What should I say and explain about a negative constant in binary logistic regression?

Question

I am finishing my undergraduate thesis and I was asked this problem of negative constant in my examination. So I am trying to find out how to explain what to say in regard to the negative constant. In my model, I have 15 predictors and one response.

And in binary logistic, the response is either 1 or 0. So the examiner insisted that the response should only be either 1 or 0, not negative. So why is the constant negative if all the predictors excluded? What should I say here? What is the problem? Much appreciated on the help.

Answer 1

I suspect your examiner was trying to trick you and/or test your knowledge of logistic regression. A negative intercept is relatively easy to interpreter as very low proportion of occurrences of the event of interest in the original sample in the absence of further influence from variables $X_1 \dots X_{15}$.

In general, with logistic regression you analyse the association of a binary outcome with a set of predictors. In particular you are modelling the $\log(\text{odds})$ of that particular outcome. The odds themselves are simply: $\text{odds} = \frac{\text{Pr(of Occurring)}}{\text{Pr(of Not Occurring)}} = \frac{\text{Pr(of Occurring)}}{1 - \text{Pr(of Occurring)}}$ where $\text{Pr}$ refers to proportions (or loosely speaking probability). From this later equality it follows that: $\text{Pr(of Occurring)} = \frac{\text{odds}}{1 + \text{odds}} $.

Now, particular to your case, a negative constant ($\beta_0$) simply means that the baseline proportion of your sample is quite low: $\exp(-4.587) / (1+ \exp(-4.587)) \approx 0.01008.$ This not catastrophic; maybe you have not centred your variable $X_i$ for example, this commonly leads to this phenomenon, but in case you need to be able to explain why your baseline is so low. If you truly have a very low occurrence of events in the original sample you may want/need to consider rare-event logistic regression (see King & Zeng's 2001 paper on Logistic regression in rare events data for a first taste).

As a quick step-through though to find the change in terms of the proportions that are modelled you need to:

1. Get the $\log(\text{odds})$ estimate.
2. Exponentiate it to get the $\text{odds}$.
3. Get the new proportions as: $\text{Pr}_{\text{new}} = \frac{\text{odds}}{1 + \text{odds}} $.

As a final comment: the statistical significance of parameters included the model you present seems relatively low so giving a solid reason as to why you included them is crucial. I am against $p$-value hunting -which is a bad thing- but a 15-variable model with not a single very strongly ($p \leq1e^{-3}$) statistical significant variable seems a bit awkward at first glance.

The user @gung has given a very good answer on the matter too here.