13 Research methods, statistics, critical review and evidence-based practice13.10 Correlation and regression
13.10.17 Quantile regression
2 min read1 figure
Written by Resident Doctor Committee · Published
New in the 2027 syllabus
The College added this topic for exams from January 2027.
Beyond the average
Ordinary least-squares regression models the conditional mean. This is the average outcome for people with a given set of predictor values. Because it only looks at the average, it can miss differences that sit mainly in the upper or lower part of the outcome distribution.
Quantile regression models a chosen quantile of the outcome instead, again conditional on the predictors. Common choices are the median and the 90th percentile.
Median regression is quantile regression at the 50th percentile. It minimises absolute residuals rather than squared residuals.
Target | What it describes |
|---|---|
25th percentile | Lower part of the outcome distribution, given the predictors |
50th percentile | Median outcome, given the predictors |
90th percentile | Upper part of the outcome distribution, given the predictors |
Worked example
Suppose a hypothetical service model gives these equations for waiting time, with severity measured in points.
Quantile | Fitted waiting days |
|---|---|
Median | 10 + 2 x severity |
90th percentile | 20 + 5 x severity |
At severity 3, the predicted median is 10 + 2 x 3 = 16 days.
The predicted 90th percentile is 20 + 5 x 3 = 35 days.
Each one-point rise in severity goes with 2 more days at the median and 5 more days at the 90th percentile, assuming the model holds.
Interpretation and traps
Slopes describe groups, not individuals. A slope at a given quantile compares outcome distributions between people with different predictor values. It does not follow one person as their predictor changes.
Skewed outcomes can make the median and the tails useful things to model, but skewness alone does not decide the best model.
Extreme quantiles need enough observations in that tail, and their estimates are often less certain.
Crossing lines can happen when fitted quantiles give an impossible order, so check the predictions together.
Dependent observations still need suitable methods for inference. Quantile regression does not remove clustering or repeated-measures dependence.
Causal claims need further design assumptions, just as they do for mean regression.