`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)

formula

an S statistical model formula. Interactions up to third order are
supported. The left hand side must be a `Surv`

object.

data,subset,na.action

the usual statistical model fitting arguments

fit

a fit created by `bj`

, required for all functions except `bj`

.

x

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.

y

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.

link

set to, for example, `"log"`

(the default) to model the log of the
response, or `"identity"`

to model the untransformed response.

control

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).

method

set to `"model.frame"`

or `"model.matrix"`

to return one of those
objects rather than the model fit.

time.inc

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.

digits

number of significant digits to print if not 4.

long

set to `TRUE`

to print the correlation matrix for parameter estimates

coefs

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.

title

a character string title to be passed to `prModFit`

object

the result of `bj`

type

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`

.

which

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).

B,bw,rule,sls,aics,force,estimates,pr,tol,rel.tolerance,maxiter

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) # }