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FORMAL MODEL

 

 

 

How the covariate density defined (CDD) mixtures of logistic regressions method is applied to control for heterogeneity in infant mortality:

The covariate density defined (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. 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.

Additional information can be found in the Working Papers section of http://www.albany.edu/csda/

This method can be applied to Wilcox's definition of causality (see http://eb.niehs.nih.gov/bwt/index.htm)

Equations to Describe Population Based Parametric Mixtures of Logistic Regressions

Joint density of birth weight and occurrence of death for the 2-subpopulation case

 

Birth-weight density:

 is the mixing proportion

is the Gaussian density with mean  and variance truncated at 0.

 

Probability of death conditioned on birth:

 

An infant of birth weight x in the ith sub-population has probability of dying given in quadratic logistic form:

 is the probability that this child is from sub-population 1 given it has birth weight x.

 

 

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