Definitions

Quantile regression

Quantile regression is a type of regression analysis used in statistics. Whereas the method of least squares results in estimates that approximate the conditional mean of the response variable given certain values of the predictor variables, quantile regression results in estimates approximating either the median or other quantiles of the response variable.

Advantages and applications

Quantile regression is used when an estimate of the various quantiles (such as the median) of a population is desired. One advantage of using quantile regression to estimate the median, rather than ordinary least squares regression to estimate the mean, is that quantile regression will be more robust in response to large outliers. Quantile regression can be seen as a natural analogue in regression analysis to the practice of using different measures of central tendency and statistical dispersion to obtain a more comprehensive and robust analysis. Another advantage to quantile regression is the fact that any quantile can be estimated.

In ecology, quantile regression has been proposed and used as a way to discover more useful predictive relationships between variables in cases where there is no relationship or only a weak relationship between the means of such variables. The need for and success of quantile regression in ecology has been attributed to the complexity of interactions between different factors leading to data with unequal variation of one variable for different ranges of another variable.

Mathematics

The mathematical forms arising from quantile regression are distinctly different from those arising in the method of least squares. The method of least squares leads to a consideration of problems in an inner product space, involving projection onto subspaces, and thus the problem of minimizing the squared errors can be reduced to a problem in numerical linear algebra. Quantile regression does not have this structure, and instead leads to problems in linear programming that can be solved by the simplex method. The fact that the algorithms of linear programming appear more esoteric to users may explain why quantile regression is not as widely used as the method of least squares..

Implementations

Some statistics packages, such as Stata, gretl and R, include implementations of quantile regression.

References

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