# Interpretation of the coefficients in quantile regression - a discrepancy between sources

In various sources you can find, that interpretation of the quantile regression is pretty much like in the linear regression, with the difference now it's about the medians, rather than means. Like here: How can I interpret the categorical variables in quantile regression?

But then I found this article: "Quantile Regression in the Study of Developmental Sciences" https://srcd.onlinelibrary.wiley.com/doi/abs/10.1111/cdev.12190 where it's said, that:

For example, if we have an equation for a linear regression as, Yi = 2.5 + 1.2(X) it is expected that Y will increase by 1.2 units for each unit increase in X. An alternative way to construct the interpretation is in terms of how much gap exists in Y when considering different values of X. To illustrate, it could also be stated that the X coefficient of 1.2 reflects the gap in estimated performance at the mean of Y for a student who is average on X compared to an individual who is 1 SD above the mean on X. Although this is a bit more complex than the traditional interpretation of a slope, it serves as a foundation to understand how the slope relates to the outcome in quantile regression. Now suppose we have two equations from a quantile regression that estimated the association between X and Y at the .25 and .75 quantiles:

.25 quantile: Yi = 0.43 + 2.5(X)
.75 quantile: Yi = 1.57 + 0.82(X)


The interpretation of the slope coefficient for the .25 quantile (i.e., 2.5) is best stated as the gap in performance on Y at the .25 quantile for individuals who were average on X compared to individuals who were 1 SD above the mean on X was 2.5 units. Similarly, the interpretation of the slope at the .75 quantile (0.82) is stated as: the gap in performance on Y at the .75 quantile for individuals who were average on X compared to those who were 1 SD above the mean on X was 0.82 units. This aspect of slope interpretation in quantile regression is necessary as it assists in avoiding confusion about the relations between the model coefficients in the analysis. It is tempting to use the traditional linear regression interpretation of a slope; however, this would potentially mislead an individual to think that at the .25 quantile, increasing the X by 1 unit leads to a predicted score of Y, which increases by 2.5 units. By increasing a score in Y by 2.5 units, an individual would no longer be at the .25 quantile, but would be at a higher quantile. Consequently, it is more appropriate to think and describe slope coefficients as reflective of gap performances in Y based on differences in performance on X.

This is out of my mind yet... Please, can anybody tell me, if this interpretation matches somehow those given by Peter Flom?

It just means to watch out about thinking that it results in an increase in average. The average may not change...but that quantile of the response distribution changes.

Remember that $$\hat{y}_i = \beta^T X_i$$ means that $$\mathbb{E}\big{[} Y_i \vert X_i \big{]} = \beta^T X_i$$. "Regular" regression is about the expected value of some distribution.

But there is a whole distribution of the response variable.

Yes, if the quantile regression is $$2.5 + 1.2(X)$$, then for every one-unit increase in $$X$$, we expect an increase of $$1.2$$ in the specified quantile of the response distribution.

• Thank you, Sir for the answer. It clarifies it very well. Mar 3, 2020 at 21:35
• To be precise, it is not quite true that $\hat{y}_i = \beta^T X_i$ means that $\mathbb{E}\big{[} Y_i \vert X_i \big{]} = \beta^T X_i$, though $\mathbb{E}\big{[} Y_i \vert X_i \big{]} = \beta^T X_i$ implies that $\hat{y}_i = \beta^T X_i$ is optimal under square loss. Mar 4, 2020 at 7:39

To illustrate, it could also be stated that the X coefficient of 1.2 reflects the gap in estimated performance at the mean of Y for a student who is average on X compared to an individual who is 1 SD above the mean on X.

This might be a point of confusion. Typically, you would interpret the 1.2 as indicating the expected change in Y when X changes by 1 unit, not by 1 standard deviation. This author seems to be using units of X and standard deviations interchangeably. Standardizing your variables so that they have a mean of 0 and a standard deviation of 1 is a common practice in some areas of social science. If the authors had done this with their X variable, then this interpretation makes more sense because a 1 unit change would equal a 1 standard deviation change.

• This raises a valid point, but the rest of the article likely gives the context that $X$ has been standardized. The confusion about what the coefficients mean remains even if the predictors are not standardized.
– Dave
Mar 3, 2020 at 21:06
• Thank you, Sir for the answer. This makes a lot of sense, I admit, I did not think this way. Mar 3, 2020 at 21:35