In Section 13.6, adjusted mean BMI values were obtained for the two nominal variables exercise and tobacco_now. A drawback of the approach used is that these four adjusted means were estimated for the four combinations of these two factors considered simultaneously, leaving one without adjusted estimates for the two levels of each variable considered separately (e.g., the adjusted "marginal" mean for exercise = 1 regardless of the level of tobacco_now, etc.). Given that no product term between exercise and tobacco_now was included in the model, the computation of these four marginal adjusted means should be possible.
One approach is to use estimates for the "average" value of the nominal factor being controlled, but it is unclear which values should be used for exercise or tobacco_now, given that these dichotomous variables take on only the values 0 and 1. Two possible options are
1. For the two levels of exercise, estimate the adjusted mean BMI assuming a value of 0.5 for tobacco_now. Similarly, for the two levels of tobacco_now, assume a value of 0.5 for exercise. Specifically, the adjusted means for both values of exercise would be estimated by Ŷ = β0 + β1(drink_days) + β2(poor_sleep_days) + β3(exercise) + β4(0.5). A criticism of this approach is that, since no person could have an observed value of 0.5 for either factor, using 0.5 may yield unrealistic estimates. A second issue is that, for population-based studies such as the BRFSS, values of 0.5 may represent unrealistic choices for the sample prevalences of these variables (see option 2 of this problem). This is because the sample distributions of exercise and tobacco_now are not perfectly balanced, with exactly 50% of the population having the value 1 for each variable. This approach for computing adjusted means is revisited in the discussion of ANOVA in Chapter 17.
2. For the two levels of exercise, estimate the adjusted mean BMI assuming that the sample prevalence for current tobacco use is 16.1%. For the two levels of tobacco_now, assume that the sample prevalence for exercise is 79.9%. This is known as the observed margins weighting, conditional, or conditional margins approach and is the simplest of a set of techniques known as standardization methods (Wilcosky and Chambless 1985). By using sample prevalences for these dichotomous predictors, one is effectively using the analog of sample means for continuous covariates. As with option 1, option 2 assigns a non-observable value to a variable taking only the values 0 and 1.
Both of the above options are readily computed in SAS, although there are even more approaches, such as stratified and marginal methods (Wilcosky and Chambless 1985; Flanders and Rhodes 1987). Using each of the two options described above and the output in Section 13.6, estimate the adjusted mean BMI for the levels of exercise and tobacco_now. How do these estimates compare to each other and to the different sets of adjusted means in Section 13.6?