Quantile Regression — β(τ) at requested quantiles (median + tails)

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Advanced & specialized

This test is implemented and is currently going through StatMinds’ production verification: every statistic is independently checked against a trusted reference (scipy / R), locked with regression tests, and the screen is exercised across assumption-met/violated and significant/non-significant scenarios before it opens up.

See what’s live now

Estimates conditional QUANTILES of a continuous outcome instead of the conditional mean.

Each requested quantile τ ∈ (0, 1) gets its own coefficient vector β(τ); together they describe how the entire conditional distribution depends on the predictors. Reports per-quantile coefficients (β, SE, t, p, 95% CI), Koenker-Machado pseudo-R¹ per quantile, the median (τ = 0.5) row as the headline outlier-robust regression line, and the tail-difference β(0.9) − β(0.1) per predictor as a heterogeneity-of-effect summary. Naturally robust to y-outliers and equivariant to monotone transformations of y.

Worked example

Does study time help low-scoring students as much as high-scoring ones?

Exam score is modelled at the 10th, 50th and 90th percentiles (quantile regression) rather than only at the mean, to see if the effect varies across the distribution.

Result

The study-time slope grew across the distribution — 10th percentile b = 1.5, 50th b = 2.1, 90th b = 2.4 — study time mattered more for higher-scoring students.

How you'd report it (APA)

Quantile regression showed the study-time effect was larger at the 90th percentile (b = 2.4) than the 10th (b = 1.5), so it benefits higher-scoring students more.

Try it yourself: Load this ready-made sample and follow the run above.

When to use it

  • Outcome where the EFFECT differs across the distribution
    Wage ~ education + experience evaluated at τ ∈ {0.1, 0.25, 0.5, 0.75, 0.9}.
  • Median regression as outlier-robust OLS alternative
    Income (right-skewed) ~ education + experience.

When NOT to — use instead

Hypotheses

H₀: β_j(τ) = 0 at the specified τ for each predictor j (per-coefficient tests). Joint H₀: β_j(τ) is constant across τ ∈ requested set (heterogeneity test).
Hₐ: at least one β_j(τ) ≠ 0 OR β_j(τ) varies across τ.

Parameter tested: vector of conditional-quantile coefficients β(τ)

Assumptions (and what to do if they fail)

Extreme quantiles (τ ≤ 0.1 or τ ≥ 0.9) need a larger sample. Rule of thumb: n × min(τ, 1−τ) ≥ 30 per requested quantile.medium

Check: min(n × τ, n × (1 − τ)) per quantile.

If violated: Extreme-quantile coefficients can be highly unstable with small effective sample sizes.

Quantile regression is naturally ROBUST to y-outliers (the loss function is L1-based), but high-leverage x-outliers can still distort fits.low

Check: Max leverage in the OLS comparison.

If violated: y-outliers are well handled by QR; x-outliers (high-leverage) can still bias coefficients at all quantiles.

VIF for each predictor in the OLS comparison is moderate (< 5 preferred; < 10 tolerable). Multicollinearity affects QR SEs the same way it affects OLS.medium

Check: Max VIF across predictors.

If violated: High VIF inflates β(τ) standard errors at every quantile.

Estimated conditional quantiles Q̂_τ should be NON-DECREASING in τ at every x. When fit per quantile (no monotonicity constraint) crossings can occur, especially in the tails or with small n.medium

Check: Compare Q̂_τ at the mean predictor vector across τ.

If violated: Crossings indicate the per-quantile linear specification is too restrictive; the predictor effect changes very fast across τ.

Ready to run a Quantile Regression — β(τ) at requested quantiles (median + tails) on your own data?

Guided setup, automatic assumption checks, effect sizes, figures and an APA write-up.

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