performing multivariate binary regression when the independent variables are subset to other

Question

say i have a data of cancer patients who have fever and i want to see what factors are associated with mortality. After performing univariate analysis (crosstab in SPSS), i have 3 factors with p-value <0.05; allso=cause of fever found, hc=cause of fever was blood infection, pneu=cause of fever was lung infection. (all are binary, value 1 indicates presence of cause) Among the 3 variables "allso" is like a big set while "hc" and "pneu" are subset in "allso". "hc" and "pneu" have some overlapping case. ( allso = hc + pneu - overlap_between_hc_and_pneu + other )

Question

1. After perform multivariate binary regression, only "allso" remains significant. It is correct for me to conclude that if a cause of fever was found, the patient will be more likely to die irrespective of what the cause is? (which make biological sense because identifiable cause may means a great load of bacteria)

2. Just for understanding, i perform multivariate regression with only "hc" and "pneu" as independent variable and the result: none of them are significant. Could you explain what happens in this situation i really don't get it.

thanks, i have tried to find a similar question but failed

Answer 1

1. Recognize that in your multiple regression ("multivariate" probably is best restricted to cases with multiple outcome variables), the influence of each independent variable is calculated with the influences of the other independent variables taken into account. So a better way to put the results would be something like "the probability of dying within the time frame under consideration is greater if a cause of fever was found, when (insert your list of variables here) were taken into account." In your case the finding of univariate significance makes it possible not to include that qualifier, but in medical studies you almost always want to correct for other potential influences, and let others know that you have done so. This problem might be better handled by survival analysis rather than logistic regression, given that we all know the probability of dying.

2. I would guess that you have a case of what's called "multicollinearity," when multiple independent variables are correlated. This comes from the overlap between hc and pneu. Heuristically, when you try to take each variable into account in the multiple regression, it's hard to know which one should get credit for the effect, making it hard to show "significance" for either. This also makes it hard to conclude that your results hold "irrespective of what the cause is"; the structure of your data might not allow you to get a good handle on that issue, as both of those variables are correlated to allso.