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
bootcov, and other functions. The
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.
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
bjplot function is a special plotting function for objects
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.
validate method for
bj validates the Somers'
correlation between predicted and observed responses, accounting for censoring.
The primary fitting function for
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)"print"(x, digits=4, long=FALSE, coefs=TRUE, latex=FALSE, title="Buckley-James Censored Data Regression", ...)"residuals"(object, type=c("censored","censored.normalized"),...)bjplot(fit, which=1:dim(X)[])"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
- the usual statistical model fitting arguments
a fit created by
bj, required for all functions except
a design matrix with or without a first column of ones, to pass
bj.fit. All models will have an intercept. For
print.bjis a result of
x=TRUEto include the design matrix in the fit object.
Survobject to pass to
bj.fitas the two-column response variable. Only right censoring is allowed, and there need not be any censoring. For
TRUEto 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-
TRUEto 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.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.incis not given and max time/default
time.incis $> 25$,
- number of significant digits to print if not 4.
TRUEto print the correlation matrix for parameter estimates
coefs=FALSEto suppress printing the table of model coefficients, standard errors, etc. Specify
coefs=nto print only the first
nregression coefficients in the model.
- a logical value indicating whether information should be formatted as plain text or as LaTeX markup
- a character string title to be passed to
- the result of
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
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
bjplotmake plots of only the variables listed in
which(names or numbers).
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).
bjreturns a fit object with similar information to what
cphwould store as well as what
Survobject. One of the components of the
fitobject produced by
bj.fit) is a vector called
statswhich contains the following names elements:
"Obs", "Events", "d.f.","error d.f.","sigma","g". Here
sigmais the estimate of the residual standard deviation.
gis the $g$-index. If the link function is
"log", the $g$-index on the anti-log scale is also returned as
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.
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)