Quantile Regression

Regression is a statistical method broadly used in quantitative modeling. Multiple linear regression is a basic and standard approach in which researchers use the values of several variables to explain or predict the mean values of a scale outcome. However, in many circumstances, we are more interested in the median, or an arbitrary quantile of the scale outcome.

Quantile regression models the relationship between a set of predictor (independent) variables and specific percentiles (or "quantiles") of a target (dependent) variable, most often the median. It has two main advantages over Ordinary Least Squares regression:

Quantile regression makes no assumptions about the distribution of the target variable.
Quantile regression tends to resist the influence of outlying observations

Quantile regression is widely used for researching in industries such as ecology, healthcare, and financial economics.

Example: What is the relationship between total household income and the proportion of income that is spent on food? Engel's law is an observation in economics stating that as income rises, the proportion of income spent on food falls, even if absolute expenditure on food rises. Applying quantile regression to these data, you can determine which food expense can cover 90% of families (for 100 families with a given income) when not interested in the mean food expense.
Statistics: Quantile Regression, Simplex approach, Frisch-Newton interior-point non-linear optimization algorithm, Barrodale and Roberts, Bofinger, Hall Sheather, bandwidth, significance level, matrix manipulations, convergence criterion, regression weights, intercept term, predicted target, prediction residuals, tabulation, prediction plots, parameter estimates, covariance matrix, correlation matrix, observed values, confidence interval.

Quantile Regression data considerations

Data: A single numeric dependent variable is required. The target variable needs to be a continuous variable. The predictors can be continuous variables or dummy variables for categorical predictors. Either the intercept term or at least one predictor is required to run an analysis.
Assumptions: Quantile regression does not make assumptions on the distribution of the target variable and resists the influence of outlying observations.
Related procedures: Quantile analysis is related to Ordinary Least Squares regression.

Obtaining a Quantile Regression analysis

This feature requires Custom Tables and Advanced Statistics.

From the menus choose:
Analyze > Regression > Quantile...

The Variables dialog allows you to specify the target, factor, covariate, and weight variables to use for quantile regression analysis. The dialog also provides the option of conserving memory for complex analysis or large datasets.
Select a numeric target variable. Only one target variable is required to run an analysis. Only numeric variables are allowed.
Optionally, select one or more factor variables. Scale variables are not allowed.
Optionally, select one or more covariate variables. String variables are not allowed.
Note: When both the Factor(s) and Covariate(s) lists are empty, and Include intercept in model is selected on the Model dialog, the following message displays:
```
No effects have been specified. Therefore, an intercept only model will be fit. 
Do you want to fit an intercept-only model?
```
Optionally, select a regression weight variable. String variables are not allowed.
Optionally, select Conserve memory for complex analysis or large datasets. This setting controls whether or not the data is held in an external file during processing. Enabling the setting can help conserve memory resources when running complex analyses, or analyses with large data sets.

This procedure pastes QUANTILE REGRESSION command syntax.