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?

The only thing wrong with this is your conclusion "we would have to predict an outcome of 0 for all cases". Logistic regression is not a classifier; it is a regression model that produces predicted probabilities. The predicted probabilities for several values of your predictor are close to .5, and that is borne out in the fact that at those levels of the predictor, there are approximately equal 0s and 1s in the outcome.
It is the case that, if you create a classifier by adding a decision rule that any probability less than .5 leads you to classify the case as a 0, you should predict all 0s for this dataset, but that's exactly the problem with using such a strict decision rule with no regard for balancing false positives and false negatives.
Also, there is nothing inherently wrong with the shape of the curve. A nearly flat curve in the domain of the variable does indeed mean the predictor is not a great predictor of the outcome, but that doesn't mean the logistic curve is not the right model to use. You can always try other models and compare them on some metric; if you think there is a pattern that the logistic model is missing, you can try a machine learning model that more flexibly adapts to nonlinear relationships. That said, nothing about this problem has to do with the restriction of the range of the predictor. The range simply is what it is; it is not restricted in any sense.
