\name{rq} \alias{rq} \title{ Quantile Regression } \description{ Returns an object of class \code{"rq"} or \code{"rq.process"} that represents a quantile regression fit. } \usage{ rq(formula, tau=.5, data, weights, na.action, method="br", contrasts, \dots) } \arguments{ \item{formula}{ a formula object, with the response on the left of a \code{~} operator, and the terms, separated by \code{+} operators, on the right. } \item{tau}{ the quantile to be estimated, this is generally a number between 0 and 1, but if specified outside this range, it is presumed that the solutions for all values of \code{tau} in (0,1) are desired. In the former case an object of class \code{"rq"} is returned, in the latter, an object of class \code{"rq.process"} is returned. } \item{data}{ a data.frame in which to interpret the variables named in the formula, or in the subset and the weights argument. If this is missing, then the variables in the formula should be on the search list. This may also be a single number to handle some special cases -- see below for details. } \item{weights}{ vector of observation weights; if supplied, the algorithm fits to minimize the sum of the weights multiplied into the absolute residuals. The length of weights must be the same as the number of observations. The weights must be nonnegative and it is strongly recommended that they be strictly positive, since zero weights are ambiguous. } \item{na.action}{ a function to filter missing data. This is applied to the model.frame after any subset argument has been used. The default (with \code{na.fail}) is to create an error if any missing values are found. A possible alternative is \code{na.omit}, which deletes observations that contain one or more missing values. } \item{method}{ the algorithmic method used to compute the fit. There are currently three options: The default method is the modified version of the Barrodale and Roberts algorithm for \eqn{l_1}{l1}-regression, used by \code{l1fit} in S, and is described in detail in Koenker and d'Orey(1987, 1994), default = \code{"br"}. This is quite efficient for problems up to several thousand observations, and may be used to compute the full quantile regression process. It also implements a scheme for computing confidence intervals for the estimated parameters, based on inversion of a rank test described in Koenker(1994). For larger problems it is advantagous to use the Frisch--Newton interior point method \code{"fn"}. And very large problems one can use the Frisch-=Newton approach after preprocessing \code{"pfn"}. Both of the latter methods are described in detail in Portnoy and Koenker(1997). } \item{contrasts}{ a list giving contrasts for some or all of the factors default = \code{NULL} appearing in the model formula. The elements of the list should have the same name as the variable and should be either a contrast matrix (specifically, any full-rank matrix with as many rows as there are levels in the factor), or else a function to compute such a matrix given the number of levels. } \item{...}{ additional arguments for the fitting routines (see \code{\link{rq.fit.br}} and \code{\link{rq.fit.fn}} and the functions they call). } } \value{ See \code{\link{rq.object}} and \code{\link{rq.process.object}} for details. } \examples{ data(stackloss) rq(stack.loss~stack.x,.5) #median (l1) regression fit for the stackloss data. rq(stack.loss~stack.x,.25) #the 1st quartile, #note that 8 of the 21 points lie exactly on this plane in 4-space rq(stack.loss~stack.x,tau=-1) #this returns the full rq process rq(rnorm(50)~1,ci=F) #ordinary sample median --no rank inversion ci rq(rnorm(50)~1,weights=runif(50),ci=F) #weighted sample median } \section{Method}{ The function computes an estimate on the tau-th conditional quantile function of the response, given the covariates, as specified by the formula argument. Like \code{lm()}, the function presumes a linear specification for the quantile regression model, i.e. that the formula defines a model that is linear in parameters. For non-linear quantile regression see the function \code{nlrq()}. [To appear real soon now on a screen near you.] The function minimizes a weighted sum of absolute residuals that can be formulated as a linear programming problem. As noted above, there are three different algorithms that can be chosen depending on problem size and other characteristics. For moderate sized problems (n << 5,000, p << 20) it is recommended that the default \code{"br"} method be used. There are several choices of methods for computing confidence intervals and associated test statistics. Using \code{"br"} the default approach produces confidence intervals for each of the estimated model parameters based on inversion of a rank test. See the documentation for \code{\link{rq.fit.br}} for further details and options. For larger problems, the \code{"fn"} and \code{"pfn"} are preferred, and there are several methods of computing standard errors and associated test statistics described in the help files for \code{\link{rq.fit.fn}}, and \code{\link{summary.rq}}. } \keyword{regression} \references{ [1] Koenker, R. W. and Bassett, G. W. (1978). Regression quantiles, \emph{Econometrica}, \bold{46}, 33--50. [2] Koenker, R.W. and d'Orey (1987, 1994). Computing regression quantiles. \emph{Applied Statistics}, \bold{36}, 383--393, and \bold{43}, 410--414. [3] Gutenbrunner, C. Jureckova, J. (1991). Regression quantile and regression rank score process in the linear model and derived statistics, \emph{Annals of Statistics}, \bold{20}, 305--330. [4] Koenker, R. W. (1994). Confidence Intervals for regression quantiles, in P. Mandl and M. Huskova (eds.), \emph{Asymptotic Statistics}, 349--359, Springer-Verlag, New York. There is also recent information available at the URL: \url{http://www.econ.uiuc.edu}. } \seealso{ \code{\link{summary.rq}}, \code{\link{rq.object}}, \code{\link{rq.process.object}} } % Converted by Sd2Rd version 0.3-3.