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I'm not very clear with binary quantile regression. As if it was ordinary quantile regression, it would divide the dependent variable's value by its ascending value into quantiles.

But I cannot imagine how it divides $y \{0;1\}$ value into quantiles. Can you explain it to me?

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Binary quantile regression does not actually classify

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It classifies some

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I.e. the (predicted) probability that the target variable (regressand, dependent variable) is 1 (or alternatively 0) based on the input variable(s) (regressor(s), independent variable(s)). Details depend greatly on the method, but one can see how a median predicted value of the variable would allow for a quantile distribution.

Further Reading

Intro To Binary Quantile Regression Discussion

Recent Binary Quantile Regression Method Research

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  • $\begingroup$ But isn't $P(Y|X) = E(Y|X)$ for a binary variable, so how does it differ from a regular GLM on a binary variable? $\endgroup$
    – Andrew M
    Dec 13, 2014 at 1:41
  • $\begingroup$ Yes it does. Depending on the method, a nonspecific binary quantile regression could simply be quantiles based on the p(y|x) = e(y|x). There are deeper methods for which this would not suffice as an explanation. $\endgroup$
    – Thomas Cleberg
    Dec 13, 2014 at 2:42
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    $\begingroup$ I looked at those references and don't think they are useful for this problem. The first reference doesn't even use the word "quantile". $\endgroup$
    – Frank Harrell
    Dec 13, 2014 at 11:34
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I don't think that "binary" and "quantile" should be used in the same sentence. Quantile regression requires not only multiple levels of $Y$ but that $Y$ be continuous. Sticking with ordinary logistic regression is more appropriate.

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Imagine an unobserved random variable Z with a logistic density with mean E[Z| X] where X may be a vector of covariates. We may ask for a predictor of the quantile of Z given predictors X.

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  • $\begingroup$ I do not think this answers the question. The binary variable is really not numeric so there iis only a two point distribution. If you designate the outcome 0 for failure and 1 for success. ordering 0<1 and say P(0)=.30 and P(1)=.70 then 0 becomes the 30th quantile and 1 is the 100th quantile. $\endgroup$
    – Michael R. Chernick
    Jan 12, 2017 at 19:28
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Quantiles are invariant to monotone transformations. Binary quantile regression uses the fact that the indicator function is monotone. You actually model the conditional quantiles of the underlying latent continuous variable of which the observed binary is just an indicator.

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  • $\begingroup$ I think you mean equivariant or more simply that there is a commutative property, so that e.g. the median of a logarithm is in principle the logarithm of a median, but the reason is not that the logarithm of x is equal to x, because it never is. $\endgroup$
    – Nick Cox
    Sep 22, 2020 at 12:29

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