SPSS command: SPSS QUANTILE REGRESSION

Quantile regression for medians and percentiles

Q: How is the median regression slope interpreted?

A slope of 1.4 at quantile 0.5 means the median of the outcome rises by 1.4 for a one-unit increase in the predictor, holding the others fixed. It says nothing about the mean.

Q: Should I tick the i.i.d. assumption?

Tick it only when the residual spread looks the same across the predictors. If slopes differ a lot between quantiles, the errors are not identically distributed, so untick it to get the Hendricks–Koenker sandwich standard errors.

Q: Bofinger or Hall-Sheather bandwidth?

Both set how wide a window of residuals is used to estimate the density at the quantile. Hall-Sheather gives a narrower window and is often preferred for confidence intervals; Bofinger is the panel default. Results usually differ little with moderate samples.

Q: Why is pseudo R² so low at extreme quantiles?

Pseudo R² compares the check-function loss with an intercept-only model at the same quantile. If a predictor barely moves that percentile, the model gains little there even when it matters elsewhere in the distribution.

Quantile regression models a chosen percentile of the outcome — the median, the 10th or the 90th — instead of the mean, so it is robust to outliers and shows whether predictors affect the low end, the middle and the high end differently.

Try with example data Run on your own file

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Worked example result

Parameter Estimates (quantile 0.1) · tenure

Coefficient = −0.055

p = .834 · Pseudo R Squared = 0.372

Parameter	Coefficient	Std. Error	Sig.
(Intercept)	36.405	1.457	<.001
tenure	−0.055	0.261	.834

Parameter Estimates (quantile 0.1)

See the full worked example

What it tells you

Crosstabs runs the SPSS QUANTILE REGRESSION procedure on your device for up to nine quantiles at once, with an exact linear-programming fit, i.i.d. or sandwich standard errors and pseudo R².

How each predictor shifts a given percentile of the outcome, with standard errors, t tests and confidence intervals.
Whether effects differ across the distribution — slopes that grow with the quantile mean the predictor widens the spread.
How well each quantile model fits: pseudo R² against the intercept-only fit at the same quantile and the mean absolute error.

When to use it

Use it when

Skewed outcomes such as income, spend, salaries or response times, where the median is more meaningful than the mean.
Outliers in the outcome that would pull an OLS line.
Questions about the tails, such as what drives the slowest 10% of delivery times or the top 10% of salaries.

Use something else when

You need the mean, for example to forecast totals: linear regression estimates the conditional mean.
Very small samples and extreme quantiles (such as 0.01 with 50 cases): there is too little data in the tail.
Outliers in the predictors rather than the outcome: quantile regression is robust to outlying Y, not to high-leverage X values.

How to run it in Crosstabs

01
Choose Quantile regression
Open your data in Statistics mode and, in the Regression group, pick Quantile regression (QUANTILE REGRESSION).
02
Set the target and quantiles
Choose the Target variable and type Quantiles (comma-separated) strictly between 0 and 1, for example 0.25, 0.5, 0.75 (up to nine).
03
Add predictors
Tick categorical predictors under Factors and numeric ones under Covariates.
04
Choose the standard errors and run
Under Estimation keep Include intercept, tick Assume errors are independent and identically distributed for sparsity-based SEs or untick it for the sandwich estimator, choose the Bandwidth (Bofinger or Hall-Sheather) and Confidence level (%), then click Run.

Run Quantile Regression in your browser

Worked example

Computed by the Crosstabs engine from the synthetic data below when this page was built — not typed in.

Try with example data

Data

120 synthetic employees with 0–20 years of tenure; salary (thousand) = 40 + 1.2·tenure plus noise whose spread widens with tenure (fixed seed). Target variable = salary, Covariates = tenure, Quantiles = 0.1, 0.5, 0.9, with the i.i.d. assumption unticked (Bofinger bandwidth).

Model Quality

Quantile	Pseudo R Squared	Mean Absolute Error
0.100	0.000	16.736
0.500	0.160	8.944
0.900	0.372	15.273

Parameter Estimates (quantile 0.1)

Parameter	Coefficient	Std. Error	Sig.	Lower Bound	Upper Bound
(Intercept)	36.405	1.457	<.001	33.519	39.291
tenure	−0.055	0.261	.834	−0.571	0.462

Parameter Estimates (quantile 0.5)

Parameter	Coefficient	Std. Error	Sig.	Lower Bound	Upper Bound
(Intercept)	38.819	1.550	<.001	35.751	41.888
tenure	1.361	0.193	<.001	0.979	1.744

Parameter Estimates (quantile 0.9)

Parameter	Coefficient	Std. Error	Sig.	Lower Bound	Upper Bound
(Intercept)	43.497	1.923	<.001	39.688	47.306
tenure	2.185	0.282	<.001	1.626	2.743

Standard errors allow non-identically distributed errors (Hendricks–Koenker sandwich, Bofinger bandwidth).

How to read it

Tenure barely moves the 10th percentile (slope −0.055, p = .834) but raises the median by 1.361 and the 90th percentile by 2.185 per year (p < .001). Slopes that grow across quantiles mean tenure widens the salary distribution as well as shifting it — something a single least-squares slope would average away. Because the spread changes with tenure, the standard errors use the non-i.i.d. (sandwich) estimator.

Output tables explained

Model Quality: For each quantile: pseudo R² (one minus the check-function objective relative to the intercept-only model) and the mean absolute error.
Parameter Estimates (quantile q): One table per quantile: coefficient, standard error, t, df, significance and confidence interval; the footnote names the standard-error method and bandwidth.

Assumptions and notes

The chosen quantile is a linear function of the predictors; the effects may differ between quantiles.
Observations are independent. The i.i.d. option additionally assumes the error distribution has the same shape for every case; untick it when the spread changes with the predictors.
Enough cases near each requested quantile for the sparsity (density) estimate behind the standard errors.

How the results are validated

Quantile regression is checked against R quantreg::rq with the Barrodale–Roberts method at τ = 0.25, 0.5 and 0.9 with a factor and a covariate: coefficients agree within 1e-8, i.i.d. standard errors (an R implementation of SPSS's sparsity formula with the Bofinger bandwidth) within 1e-8, Hendricks–Koenker sandwich standard errors with the Hall–Sheather bandwidth within 1e-5, and pseudo R² and mean absolute error within 1e-9. Validation evidence

Questions

How is the median regression slope interpreted?

A slope of 1.4 at quantile 0.5 means the median of the outcome rises by 1.4 for a one-unit increase in the predictor, holding the others fixed. It says nothing about the mean.

Should I tick the i.i.d. assumption?

Tick it only when the residual spread looks the same across the predictors. If slopes differ a lot between quantiles, the errors are not identically distributed, so untick it to get the Hendricks–Koenker sandwich standard errors.

Bofinger or Hall-Sheather bandwidth?

Both set how wide a window of residuals is used to estimate the density at the quantile. Hall-Sheather gives a narrower window and is often preferred for confidence intervals; Bofinger is the panel default. Results usually differ little with moderate samples.

Why is pseudo R² so low at extreme quantiles?

Pseudo R² compares the check-function loss with an intercept-only model at the same quantile. If a predictor barely moves that percentile, the model gains little there even when it matters elsewhere in the distribution.

Run Quantile Regression on your own data

Open a .sav, CSV or Excel file. Everything runs in your browser; the file never leaves your device.

Open the workspace

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