bj

Buckley-James Multiple Regression Model

bj fits the Buckley-James distribution-free least squares multiple regression model to a possibly right-censored response variable. This model reduces to ordinary least squares if there is no censoring. By default, model fitting is done after taking logs of the response variable. bj uses the rms class for automatic anova, fastbw, validate, Function, nomogram, summary, plot, bootcov, and other functions. The bootcov function may be worth using with bj fits, as the properties of the Buckley-James covariance matrix estimator are not fully known for strange censoring patterns.

For the print method, format of output is controlled by the user previously running options(prType="lang") where lang is "plain" (the default), "latex", or "html".

The residuals.bj function exists mainly to compute residuals and to censor them (i.e., return them as Surv objects) just as the original failure time variable was censored. These residuals are useful for checking to see if the model also satisfies certain distributional assumptions. To get these residuals, the fit must have specified y=TRUE.

The bjplot function is a special plotting function for objects created by bj with x=TRUE, y=TRUE in effect. It produces three scatterplots for every covariate in the model: the first plots the original situation, where censored data are distingushed from non-censored data by a different plotting symbol. In the second plot, called a renovated plot, vertical lines show how censored data were changed by the procedure, and the third is equal to the second, but without vertical lines. Imputed data are again distinguished from the non-censored by a different symbol.

The validate method for bj validates the Somers' Dxy rank correlation between predicted and observed responses, accounting for censoring.

The primary fitting function for bj is bj.fit, which does not allow missing data and expects a full design matrix as input.

Usage

bj(formula=formula(data), data, subset, na.action=na.delete,
   link="log", control, method='fit', x=FALSE, y=FALSE, 
   time.inc)
# S3 method for bj
print(x, digits=4, long=FALSE, coefs=TRUE, 
title="Buckley-James Censored Data Regression", …)
# S3 method for bj
residuals(object, type=c("censored","censored.normalized"),…)
bjplot(fit, which=1:dim(X)[[2]])
# S3 method for bj
validate(fit, method="boot", B=40,
         bw=FALSE,rule="aic",type="residual",sls=.05,aics=0,
         force=NULL, estimates=TRUE, pr=FALSE,
		 tol=1e-7, rel.tolerance=1e-3, maxiter=15, …)
bj.fit(x, y, control)

Details

The program implements the algorithm as described in the original article by Buckley & James. Also, we have used the original Buckley & James prescription for computing variance/covariance estimator. This is based on non-censored observations only and does not have any theoretical justification, but has been shown in simulation studies to behave well. Our experience confirms this view. Convergence is rather slow with this method, so you may want to increase the number of iterations. Our experience shows that often, in particular with high censoring, 100 iterations is not too many. Sometimes the method will not converge, but will instead enter a loop of repeating values (this is due to the discrete nature of Kaplan and Meier estimator and usually happens with small sample sizes). The program will look for such a loop and return the average betas. It will also issue a warning message and give the size of the cycle (usually less than 6).

References

Buckley JJ, James IR. Linear regression with censored data. Biometrika 1979; 66:429--36.

Miller RG, Halpern J. Regression with censored data. Biometrika 1982; 69: 521--31.

James IR, Smith PJ. Consistency results for linear regression with censored data. Ann Statist 1984; 12: 590--600.

Lai TL, Ying Z. Large sample theory of a modified Buckley-James estimator for regression analysis with censored data. Ann Statist 1991; 19: 1370--402.

Hillis SL. Residual plots for the censored data linear regression model. Stat in Med 1995; 14: 2023--2036.

Jin Z, Lin DY, Ying Z. On least-squares regression with censored data. Biometrika 2006; 93:147--161.

See Also

rms, psm, survreg, cph, Surv, na.delete, na.detail.response, datadist, rcorr.cens, GiniMd, prModFit, dxy.cens

Aliases

• bj
• bj.fit
• residuals.bj
• print.bj
• validate.bj
• bjplot

Examples

# NOT RUN {
suppressWarnings(RNGversion("3.5.0"))
set.seed(1)
ftime  <- 10*rexp(200)
stroke <- ifelse(ftime > 10, 0, 1)
ftime  <- pmin(ftime, 10)
units(ftime) <- "Month"
age <- rnorm(200, 70, 10)
hospital <- factor(sample(c('a','b'),200,TRUE))
dd <- datadist(age, hospital)
options(datadist="dd")

f <- bj(Surv(ftime, stroke) ~ rcs(age,5) + hospital, x=TRUE, y=TRUE)
# add link="identity" to use a censored normal regression model instead
# of a lognormal one
anova(f)
fastbw(f)
validate(f, B=15)
plot(Predict(f, age, hospital))
# needs datadist since no explicit age,hosp.
coef(f)               # look at regression coefficients
coef(psm(Surv(ftime, stroke) ~ rcs(age,5) + hospital, dist='lognormal'))
                      # compare with coefficients from likelihood-based
                      # log-normal regression model
                      # use dist='gau' not under R 


r <- resid(f, 'censored.normalized')
survplot(npsurv(r ~ 1), conf='none') 
                      # plot Kaplan-Meier estimate of 
                      # survival function of standardized residuals
survplot(npsurv(r ~ cut2(age, g=2)), conf='none')  
                      # may desire both strata to be n(0,1)
options(datadist=NULL)
# }