## Macros

- NLMIXED UNIVARIATE
- NLMIXED BIVARIATE
- PREDICT_INTAKE_DENSITY

## Procedure

First, call the NLMIXED_UNIVARIATE macro twice (once for each variable) to get starting estimates for subsequent NLMIXED_BIVARIATE calls

Because replication methods (bootstrap or BRR) are used to estimate standard errors of calculated statistics, the following tasks must be performed repeatedly – once for the original data set (or using the base sampling weight variable) to obtain point estimates and again for each resampled data set (or using each of the bootstrap/BRR weight variables in turn):

- Use the NLMIXED_BIVARIATE macro to fit the measurement error model and store parameter estimates, then

- Use the parameter estimates as input to the PREDICT_INTAKE_DENSITY macro to compute the conditional expectation of the ratio of usual intakes of each individual given their FFQ response, then

- Fit an appropriate health outcome-exposure model, using the conditional expectations as the dietary exposures

After calculating desired statistics for all data sets/sampling weights, use the appropriate bootstrap/BRR algorithms to estimate standard errors for the coefficients in the health outcome–exposure model by taking the square root of the (adjusted, if BRR) variance across replicates.

## Notes

- The conditional expectations produced by PREDICT_INTAKE_DENSITY are not true intakes for a particular individual. The computations involve averaging over an assumed (i.e., not observable) distribution of individual effects. Two individuals may have very different true usual intakes, yet report the same on FFQ. Their corresponding output from the PREDICT_INTAKE_DENSITY macro would be the same. Thus, categorizing the two individuals based on their PREDICT_INTAKE_DENSITY output would be subject to potentially extreme misclassification. However, under the assumptions required of the regression calibration method, using the output from PREDICT_INTAKE_DENSITY yields a measurement-error-corrected estimate of the regression slopes in a health outcome-exposure model.

- Using resampling methods to calculate standard errors of the coefficients for exposures in the health outcome-exposure model properly accounts for variability in all stages of the estimation.

- PREDICT_INTAKE_DENSITY can estimate conditional expectations of (Box-Cox) transformed ratios of usual intakes, if the health outcome-exposure model is nonlinear in the exposure. Second-degree and higher polynomial terms of exposure can be obtained by repeated calls to the PREDICT_INTAKE_DENSITY macro and some algebra.

## Example Code

This application is similar to the following application: Estimation of the association between a dietary intake and a health outcome; FFQ is the main instrument; Two regularly-consumed or one regularly-consumed and one episodically-consumed foods or nutrients. The crucial difference is that only one call to PREDICT_INTAKE_DENSITY is required for a model with a single ratio as in this application.