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.
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)
an S statistical model formula. Interactions up to third order are
supported. The left hand side must be a Surv object.
the usual statistical model fitting arguments
a fit created by bj, required for all functions except bj.
a design matrix with or without a first column of ones, to pass
to bj.fit. All models will have an intercept. For
print.bj is a result of bj. For bj, set
x=TRUE to include the design matrix in the fit object.
a Surv object to pass to bj.fit as the two-column response
variable. Only right censoring is allowed, and there need not be any
censoring. For bj, set y to TRUE to include the
two-column response matrix, with the
event/censoring indicator in the second column. The first column will
be transformed according to link, and depending on
na.action, rows with missing data in the predictors or the
response will be deleted.
set to, for example, "log" (the default) to model the log of the
response, or "identity" to model the untransformed response.
a list containing any or all of the following components: iter.max
(maximum number of iterations allowed, default is 20),
eps (convergence criterion: concergence is assumed when the ratio of
sum of squared errors from one iteration to the next is between
1-eps and 1+eps), trace (set to TRUE to monitor iterations),
tol (matrix singularity criterion, default is 1e-7), and 'max.cycle'
(in case of nonconvergence the program looks for a cycle that repeats itself,
default is 30).
set to "model.frame" or "model.matrix" to return one of those
objects rather than the model fit.
setting for default time spacing.
Default is 30 if time variable has units="Day", 1 otherwise, unless
maximum follow-up time \(< 1\). Then max time/10 is used as time.inc.
If time.inc is not given and max time/default time.inc is
\(> 25\), time.inc is increased.
number of significant digits to print if not 4.
set to TRUE to print the correlation matrix for parameter estimates
specify coefs=FALSE to suppress printing the table
of model coefficients, standard errors, etc. Specify coefs=n
to print only the first n regression coefficients in the
model.
a character string title to be passed to prModFit
the result of bj
type of residual desired. Default is censored unnormalized residuals,
defined as link(Y) - linear.predictors, where the
link function was usually the log function. You can specify
type="censored.normalized" to divide the residuals by the estimate
of sigma.
vector of integers or character strings naming elements of the design
matrix (the names of the original predictors if they entered the model
linearly) for which to have bjplot make plots of only the variables listed in which (names or numbers).
see
predab.resample
ignored for print; passed through to
predab.resample for validate
bj returns a fit object with similar information to what survreg,
psm, cph would store as
well as what rms stores and units and time.inc.
residuals.bj returns a Surv object. One of the components of the
fit object produced by bj (and bj.fit) is a vector called
stats which contains the following names elements:
"Obs", "Events", "d.f.","error d.f.","sigma","g". Here
sigma is the estimate of the residual standard deviation.
g is the \(g\)-index. If the link function is "log",
the \(g\)-index on the anti-log scale is also returned as gr.
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).
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.
rms, psm, survreg,
cph, Surv,
na.delete,
na.detail.response, datadist,
rcorr.cens, GiniMd,
prModFit, dxy.cens
# 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)
# }