Quantile Regression: follow up methods to have a more fine-grained understanding of what the results really mean?

Question

I recently employed multiple quantile regression in my area of research and found some interesting quantile differences across the distribution of Y, but I don't quite understand what they all really mean. Unlike the traditional methods such as dividing the sample into multiple groups where I have access to the groups' data on various variables which then allows me to make sense of, for example, why the correlation between X and Y is 0 for group 1 and .7 for group 2, I feel like I have no idea where those quantile regression estimates come from, especially when there are more than 2 predictors in the QR model. Another way of putting this is I don't know which specific data points contribute heavily to a given quantile regression estimate and so this makes it very difficult for me to understand what the quantile differences really mean.

Based on my understanding of QR, it uses all the data points in the full sample but weights the data points that are farther from a quantile of interest less heavily than the data points that are closer to that same quantile of interest, is this correct? If so, as a follow up, can I divide my full sample into 10 groups, e.g., 10th quantile, 20th quantile, 30th quantile group, and then examine how the 10 groups differ on various variables of interest in order to make sense of the 10 quantile regression estimates that I got? I know the subgroups approach is not ideal, which is why I used QR, but if you think this is a terrible idea, please let me know why. And if you know of any other methods that allow me to have a more fine-grained understanding of my results, please help. I conducted QR using the quantreg package in R.

Answer 1

You have alot of questions here, I will attempt to answer some of them to the best of my understanding. In my opinion the best way to understand what quantile regression is doing is to consider what regular linear regression does and compare it to quantile regression on the median (special case). Regular linear regression asks how does X affect Y on average (mean), while median quantile regression asks how does X affect median of Y. This is initially how quantile regression was developed, and it was later extended to any quantile of interest. So lets say im interested in the 10th percentile and the 90th percentile, quantile regression allows to me ask, how does the variable X affect the position (or estimate or value) of the 10th percentile of Y and the 90th percentile of Y.

So quantile regression is not weighting your samples, it is simply estimating the effect of X on Y, but rather than the effect operating on the mean of Y (as in regular linear regression) it is estimating the effect of X on the quantile (eg 10th or 90th) of Y.

The main problem with subgroup analysis is the dramatic reduction in power, you are basically reducing sample size by considering smaller and smaller groups. If you have extremely large sample sizes and/or large effect sizes, then subgroup analysis is not so bad, but if not you will not have to the power to detect effects in the subgroups.

Answer 2

QR does not weigh in the data points differently. Here is how I would think about QR:

In the QR, quantiles are defined on the response variable (y).

What QR tells you is the effect of your predictor variable (X) at different quantiles of your response (Y) variable. Consider the following regression (Don’t worry too much about other things that might come into play in this regression):

Wage = b0 +b1*age+……

In a linear regression setup the coefficient on age tells you the effect of age on salary. Whereas, in the QR setup you will get to see the effect of age on different quantiles of the response variable.

Ex: QR will tell you whether age has a different effect on the top 10% of the wage earners than the bottom 10% of the wage earners