Model Description: The Covariate Density Defined (CDD) Mixture of Logistic Regressions Method




The method used to model and control for heterogeneity in a birth cohort is the covariate density defined (CDD) mixture of logistic regressions. The CDD mixture of regression method is different from both the conventional finite mixture and the growth mixture methods. Unlike finite mixtures of logistic regressions, the CDD approach is usually identified and is probably generalizable to most regression like procedures. CDD mixtures use the marginal density of a covariate y (birth weight in this case) to assign probabilistic (latent) subpopulation membership to separate logistic probabilities. These subpopulations are usually, but not restricted to, normal distributions and are defined as the primary and secondary subpopulations. For each subpopulation, there is a regression on the outcome u (infant mortality in this case) weighted by the latent membership determined by the mixing submodel. This method can identify direct and indirect effects (mediated through the density of y) of covariate x on the outcome (indicated by the proximate determinates model described here). Graphically, a direct effect is seen as a vertical shift in the mortality curve while an indirect effect is seen as horizontal shift.

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Characteristic changes in mortality with respect to changes in birth weight based on the Wilcox-Russell “causality” theory (A) no indirect effect, shift in birth weight (bold lines, solid to dashed) and corresponding shift in birth weight-specific mortality curve (thin lines, solid to dashed), no overall change in infant mortality, (B) no indirect effect plus a direct effect, shift in birth weight (bold lines, solid to dashed) and corresponding shift in birth weight-specific mortality curve (thin lines, solid to dashed), no overall change in infant mortality due to shift in birth weight but direct effect increases mortality at all birth weights and overall, and (C) indirect effect but no direct effect, shift in birth weight (bold lines, solid to dashed) not identical to the shift in birth weight-specific mortality curve (thin lines, solid to dashed), infant mortally changes due to shift in birth weight. Only panel (C) suggests that birth weight is on the “causal” pathway. See Wilcox’s website for additional details.

The procedure appears to be unbiased, and consistent. The method identifies significant heterogeneity, which influences birth weight specific infant mortality, and is consistent across populations. The CDD method will be applicable in many other settings where heterogeneity is present and cannot be identified by conventional methods. Applications with additional covariates could identify the ultimate causes of this heterogeneity.