# Should a predictor, significant on its own but not with other predictors, be included in an overall multinomial logistic regression?

I constructed a model via multinominal logistic regression analysis. The final model contains three predictors. All predictors are significant when they are the only predictors. However, the coefficient of one of the predictors is not significant when included all three predictors are included in the model.

Should I include this predictor in the final multinomial logistic regression equation?

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have you checked the standard diagnostics for collinearity? –  Macro Sep 28 '11 at 2:32
This is almost the same as the question at stats.stackexchange.com/q/14500/919 which concerns the same phenomenon in multivariate regression generally. The answers there apply here, too. –  whuber Sep 28 '11 at 21:09

It depends whether you are doing... a) predictive research, where you don't care about what is causally responsible, only what serves as an efficient set of indicators, or b) explanatory research, where you want to disentangle causal relationships as much as you can.

In the latter, when multiple correlated predictors vie for a role in your equation, you would care about such things as giving "causal credit" to earlier factors over later ones, since what comes later could never cause what came before, but sometimes the reverse is true. You would care about giving more "credit" to relatively objective, relatively fixed variables such as marital status or ethnicity than to relatively subjective, changeable ones such as attitudes and opinions. And (and here I'm paraphrasing James Davis's The Logic of Causal Order) you would want to choose more generative factors such as socioeconomic status over less generative ones such as what brand of toothpaste a person uses.

When your candidate predictors are correlated, no statistical algorithm (such as a stepwise regression) can deal with these issues of explanation. It is up to you as a researcher to think through your candidate variables and choose those that will best serve your purpose. It is only in pure predictive research that you can ignore such issues and simply choose those predictors that account for the most variance in the outcome--or, in your case, produce the highest pseudo-r-squared.

Your question gets to the heart of important issues in multivariate modelling of many types, and if more than 5 tags were allowed I would have also listed multicollinearity, model-building, and/or variable selection.

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+1 I'd always wondered what the right/least-wrong way to do this is. –  Thomas Levine Sep 28 '11 at 4:09
I don't think that maximizing pseudo-r-squared is a good strategy, although i agree that model goodness-of-fitting is important to assess the model... –  Manoel Galdino Sep 28 '11 at 12:11

As @rolando2 mentioned, this depends very much on what your trying to accomplish or what question(s) you are trying to answer.

If you are trying to find a good model for prediction then rather than just deciding on whethere to include a term or not, it is better to use some type of shrinkage method such as penalized regression, ridge regression, lasso/lars, or model averaging.

You should also take into account outside knowledge about the variables. If my doctor had a choice of 2 predictive models to help in diagnosing me I would prefer that he use the one that uses blood pressure as a predictor rather than the one that uses the results from an exploratory surgery, even if it has a slightly smaller $R^2$ value.

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Surgery vs. blood pressure: nice example. –  rolando2 Sep 28 '11 at 21:14