Regression of binomial response when the predictor range is limited

I am looking into a dataset for which I will be doing a regression. When considering the option of a logistic regression, I started doing univariate regressions to get a feeling for the possible relationship of my predictors, and happened across a case I haven't seen before:

The predictor is a positive rational number - there can never be observations that are below zero - and for practical purposes, we could assume it won't get over 10. You can see a bubble plot of the actual observations, on the 0 and 1 horizontal lines.

But of course, the logistic function is not restricted to that range - and it so happens, that with the current dataset, the predicted line crosses the zero line at below 50% probability.

I suppose that a very literal interpretation would be that the variable is a poor predictor (at least when taken alone) and if we had to stay with this model, we would have to predict an outcome of 0 for all cases. But I am unhappy with that idea, not only because from a domain point of view, the variable is interesting, but also because mathematically, we are trying to fit to a function of the wrong shape.

Is there a subtype of logistic regression, or maybe a completely different type of regression, that deals properly with a restricted domain of the link?

• How did you get that logistic curve if none of your data for var1 are below 0? Commented May 17, 2022 at 16:27
• @jbowman The curve is for illustration only - what the model gives are coefficients for the intercept and the predictor, so I calculated the predicted values of equidistant points between -6 and +6 to draw the fit graphically. Commented May 17, 2022 at 16:29
• I really fail to see what your problem is. 1. Just because your predictor variable $x\geq 0$ doesn't mean that $a + bx\geq 0$, after all, unless your parameter estimates are both $\geq 0$ as well. 2. A very literal interpretation would be that over the range of the predictor variable it's pretty much always the case that the probability of the event $< 0.5$. That's certainly possible! Commented May 17, 2022 at 16:44
• @jbowman yes, I know it is possible. But we should not forget that there is no actual logistic process driving the occurrence of the event; we use a logistic regression for binary outcomes, because it is mathematically workable and well-studied. I am pretty sure that there is a relationship between the predictor and the outcome, both because it is observable in daily life and because the logistic regression shows it too, with a very significant coefficient of -0.715; so there is a good chance that there is more information in that predictor, that can be teased out with better methods. Commented May 18, 2022 at 6:34
• The bands around your predicted curve appear extremely deceptive, because to the left of zero they are gross extrapolations. You shouldn't be making predictions in this region at all--and even if, for the sake of illustration, you do, then you shouldn't draw confidence or prediction bands, because they are meaningless. I don't follow your reasoning at "if we had to stay with this model, we would have to predict an outcome of 0 for all cases:" nothing in what you show suggests this and everything suggests the opposite: the predictions will be low but nonzero probabilities.
– whuber
Commented May 18, 2022 at 19:37