Nice discussion!
With respect to your comment: "in a multivariable model by looking at the SE composite score (categorized into quartiles), type of medical insurance and race/ethnicity. In this model, they found that only private insurance was associated with an increase in the odds of having a prenatal diagnosis of CHD. I find this interesting and confusing, since all of these factors are related to one another"
the multivariable models provide effect estimates for each predictor variable after accounting for all other predictor variables. Thus, if multiple highly correlated predictors are included in the model, it is actually possible that none of them are significant, even if each of them would be when entered as the only predictors in the model. But in this case, private insurance may be the only predictor after conditioning on SE composite score and race/ethnicity if, for example, insurance fully mediates the effects of SES or race/ethnicity on the outcome. There are other explanations too, but this is a simple and plausible one.
Maria