Is the logit function always the best for regression modeling of binary data?

I’ve been thinking about this problem. The usual logistic function for modeling binary data is:
log(p1p)=β0+β1X1+β2X2+
However is the logit function, which is an S-shaped curve, always the best for modeling the data? Maybe you have reason to believe your data does not follow the normal S-shaped curve but a different type of curve with domain (0,1).

Is there any research into this? Maybe you can model it as a probit function or something similar, but what if it is something else entirely? Could this lead to better estimation of the effects? Just a thought I had, and I wonder if there is any research into this.

Answer

People use all sorts of functions to keep their data between 0 and 1. The log-odds fall out naturally from the math when you derive the model (it’s called the “canonical link function”), but you’re absolutely free to experiment with other alternatives.

As Macro alluded to in his comment on your question, one common choice is a probit model, which uses the quantile function of a Gaussian instead of the logistic function. I’ve also heard good things about using the quantile function of a Student’s t distribution, although I’ve never tried it.

They all have the same basic S-shape, but they differ in how quickly they saturate at each end. Probit models approach 0 and 1 very quickly, which can be dangerous if the probabilities tend to be less extreme. t-based models can go either way, depending on how many degrees of freedom the t distribution has. Andrew Gelman says (in a mostly unrelated context) that t7 is roughly like the logistic curve. Lowering the degrees of freedom gives you fatter tails and a broader range of intermediate values in your regression. When the degrees of freedom go to infinity, you’re back to the probit model.

Hope this helps.

Edited to add: The discussion @Macro linked to is really excellent. I’d highly recommend reading through it if you’re interested in more detail.

Attribution
Source : Link , Question Author : Glen , Answer Author : Community

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