ctqr (version 1.0)

ctqr: Censored and Truncated Quantile Regression

Description

Fits a quantile regression model to possibly censored and truncated data, e.g., survival data.

Usage

ctqr(formula, data, weights, p = 0.5, CDF, control = ctqr.control(), ...)

Arguments

formula

an object of class “formula”: a symbolic description of the regression model. The response must be a Surv object as returned by Surv (see ‘Details’).

data

an optional data frame containing the variables in the model.

weights

an optional vector of weights to be used in the fitting process.

p

numerical vector indicating the order of the quantile(s) to be fitted.

CDF

an object of class “pch”, i.e., the result of a call to pchreg. If missing, it will be computed internally with default settings. See ‘Details’.

control

a list of operational parameters for the optimization algorithm, usually passed via ctqr.control.

...

for future arguments.

Value

An object of class “ctqr”, which is a list with the following items: , which is a list with the following items:Note that the dimension of all items, except call, terms, and mf, is the same as the dimension of p. For example, if p = c(0.25,0.5,0.75), coefficients and fitted will be 3-columns matrices; n.it and converged will be vectors of 3 elements; and covar will be a list of three covariance matrices.The generic accessor functions summary, plot, predict, coef, terms, nobs, can be used to extract information from the model. The functions waldtest (from the package lmtest), and linearHypothesis (from the package car) can be used to perform Wald test, and to test for linear restrictions. These functions, however, will only work if p is scalar.

Details

This function implements the method described by Frumento and Bottai (2016) for censored, truncated quantile regression.

The left side of formula must be of the form Surv(time, event) if the data are right-censored, and Surv(time0, time, event) if the data are right-censored and left-truncated (time0 < time, time0 can be -Inf). Using Surv(time) is also allowed and indicates that the data are neither censored nor truncated.

The conditional distribution function (CDF) of the response variable represents a nuisance parameter and is estimated preliminarly via pchreg. If missing, CDF = pchreg(formula, splinex = splinex()) is used as default. See also “Note” and the documentation of pchreg and splinex.

Estimation is carried out using an algorithm for gradient-based optimization. To estimate the asymptotic covariance matrix, standard two-step procedures are used (e.g., Ackerberg, 2012).

References

Frumento, P., and Bottai, M. (2016). An estimating equation for censored and truncated quantile regression. Ackerberg, D., Chen, X., and Hahn, J. (2012). A practical asymptotic variance estimator for two-step semiparametric estimators. The Review of Economics and Statistics, 94 (2), 481-498.

See Also

plot.ctqr, predict.ctqr, pchreg

Examples

# Using simulated data

n <- 1000
x1 <- runif(n)
x2 <- runif(n)

# Example 1 - censored data ######################################

t <- runif(n, 0, 1 + x1 + x2) # time variable (e.g., time to death)
c <- runif(n,0,5)             # censoring variable (e.g., end of follow-up)
y <- pmin(t,c) # observed variable = min(t,c)
d <- (t <= c)  # 1 = event (e.g., death), 0 = censored

CDF1 <- pchreg(Surv(y,d) ~ x1 + x2, splinex = splinex())
model1 <- ctqr(Surv(y,d) ~ x1 + x2, p = 0.5, CDF = CDF1)
model2 <- ctqr(Surv(y,d) ~ x1, p = 0.5, CDF = CDF1)

# model1 is identical to ctqr(Surv(y,d) ~ x1 + x2, p = 0.5)
# model2 is NOT identical to ctqr(Surv(y,d) ~ x1, p = 0.5), 
  # which would have default CDF = pchreg(Surv(y,d) ~ x1, splinex = splinex())


# Example 2 - censored and truncated data #########################

z <- rnorm(n) # truncation variable (e.g., time at enrollment)
w <- which(y > z) # data are only observed when y > z
z <- z[w]; y <- y[w]; d <- d[w]; x1 <- x1[w]; x2 <- x2[w]

# implement various CDFs and choose the model with smallest AIC

CDFs <- list(
  pchreg(Surv(z,y,d) ~ x1 + x2, breaks = 5),
  pchreg(Surv(z,y,d) ~ x1 + x2, breaks = 10),
  pchreg(Surv(z,y,d) ~ x1 + x2 + x1:x2, breaks = 5),
  pchreg(Surv(z,y,d) ~ x1 + x2 + x1^2 + x2^2, breaks = 10)
)

CDF <- CDFs[[which.min(sapply(CDFs, function(obj) AIC(obj)))]]
summary(ctqr(Surv(z,y,d) ~ x1 + x2, p = 0.5, CDF = CDF))