# Functions to fit censored quantile regression models Fits a conditional quantile regression model for censored data. There are three distinct methods: the first is the fixed censoring method of Powell (1986) as implemented by Fitzenberger (1996), the second is the random censoring method of Portnoy (2003). The third method is based on Peng and Huang (2008). ```r crq(formula, taus, data, subset, weights, na.action, method = c("Powell", "Portnoy", "Portnoy2", "PengHuang"), contrasts = NULL, ...) crq.fit.pow(x, y, yc, tau=0.5, weights=NULL, start, left=TRUE, maxit = 500) crq.fit.pen(x, y, cen, weights=NULL, grid, ctype = "right") crq.fit.por(x, y, cen, weights=NULL, grid, ctype = "right") crq.fit.por2(x, y, cen, weights=NULL, grid, ctype = "right") Curv(y, yc, ctype=c("left","right")) ## S3 method for class 'crq' print(x, ...) ## S3 method for class 'crq' print(x, ...) ## S3 method for class 'crq' predict(object, newdata, ...) ## S3 method for class 'crqs' predict(object, newdata, type = NULL, ...) ## S3 method for class 'crq' coef(object,taus = 1:4/5,...) ``` ## Arguments - `formula`: A formula object, with the response on the left of the `~' operator, and the terms on the right. The response must be a `Surv` object as returned by either the `Curv` or `Surv` function. For the Powell method, the Surv object should be created by `Curv` and have arguments (event time, censoring time,type), where "type" can take values either "left" or "right". The default (for historical reasons) for type in this case is "left". For the Portnoy and Peng and Huang methods the `Surv` should be created with the usual `Surv` function and have (event time, censoring indicator). - `y`: The event time. - `newdata`: An optional data frame in which to look for variables with which to predict. If omitted, the fitted values are used. - `grid`: A vector of taus on which the quantile process should be evaluated. This should be monotonic, and take values in (0,1). For the "Portnoy" method, grid = "pivot" computes the full solution for all distinct taus. The "Portnoy" method also enforces an equally spaced grid, see the code for details. - `x`: An object of class `crq` or `crq`. - `object`: An object of class `crq` or `crq`. - `yc`: The censoring times for the "Powell" method. - `ctype`: Censoring type: for the "Powell" method, used in `Curv`, by default "left". If you don't like "left", maybe you will like "right". Note that for fixed censoring assumed in the "Powell" method, censoring times `yc` must be provided for all observations and the event times `y` must satisfy the (respective) inequality constraints. For the Portnoy and Peng-Huang methods ctype is determined by the specification of the response as specified in `Surv`. - `type`: specifies either "left" or "right" as the form of censoring in the `Surv` function for the "Portnoy" and "PengHuang" methods. - `cen`: The censoring indicator for the "Portnoy" and "PengHuang" methods. - `maxit`: Maximum number of iterations allowed for the "Powell" methods. - `start`: The starting value for the coefs for the "Powell" method. Because the Fitzenberger algorithm stops when it achieves a local minimum of the Powell objective function, the starting value acts as an a priori "preferred point". This is advantageous in some instances since the global Powell solution can be quite extreme. By default the starting value is the "naive rq" solution that treats all the censored observations as uncensored. If `start` is equal to "global" then an attempt is made to compute to global optimum of the Powell objective. This entails an exhaustive evaluation of all n choose p distinct basic solution so is rather impractical for moderately large problems. Otherwise, the starting value can specify a set of p indices from 1:n defining an initial basic solution, or it may specify a p-vector of initial regression coefficients. In the latter case the initial basic solution is the one closest to the specified parameter vector. - `left`: A logical indicator for left censoring for the "Powell" method. - `taus`: The quantile(s) at which the model is to be estimated. - `tau`: The quantile at which the model is to be estimated. - `data`: A data.frame in which to interpret the variables named in the `formula', in the `subset', and the `weights' argument. - `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 vector 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. - `method`: The method used for fitting. There are currently two options: method "Powell" computes the Powell estimator using the algorithm of Fitzenberger (1996), method "Portnoy" computes the Portnoy (2003) estimator. The method is "PengHuang" uses the method of Peng and Huang (2007), in this case the variable "grid" can be passed to specify the vector of quantiles at which the solution is desired. - `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 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. - `...`: additional arguments for the fitting routine, for method "Powell" it may be useful to pass starting values of the regression parameter via the argument "start", while for methods "Portnoy" or "PengHuang" one may wish to specify an alternative to the default grid for evaluating the fit. ## Details The Fitzenberger algorithm uses a variant of the Barrodale and Roberts simplex method. Exploiting the fact that the solution must be characterized by an exact fit to p points when there are p parameters to be estimated, at any trial basic solution it computes the directional derivatives in the 2p distinct directions and choses the direction that (locally) gives steepest descent. It then performs a one-dimensional line search to choose the new basic observation and continues until it reaches a local mimumum. By default it starts at the naive `rq` solution ignoring the censoring; this has the (slight) advantage that the estimator is consequently equivariant to canonical transformations of the data. Since the objective function is no longer convex there can be no guarantee that this produces a global minimum estimate. In small problems exhaustive search over solutions defined by p-element subsets of the n observations can be used, but this quickly becomes impractical for large p and n. This global version of the Powell estimator can be invoked by specifying `start = "global"`. Users interested in this option would be well advised to compute `choose(n,p)` for their problems before trying it. The method operates by pivoting through this many distinct solutions and choosing the one that gives the minimal Powell objective. The algorithm used for the Portnoy method is described in considerable detail in Portnoy (2003). There is a somewhat simplified version of the Portnoy method that is written in R and iterates over a discrete grid. This version should be considered somewhat experimental at this stage, but it is known to avoid some difficulties with the more complicated fortran version of the algorithm that can occur in degenerate problems. Both the Portnoy and Peng-Huang estimators may be unable to compute estimates of the conditional quantile parameters in the upper tail of distribution. Like the Kaplan-Meier estimator, when censoring is heavy in the upper tail the estimated distribution is defective and quantiles are only estimable on a sub-interval of (0,1). The Peng and Huang estimator can be viewed as a generalization of the Nelson Aalen estimator of the cumulative hazard function, and can be formulated as a variant of the conventional quantile regression dual problem. See Koenker (2008) for further details. This paper is available from the package with `vignette("crq")`. ## Returns An object of class `crq`. ## References Fitzenberger, B. (1996): ``A Guide to Censored Quantile Regressions,'' in **Handbook of Statistics**, ed. by C.~Rao, and G.~Maddala. North-Holland: New York. Fitzenberger, B. and P. Winker (2007): ``Improving the Computation of Censored Quantile Regression Estimators,'' CSDA, 52, 88-108. Koenker, R. (2008): ``Censored Quantile Regression Redux,'' **J. Statistical Software**, 27, [https://www.jstatsoft.org/v27/i06](https://www.jstatsoft.org/v27/i06). Peng, L and Y Huang, (2008) Survival Analysis with Quantile Regression Models, **J. Am. Stat. Assoc.**, 103, 637-649. Portnoy, S. (2003) ``Censored Quantile Regression,'' **JASA**, 98,1001-1012. Powell, J. (1986) ``Censored Regression Quantiles,'' **J. Econometrics**, 32, 143--155. ## Author(s) Steve Portnoy and Roger Koenker ## See Also `summary.crq` ## Examples ```r # An artificial Powell example set.seed(2345) x <- sqrt(rnorm(100)^2) y <- -0.5 + x +(.25 + .25*x)*rnorm(100) plot(x,y, type="n") s <- (y > 0) points(x[s],y[s],cex=.9,pch=16) points(x[!s],y[!s],cex=.9,pch=1) yLatent <- y y <- pmax(0,y) yc <- rep(0,100) for(tau in (1:4)/5){ f <- crq(Curv(y,yc) ~ x, tau = tau, method = "Pow") xs <- sort(x) lines(xs,pmax(0,cbind(1,xs)%*%f$coef),col="red") abline(rq(y ~ x, tau = tau), col="blue") abline(rq(yLatent ~ x, tau = tau), col="green") } legend(.15,2.5,c("Naive QR","Censored QR","Omniscient QR"), lty=rep(1,3),col=c("blue","red","green")) # crq example with left censoring set.seed(1968) n <- 200 x <-rnorm(n) y <- 5 + x + rnorm(n) plot(x,y,cex = .5) c <- 4 + x + rnorm(n) d <- (y > c) points(x[!d],y[!d],cex = .5, col = 2) f <- crq(survival::Surv(pmax(y,c), d, type = "left") ~ x, method = "Portnoy") g <- summary(f) for(i in 1:4) abline(coef(g[[i]])[,1]) ```

crq function