## Task 4: Key Concepts about Estimating Population Distributions of Ratios of Usual Intakes of Two Dietary Constituents that are Ubiquitously Consumed

With daily intake data, such as that collected on 24-hour recalls, the ratio of dietary components may be constructed in one of two ways:

1. The ratio of intakes may be constructed for each day and the “usual” such ratio determined, or
2. The “usual” intake for each component may be computed, and the ratio of these usual intakes calculated.

The first way is termed the “usual ratio of intakes”, and the second the “ratio of usual intakes.”  (For more information about ratios, see Module 15 and Freedman et al. 2010a in the Key References page). The NCI method may be used to calculate either method.

The usual ratio of intakes may be computed by calculating the ratio of the dietary components of interest each day, and applying the methods for a single ubiquitously consumed dietary component, as described in Module 20, Task 1.  The ratio of usual intakes can be estimated by modeling both ubiquitously consumed nutrients simultaneously in a bivariate model.  Modeling the ratio of usual intakes will be described in the remainder of this Task.

The method for estimating the ratio of usual intakes of two ubiquitously-consumed dietary components is similar to the approach used for the NCI method for estimating a single ubiquitously-consumed dietary component (see Module 20, Task 1), except that the two dietary components are modeled simultaneously.  The model assumes that, after appropriate Box-Cox transformations, the 24-hour recall reported values for the two dietary components follow a bivariate linear mixed effects model.  The model includes an intercept and covariates for each dietary component, plus two person-specific random effects, one for each of the two ubiquitously-consumed dietary components.  The two person-specific effects are assumed to have a bivariate normal distribution and be independent of the person-specific terms.

In this task, we reproduce an analysis of the ratio of saturated fat to energy, described in Freedman et al. (2010a). This task demonstrates an alternative method to estimate Box-Cox transformation parameters.  In Task 1 of this module, the Box-Cox parameter was estimated simultaneously with the other model parameters using maximum likelihood estimation for fitting a nonlinear model; the nonlinearity occurred due to the inclusion of the Box-Cox parameter in the model.

In this task, we choose the Box-Cox transformation that minimizes the mean squared error around a straight line fit to a weighted QQ plot, using the sampling survey weights of each participant, before fitting the model. After choosing the Box-Cox parameter and transforming the 24-hour recalls, the other parameters can be estimated using a bivariate linear mixed effects model.  (Because nonlinear mixed models can sometimes be numerically unstable, the “transform to linearity” approach could potentially produce stable estimates when the maximum likelihood approach fails.)

In the bivariate linear mixed effects model, there are two random person-specific effects, corresponding to each ubiquitously-consumed dietary component, which have a joint normal distribution, and two within-person random errors, corresponding to each each ubiquitously-consumed dietary component, which also have a joint normal distribution, and are independent of the random effects, and independent across repeats.

The model for each nutrient includes an indicator for whether the reported day was a weekday or a weekend, an indicator for the sequence number (first versus second) of the report, indicators for each 5-year age group, a person-specific random effect and a within-subject error term. For children, an extra covariate for sex was included and, for adults, men and women were analyzed separately. In this model, the covariate for age group was included to allow estimation of subpopulations by age, similar to the model fit in Module 20, Task 2. (See Module 18, Task 3 for more information on covariate adjustment.)

As in the NCI method for single nutrients, Monte Carlo simulations are generated using the parameter values estimated from the bivariate model. The ratio of usual intakes is calculated for each pseudo-individual generated from the Monte Carlo simulations, and using the sampling weights, the percentiles of the distribution of the ratio of usual intakes are estimated. The balanced repeated replication method (BRR, see Module 18, Task 4) is used to estimate standard errors that account for the complex sampling of NHANES.

The method described in Freedman et al (2010a) was extended to estimate the distribution of the HEI-2005 score.  This requires an extra calculation to convert the ratio to the HEI-2005 score in the Monte-Carlo step of the procedure.

The macros to fit the NCI method may be downloaded from the NCI website.