Quantile regression (median and other quantiles)

In brief

Quantile regression does not predict the mean but the median or another quantile of a number (for example the lower and the upper income quarter depending on education). It is chosen under "More options" → "Model" in the question "Can something be predicted?"; it applies to all subjects and to a target with numbers.

What for

It shows whether the relationship behaves differently at the upper and lower edge of the distribution than in the middle. It is robust against outliers in the target and against skewed distributions and assumes no equal spread. It gives no protection against outliers in the predictors.

Requirements

• Target: a number with more than two values; at least one predictor; at least 20 cases and five per coefficient. With fewer than 50 cases the intervals are uncertain.
• The closer τ is to 0 or 1, the fewer cases determine the line. If fewer cases lie at the edge than the line has coefficients, τ is blocked.
• If the target has only a few distinct values (a scale with few steps), many quantiles coincide; Statable points this out.
• Quantile regression does not work with a case weight: the intervals would be wrong.

Reading the result

• b per τ: the change of the conditional quantile per unit of the predictor. "Conditional" means: for people with the same value of the predictor; a person does not stand "in the 75th percentile", but the 75% limit belongs to all people with that value.
• Intervals: up to 1000 cases the case bootstrap (2000 draws, fixed seed, percentile interval, 95%), above that the nid approximation. The bootstrap gives no p; the interval shows whether zero lies in it. On request the rank inversion (asymmetric, without standard error).
• Other programs often show a 90% interval with equal spread (default of summary.rq in R). Statable sets 95% and unequal spread explicitly.
• The estimates are the same as in SPSS and R (quantreg). Standard errors and intervals can differ from SPSS: SPSS estimates them via the density (IID or NID), Statable via bootstrap, the nid approximation or rank inversion.
• The ordinary regression (mean) is shown below for comparison. Quantile regression has no R²; the R² of the ordinary regression belongs to it only.
• If the lines of different quantiles cross, a sentence says so: that is impossible and shows the estimate is uncertain there (usually at the edge of the data). Statable sorts nothing quietly.
• With an even n the median of the regression is not always unique; R then names a corner of the solution. Statable says so in a sentence.

If the intervals of different τ overlap, that neither shows nor rules out a difference of the slopes. This comparison is not a test of equal slopes.

Example (Engel data, Koenker and Bassett 1982, 235 households): food expenditure from income. At the median expenditure rises by 0.56 per unit of income, at the lowest twentieth by 0.34 and at the highest by 0.71.

Reporting the result

Quantile regression (median), n = 235; 95% bootstrap interval: income: b = 0.56 [0.47, 0.61]

In the text, name the quantile, the kind of interval and the number of cases. Do not word it causally.