Intercept in Multiple logistic regression

Question

I've run a multivariable logistic regression on 8 variables and my results are a bit puzzling. The intercept (that is the log odds when the other covariates = 0) is significant (p<0.001), but the p-values of all my other covariates is non-significant.

My question for performing this regression is to answer the question: what variables predict the use of Tool X? The nonsignificant coefficients for all of the variables mean they don't predict the use, but it doesn't make sense that if you don't have these variables, it will suddenly be able to predict usage?

I don't think I have a strong grasp of multivariable logistic regression, so I would be very grateful if anyone can give some advice on this matter.

Answer 1

As you put it, the intercept is the log-odds when all covariate values equal 0. See this page, for example.

The standard test for "significance" of an intercept is whether its value differs significantly from 0. So what you have found is that the log-odds of using Tool X are different from 0--that is, odds are different from 1--when all covariates in your model have values of 0. So to that extent you can make a reasonable prediction about log-odds--not "without" those variables, but with their values all at 0. As the page linked in the previous paragraph notes, this is different from some "average" odds of using Tool X absent information about the covariates.

Note, however, that even variables with "nonsignificant" coefficients can be helpful in making predictions. This is even the case in standard linear regression, and it's particularly so in logistic regression with its inherent omitted-variable bias.

Answer 2

It can be illuminating to look at a logistic regression for, say, disease ($D$) with two dichotomous explanatory variables, say $E$ [1=exposed, 0=unexposed] and $G$ [1=female, 0=male].

If one considers : $p=pr(D) : logit(p) = \beta_0 +\beta_1 E + \beta_2 G + \beta_3 GE$, then $\beta_0$ is the log odds of disease for unexposed men and $\beta_1$ is the log odds ratio specific to men.

If however, one considers $logit(p) = \beta_0 +\beta_1 E + \beta_2 G$ then $\beta_0$ does not have a simple or useful interpretation. Here $\beta_1$ is the log odds ratio assumed common to men and woman.

Additive models [i.e. models without products of the explanatory variables] can have many caveats in the interpretations of the regression coefficients.

As well, the fit of your model with 8 explanatory variables may be facing issues of multicollinearity. Sometimes, it can be best to try a much simpler non-additive model to understand the issues at play.