naturally accommodates multiple correlated nutrients with correlated errors as discussed in Day et al.’s paper in this issue of the IJE. Fraser & Stram2 discuss study design in this setting, and show that validation studies of 1000 participants or more are needed when correlations of either the errors or the underlying true values of nutrients are high. Concerns have been raised about the validity of the assumptions made in the original regression calibration method. First, it was suggested that when multiple nutrients are being validated simultaneously by ‘alloyed’ gold standards, the errors in the reference instruments may be correlated with the errors in the surrogates. When this is the case, regression calibration will fail to remove all of the bias due to measurement error— the surrogates will appear to be better measured than they are. Spiegelman et al.13 developed an augmented validation study design which includes a third method of exposure assessment (e.g. a biomarker), the errors of which can reasonably be assumed to be uncorrelated with the errors in the reference instrument (e.g. diet records) and surrogate (e.g. food frequency questionnaires). With these measurements in hand, they extended the regression calibration procedure to allow for correlated errors between the reference and surrogate measures. Error correlations were small in examples of vitamin E and physical activity. Next, Kipnis et al.14 suggested that in addition to the presence of correlated random within-person error of the reference and surrogate exposure measures, there may also be systematic correlated within-person error between these two instruments. In response, Spiegelman et al.15 developed a class of augmented main study/validation study designs which permit valid regression calibration estimation of relative risks and other regression slopes, requiring either replicated biomarkers or an additional ‘instrumental variable’ which has some correlation with the underlying exposure of interest with error jointly uncorrelated with all other errors. It should be pointed out that these extended regression calibration methods have not been expanded to handle multiple regression covariates measured with error, as Day et al. focus upon here.1 Should there be sufficient interest, it is straightforward to derive these further generalizations and produce user-friendly SAS macros to facilitate use, as has already been done with standard multivariate regression calibration (www.hsph.harvard.edu/faculty/spiegelman/software.html). The truly limiting issue in practice is the one underscored by Fraser & Stram—we nutritional epidemiologists have not been conducting validation studies of sufficient size to make the necessary bias corrections with meaningful precision. To my knowledge, there is not yet a single validation study available for which it would be informative to fully investigate the independent effects of several nutrients together, allowing for correlated random and possibly systematic errors of the reference instrument (diaries) with the surrogate (food frequency questionnaire or diet recall). Let’s do it, and do the job right!