## Macros

- MULTIVAR_MCMC
- BOXCOX_SURVEY
- STD_COV_BOXCOX24HR_CONDAY_MINAMT
- MULTIVAR_DISTRIB

## Procedure

First, choose a Box-Cox transformation parameter for the nonzero amounts of each dietary variable. The BOXCOX_SURVEY macro or PROC TRANSREG in SAS can be used to perform this task.

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 STD_COV_BOXCOX24HR_CONDAY_MINAMT macro to prepare each raw data set for the MULTIVAR_MCMC macro by applying the chosen Box-Cox transformation to nonzero 24-hour recall amounts, then standardizing the results and other covariates. Because the standardizing constants differ for different bootstrap samples or BRR/bootstrap weight sets, this step must be repeated.

- Use the MULTIVAR_MCMC macro to fit the measurement error model and store parameter estimates and repeated draws from the conditional distribution of individual effects given observed recalls, then

- Use the parameter estimates and conditional draws as input to the MULTIVAR_DISTRIB macro to simulate a sample of usual intakes based on each individual’s 24-hour responses, then average across the sample for each individual to obtain estimated conditional expectations of usual intake, then

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

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

## Notes

- The conditional expectations produced in step 3 above 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 multiple 24HRs. Their corresponding output from Step 3 would be the same. Thus, categorizing the two individuals based on their results of Step 3 would be subject to potentially extreme misclassification. However, under the assumptions required of the regression calibration method, using the output from Step 3 yields measurement-error-corrected estimates 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.

- If the health outcome-exposure model is nonlinear in exposures, or polynomial terms of exposure are desired, Step 3 can be modified to produce functions of simulated intakes before averaging across the simulated intakes per individual.

## Example Code

This application is a special case of the following application: Estimation of the association between a dietary intake and a health outcome; 24-hour recall is the main instrument; Several regularly-consumed or episodically-consumed foods or nutrients, one of which has never consumers.