rq {quantreg}  R Documentation 
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 ~ operator,
and the terms, separated by + operators, on the right.

tau 
the quantile(s) 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 tau in (0,1) are desired. In the former case an
object of class "rq" is returned, in the latter,
an object of class "rq.process" is returned. As of version 3.50
tau can also be a vector of values between 0 and 1; in this case an
object of class "rqs" is returned containing among other things
a matrix of coefficient estimates at the specified quantiles.

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 na.fail ) is to create an error if any missing values are
found. A possible alternative is na.omit , which
deletes observations that contain one or more missing values.

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 currently
four options: The default method is the modified version of the
Barrodale and Roberts algorithm for l1regression,
used by l1fit in S, and is described in detail in
Koenker and d'Orey(1987, 1994), default = "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 "fn" .
And very large problems one can use the Frisch–Newton approach after
preprocessing "pfn" . Both of the latter methods are
described in detail in Portnoy and Koenker(1997). Finally, there
is a fourth option "fnc" that enables the user to specify
linear inequality constraints on the fitted coefficients; in this
case one needs to specify the matrix R and the vector r
representing the constraints in the form $Rb >=q r$. See the
examples

contrasts 
a list giving contrasts for some or all of the factors
default = 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 fullrank
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.

... 
additional arguments for the fitting routines
(see rq.fit.br and rq.fit.fn
and the functions they call).

See rq.object
and rq.process.object
for details.
Inferential matters are handled with summary
.
The function computes an estimate on the tauth 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 nonlinear 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 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 "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] Koenker, R. W. (1994). Confidence Intervals for regression quantiles, in P. Mandl and M. Huskova (eds.), Asymptotic Statistics, 349–359, SpringerVerlag, New York.
[5] Koenker, R. and S. Portnoy (1997) The Gaussian Hare and the Laplacean Tortoise: Computability of Squarederror vs Absolute Error Estimators, (with discussion). Statistical Science, 12, 279300.
There is also recent information available at the URL: http://www.econ.uiuc.edu.
summary.rq
, rq.object
,
rq.process.object
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 4space! 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(x,y,xlab="Household Income",ylab="Food Expenditure",type = "n", cex=.5) points(x,y,cex=.5,col="blue") taus < c(.05,.1,.25,.75,.9,.95) xx < seq(min(x),max(x),100) f < coef(rq((y)~(x),tau=taus)) yy < cbind(1,xx)%*%f for(i in 1:length(taus)){ lines(xx,yy[,i],col = "gray") } abline(lm(y~x),col="red",lty = 2) abline(rq(y~x),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(y~x,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")