# How can I refute a monotonic correlation

I have a binary target T and a continuous variable I expect to correlate with it. I expect some monotonic correlation, as the explaining variable increases I expect P(T) to increase. I collect some (independent) samples and I see in some binning what appears like a dip in the measured P(T), how can I test if this is a sampling fluke or if my monotonic assumption is likely false. I'm considering using isotonic regression to convert my explaining variable to a probability estimate, but I'm wondering when I should not do this, maybe pre-select bins and average in them and not assume montonicity. Essentially I would like to assume monotonicity unless the data tells me this very unlikely.

## 2 Answers

First, if you have an "explaining" variable, then you should probably be thinking about regression.

Second, "in some binnings" makes me leery. Why are you binning a continuous variable at all, much less several different ways? This is almost always a bad idea.

Third, refuting a monotonic relationship should be only partly a statistical issue; it should also be substantive. Is there any practical reason for a non-monotonic relationship? Can you explain it in ways that are interesting and make senses?

Finally, you can use a spline regression and compare it to a linear regression and see whether you think the additional complexity of the spline model is worth the increase in fit. You could compare AICs, even though the models are not nested, although this is controversial (see this thread).

• What I have is actually a score from a model I don't trust too much, I normally would apply isotonic regression to convert the model score to probability and assuming montonicity this is optimal. But my model may have some artifact making this a bad idea. I can see in a specific binning, I can do a p-value for getting at least/most that score given the isotonic regression hypothesis, but that is limited to a specific bin. I'm looking for something better. – Meir Maor May 8 '17 at 11:18

I would model this using logistic regression, and definitely not binning! I will denote the continuous variable by $$x$$. So then we use a logistic model for $$\DeclareMathOperator{\P}{\mathbb{P}} \P\left\{ Y=1 \mid X=x\right\} = \frac1{1+e^{-\eta(x)}}$$ where $$\eta(x)$$ is some linear predictor. The "usual" logistic regression is $$\eta(x)=\beta^T x$$ but you could also use a spline function. An interesting idea, giving a non-parametric fit, is to use a spline representation. To force a monotone solution, you could use a monotone spline, see Looking for function to fit sigmoid-like curve. Then such a model could be compared with a fit with regression splines, as in the answer by @Peter Flom (and with the problems he mentions, as this would be non-nested models.)