coxrobust (version 1.0)

coxr: Fit Robustly Proportional Hazards Regression Model

Description

Fits efficiently and robustly Cox proportional hazards regression model in its basic form, where explanatory variables are time independent with one event per subject. Method is based on a smooth modification of the partial likelihood.

Usage

coxr(formula, data, subset, na.action, trunc = 0.95,
     f.weight = c("linear", "quadratic", "exponential"),
     singular.ok = TRUE, model = FALSE)

Arguments

formula

a formula object, with the response on the left of a ~ operator, and the terms on the right. The response must be a survival object as returned by the Surv function.

data

a data frame in which to interpret the variables named in the formula, or in the subset.

subset

expression saying that only a subset of the rows of the data should be used in the fit.

na.action

a missing-data filter function, applied to the model.frame, after any subset argument has been used.

trunc

roughtly, quantile of the sample \(T_i exp(\beta'Z_i)\), it determines the trimming level for the robust estimator

f.weight

type of weighting function, default is "quadratic"

singular.ok

logical value indicating how to handle collinearity in the model matrix. If TRUE, the program will automatically skip over columns of the X matrix that are linear combinations of earlier columns. In this case the coefficients for such columns will be NA, and the variance matrix will contain zeros. For ancillary calculations, such as the linear predictor, the missing coefficients are treated as zeros.

model

a logical value indicating whether model frame should be included as a component of the returned value.

Value

An object of class coxr. See coxr.object for details.

Details

The method consists in maximization of an objective function which is a smooth modification of the partial likelihood. Observations with excessive values of \(\Lambda(T) exp(\beta'Z)\), where \(\Lambda\) is the cumulated hazard, \(\beta\) vector of parameters, \(Z\) explanatory variables and \(T\) possibly censored survival time, are down-weighted. Both \(\Lambda\) and \(\beta\) are iteratively robustly estimated.

Numerical results are supported by a graphical tool plot, which in a series of 5 graphs let us compare how well data are explained by the estimated proportional hazards model with non-robust (black color) and robust method (green color). The first graph shows standardized difference of two estimated survival functions; one via the Cox model and the other via Kaplan Meier estimator. The following four graphs show the same differences for four strata, defined by the quartiles of the estimated linear predictor. Comparison of estimation results along with analysis of the graphs leads frequently to a very detailed information about the model fit (see examples).

References

Bednarski, T. (1993). Robust estimation in Cox's regression model. Scandinavian Journal of Statistics. Vol. 20, 213--225.

Bednarski, T. (1989). On sensitivity of Cox's estimator. Statistics and Decisions. 7, 215--228.

Grzegorek, K.(1993). On robust estimation of baseline hazard under the Cox model and via Frechet differentiability. Preprint of the Institute of Mathematics of the Polish Academy of Sciences.518.

Minder, C.E. & Bednarski, T. (1996). A robust method for proportional hazards regression. Statistics in Medicine Vol. 15, 1033--1047.

See Also

coxph, coxr.object, gen_data, plot.coxr, Surv

Examples

# NOT RUN {
# Create a simple test data set using the attached function gen_data
a <- gen_data(200, c(1, 0.1, 2), cont = 0.05, p.censor = 0.30)
result <- coxr(Surv(time, status) ~ X1 + X2 + X3, data = a , trunc = 0.9)
result
plot(result)

#use the lung cancer data at Mayo Clinic to
#compare results of non-robust and robust estimation
result <- coxr(Surv(time, status) ~ age + sex + ph.karno + meal.cal + wt.loss, data = lung)
result
plot(result)

#use the Veteran's Administration Lung Cancer Data
#to compare results of non-robust and robust estimation
result <- coxr(Surv(time,status) ~ age + trt + celltype + karno + diagtime + prior, data = veteran)
result
plot(result)
# }