Problem with same independent variable(s) in two separate Negative Binomial Regression models

[Tobias Dahl]

Hi, I´m running negative binomial regressions on two models. Model 1 is made on five independent variables (IV), and model 2 of two IV. Four of the five IV in model 1 are actually made from one of the IV in model 2; one IV in model 2 are the total profit (four sub-profits combined), and the four IV in model 1 is those four parts which in total gives the one IV in model 2. The fifth IV in model 1, and the second IV in model 2, are the same. The thing is thus that; when running the same IV (lets call it 'migration') in respectively model 1 (up against the four parts) and in model 2 (up against the total of the four parts) then I would expect to get the same p-value for 'migration', as it is the same in both models, which further should be two ways of running the same, but instead the 'migration' IV is significant in one of the models, but not in the other. I would think the p-value for the 'migration' IV should be the same - at least either significant or not in both...!? I would like to use both models, to thus give respectively a kind of macro and micro perspective on same factors, although I this way find it hard to conclude on 'migration' being significant or not... Can someone please help me understand what I do not seem to quite get in my tired head...? (Please look at the attached pictures). Despite it is a bit complex to write in few words, I hope I made it understandable what I mean...!?

[Clyde Schechter]

There is nothing wrong with what you've done. What's wrong is your expectation that the findings in both models should be the same for Migration. In the 5-variable model, there are opportunities for the various profit subtype variables to steal (share, if you prefer) variance with Migration. But in the second model, those distinct profits are simply summed into a single variable and the distinctive association of one (or more) of them with Migration gets washed out by the summation. There is really no reason to expect the findings for Migration to remain the same. In fact they could be quite dissimilar, and even be of opposite signs.

[Clyde Schechter]

Well, first of all, don't get hung up on "significant" vs "non-significant." The p < 0.05 cutoff is just an arbitrary threshold on what is in fact a continuous measure of compatibility of the data with having been generated by a particular random number generator. Your second model clearly shows a large association (8 is a pretty large coefficient in a negative binomial model, unless the units of the Migration variable make its values very, very small) between your migration measure and this outcome. However, when the various drug profits are introduced into the model, much of the explanatory power of Migration evaporates. While a coefficient of 3 in a negative binomial model is nothing to sneeze at (even if it's not "significant"), it's clearly a great deal smaller than 8. So one or more of the drug profit variables seems to account for the apparent importance of Migration in the simpler model. You might want to figure out which of the drug profit variables is doing the work here. You might be able to see it by just looking at the correlation matrix among the drug variables and the migration variable. If that doesn't make it clear, try other models where you eliminate the drug variables one at a time, and see which one's elimination leads to a jump up in the migration coefficient. If none of them do, try eliminating pairs of the variables. The story that the apparent effects of migration are largely accounted for by specific drug profits may well be more interesting and important than an artificial "up or down" verdict on the migration variable itself.

[Clyde Schechter]

Well, it is not possible to state that one variable is more strongly related to an outcome than another predictor is, except under very unusual conditions that almost never happen in real life. But what you can say is that there is a strong association between the migration and heroin profit variables themselves, and both are related strongly to the DV, so that we have confounding of the relationships. There is, if you will, shared variance between the migration and heroin profit variables, so that while migration, in a crude analysis, is strongly associated with DV, when the analysis is adjusted for heroin profits, heroin itself becomes a strong predictor and the association of the DV with migration decreases sharply.

Putting it in simpler but approximate language, in the model where only total profit is considered, the migration variable "speaks for" both itself and the heroin profits (with which it is strongly correlated, but whose influence is muted by being wrapped up with other, not strongly related, sources of profit). Once you "take the gag off" of heroin and let it speak for itself, it becomes the driver of the DV, and migration's adjusted association to the DV is much reduced.