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
Design
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.
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 class 'bj': print(x, digits=4, long=FALSE, ...) ## S3 method for class 'bj': predict{...} ## S3 method for class 'bj': residuals(object, type=c("censored","censored.normalized"),...) bjplot(fit, which=1:dim(X)[[2]]) ## S3 method for class 'bj': validate(fit, method="boot", B=40, bw=FALSE,rule="aic",type="residual",sls=.05,aics=0,pr=FALSE, dxy=TRUE, tol=1e-7, rel.tolerance=1e-3, maxiter=15, ...) bj.fit(x, y, control)
Surv
object.
bj
, required for all functions except
bj
.
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.
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.
"log"
(the default) to model the log of the
response, or
"identity"
to model the untransformed response.
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).
"model.frame"
or
"model.matrix"
to return one of those
objects rather than the model fit.
FALSE
to prevent Somers' D_{xy} from
being computed by
validate
(saves time for very large datasets)
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.
TRUE
to print the correlation matrix for parameter estimates
bj
type="censored.normalized"
to divide the residuals by the estimate
of
sigma
.
bjplot
make plots of only the variables listed in
which
(names or numbers).
print
; passed through to
predab.resample
for
validate
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).
bj
returns a fit object with similar information to what
survreg
,
psm
,
cph
would store as
well as what
Design
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"
. Here
sigma
is the
estimate of the residual standard deviation.
Janez Stare
Department of Biomedical Informatics
Ljubljana University
Ljubljana, Slovenia
janez.stare@mf.uni-lj.si
Harald Heinzl
Department of Medical Computer Sciences
Vienna University
Vienna, Austria
harald.heinzl@akh-wien.ac.at
Frank Harrell
Department of Biostatistics
Vanderbilt University
f.harrell@vanderbilt.edu
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.
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(f, age=NA, hospital=NA) # 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(survfit(r), conf='none') # plot Kaplan-Meier estimate of # survival function of standardized residuals survplot(survfit(r ~ cut2(age, g=2)), conf='none') # may desire both strata to be n(0,1) options(datadist=NULL)