Quantile Regression
Returns an object of class "rq" "rqs"
or "rq.process" that represents a quantile regression fit.
rq(formula, tau=.5, data, subset, weights, na.action, method="br", model = TRUE, contrasts, ...)
formula |
a formula object, with the response on the left of a |
tau |
the quantile(s) to be estimated, this is generally a number strictly between 0 and 1,
but if specified strictly outside this range, it is presumed that the solutions
for all values of |
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. |
subset |
an optional vector specifying a subset of observations to be used in the fitting process. |
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. |
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 |
model |
if TRUE then the model frame is returned. This is essential if one wants to call summary subsequently. |
method |
the algorithmic method used to compute the fit. There are several options:
|
contrasts |
a list giving contrasts for some or all of the factors
default = |
... |
additional arguments for the fitting routines
(see |
For further details see the vignette available from R with
vignette("rq",package="quantreg") and/or the Koenker (2005).
For estimation of nonlinear (in parameters) quantile regression models
there is the function nlrq and for nonparametric additive
quantile regression there is the function rqss.
Fitting of quantile regression models with censored data is handled by the
crq function.
See rq.object and rq.process.object for details.
Inferential matters are handled with summary. There are
extractor methods logLik and AIC that are potentially
relevant for model selection.
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 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 (in parameters)
quantile regression see the package nlrq().
The function minimizes a weighted sum of absolute
residuals that can be formulated as a linear programming problem. As
noted above, there are several 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 "br" method be used. There are several choices of methods for
computing confidence intervals and associated test statistics.
See the documentation for summary.rq for further details
and options.
[1] Koenker, R. W. and Bassett, G. W. (1978). Regression quantiles, Econometrica, 46, 33–50.
[2] Koenker, R.W. and d'Orey (1987, 1994). Computing regression quantiles. Applied Statistics, 36, 383–393, and 43, 410–414.
[3] Gutenbrunner, C. Jureckova, J. (1991). Regression quantile and regression rank score process in the linear model and derived statistics, Annals of Statistics, 20, 305–330.
[4] Xuming He and Xiaoou Pan and Kean Ming Tan and Wen-Xin Zhou, (2020) conquer: Convolution-Type Smoothed Quantile Regression, https://CRAN.R-project.org/package=conquer
[4] Koenker, R. W. (1994). Confidence Intervals for regression quantiles, in P. Mandl and M. Huskova (eds.), Asymptotic Statistics, 349–359, Springer-Verlag, New York.
[5] Koenker, R. and S. Portnoy (1997) The Gaussian Hare and the Laplacean Tortoise: Computability of Squared-error vs Absolute Error Estimators, (with discussion). Statistical Science, 12, 279-300.
[6] Koenker, R. W. (2005). Quantile Regression, Cambridge U. Press.
There is also recent information available at the URL: http://www.econ.uiuc.edu/~roger/.
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=FALSE) #ordinary sample median --no rank inversion ci
rq(rnorm(50) ~ 1, weights=runif(50),ci=FALSE) #weighted sample median
#plot of engel data and some rq lines see KB(1982) for references to data
data(engel)
attach(engel)
plot(income,foodexp,xlab="Household Income",ylab="Food Expenditure",type = "n", cex=.5)
points(income,foodexp,cex=.5,col="blue")
taus <- c(.05,.1,.25,.75,.9,.95)
xx <- seq(min(income),max(income),100)
f <- coef(rq((foodexp)~(income),tau=taus))
yy <- cbind(1,xx)%*%f
for(i in 1:length(taus)){
lines(xx,yy[,i],col = "gray")
}
abline(lm(foodexp ~ income),col="red",lty = 2)
abline(rq(foodexp ~ income), col="blue")
legend(3000,500,c("mean (LSE) fit", "median (LAE) fit"),
col = c("red","blue"),lty = c(2,1))
#Example of plotting of coefficients and their confidence bands
plot(summary(rq(foodexp~income,tau = 1:49/50,data=engel)))
#Example to illustrate inequality constrained fitting
n <- 100
p <- 5
X <- matrix(rnorm(n*p),n,p)
y <- .95*apply(X,1,sum)+rnorm(n)
#constrain slope coefficients to lie between zero and one
R <- cbind(0,rbind(diag(p),-diag(p)))
r <- c(rep(0,p),-rep(1,p))
rq(y~X,R=R,r=r,method="fnc")Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.