# Interpreting the results of a binary logistic regression

This is a basic question. I have been handed a binary logistic regression. The model has significant terms, but the goodness of fit tests indicates the logit model is not appropriate. The author of the study indicates that the goodness of fit data does not invalidate the relationship between the dependent variable and the predictors, only the ability of the model to accurately predict outcomes. the argument is that since we were only interested in verifying a relationship and not the magnitude, the result is conclusive.

I'm skeptical of this. Wouldn't it be more appropriate to say that the lack of fit does not necessarily invalidate the relationships? With a different link function, couldn't the observed and expected counts change enough to move some insignificant term to significance or vice versa?

• Can you please share the sample size, the structure of the model (# of predictors/interactions, etc.) and how you concluded that the model does not fit? – Macro Feb 18 '13 at 16:23

One way to see this is to think of your logistic regression not as a model in its own right, but just an arbitrary data transformation. Suppose someone came along and handed you the responses and the binary predictions of the logistic regression (say thresholded at 0.5). You now just have a single binary predictor for a binary response - a $2 \times2$ contingency table. There is no "goodness-of-fit" to worry about - the only possible model is to $Y=X$ and the fit must be good since it is either right or wrong (all structure has been removed by construction!). However, by virtue of the fact that the original logistic regression was significant, it must be the case that the contingency table is significant, there $Y$ is related to $X$. Since $X$ is a function of your original predictor variables, it must also be the case that $Y$ is related to those original predictors.
• Your points are well taken but, if the model you've chosen is a very bad approximation, it becomes less clear what the parameter estimates even mean, beyond saying that they're the closest approximation to the data within this model space. For example, if the logistic model is a bad approximation, the interpretation of a logistic regression coefficient as the odds ratio associated with a one unit increase in the predictor (and that this odds ratio doesn't depend on $x$) could be way off. – Macro Feb 18 '13 at 16:40