  In the United States the preferred method of obtaining dietary intake data is
the 24-hour dietary recall, yet the measure of most interest is usual or
long-term average daily intake, which is impossible to measure. Thus, usual
dietary intake is assessed with considerable measurement error. We were
interested in estimating the population distribution of the Healthy Eating
Index-2005 (HEI-2005), a multi-component dietary quality index involving ratios
of interrelated dietary components to energy, among children aged 2-8 in the
United States, using a national survey and incorporating survey weights. We
developed a highly nonlinear, multivariate zero-inflated data model with
measurement error to address this question. Standard nonlinear mixed model
software such as SAS NLMIXED cannot handle this problem. We found that taking a
Bayesian approach, and using MCMC, resolved the computational issues and doing
so enabled us to provide a realistic distribution estimate for the HEI-2005
total score. While our computation and thinking in solving this problem was
Bayesian, we relied on the well-known close relationship between Bayesian
posterior means and maximum likelihood, the latter not computationally
feasible, and thus were able to develop standard errors using balanced repeated
replication, a survey-sampling approach.
