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#
# kmrs.S
#
# S code for Kaplan-Meier, Reduced Sample and Hanisch
# estimates of a distribution function
# from _histograms_ of censored data.
#
# kaplan.meier()
# reduced.sample()
# km.rs()
#
# $Revision: 3.28 $ $Date: 2022/05/17 07:11:07 $
#
# The functions in this file produce vectors `km' and `rs'
# where km[k] and rs[k] are estimates of F(breaks[k+1]),
# i.e. an estimate of the c.d.f. at the RIGHT endpoint of the interval.
#
"kaplan.meier" <-
function(obs, nco, breaks, upperobs=0) {
# obs: histogram of all observations : min(T_i,C_i)
# nco: histogram of noncensored observations : T_i such that T_i <= C_i
# breaks: breakpoints (vector or 'breakpts' object, see breaks.S)
# upperobs: number of observations beyond rightmost breakpoint
#
breaks <- as.breakpts(breaks)
n <- length(obs)
if(n != length(nco))
stop("lengths of histograms do not match")
check.hist.lengths(nco, breaks)
#
#
# reverse cumulative histogram of observations
d <- revcumsum(obs) + upperobs
#
# product integrand
s <- ifelseXB(d > 0, 1 - nco/d, 1)
#
km <- 1 - cumprod(s)
# km has length n; km[i] is an estimate of F(r) for r=breaks[i+1]
#
widths <- diff(breaks$val)
lambda <- numeric(n)
pos <- (s > 0)
lambda[pos] <- -log(s[pos])/widths[pos]
# lambda has length n; lambda[i] is an estimate of
# the average of \lambda(r) over the interval (breaks[i],breaks[i+1]).
#
return(list(km=km, lambda=lambda))
}
"reduced.sample" <-
function(nco, cen, ncc, show=FALSE, uppercen=0)
# nco: histogram of noncensored observations: T_i such that T_i <= C_i
# cen: histogram of all censoring times: C_i
# ncc: histogram of censoring times for noncensored obs:
# C_i such that T_i <= C_i
#
# Then nco[k] = #{i: T_i <= C_i, T_i \in I_k}
# cen[k] = #{i: C_i \in I_k}
# ncc[k] = #{i: T_i <= C_i, C_i \in I_k}.
#
# The intervals I_k must span an interval [0,R] beginning at 0.
# If this interval did not include all censoring times,
# then `uppercen' must be the number of censoring times
# that were not counted in 'cen'.
{
n <- length(nco)
if(n != length(cen) || n != length(ncc))
stop("histogram lengths do not match")
#
# denominator: reverse cumulative histogram of censoring times
# denom(r) = #{i : C_i >= r}
# We compute
# cc[k] = #{i: C_i > breaks[k]}
# except that > becomes >= for k=0.
#
cc <- revcumsum(cen) + uppercen
#
#
# numerator
# #{i: T_i <= r <= C_i }
# = #{i: T_i <= r, T_i <= C_i} - #{i: C_i < r, T_i <= C_i}
# We compute
# u[k] = #{i: T_i <= C_i, T_i <= breaks[k+1]}
# - #{i: T_i <= C_i, C_i <= breaks[k]}
# = #{i: T_i <= C_i, C_i > breaks[k], T_i <= breaks[k+1]}
# this ensures that numerator and denominator are
# comparable, u[k] <= cc[k] always.
#
u <- cumsum(nco) - c(0,cumsum(ncc)[1:(n-1)])
rs <- u/cc
#
# Hence rs[k] = u[k]/cc[k] is an estimator of F(r)
# for r = breaks[k+1], i.e. for the right hand end of the interval.
#
if(!show)
return(rs)
else
return(list(rs=rs, numerator=u, denominator=cc))
}
"km.rs" <-
function(o, cc, d, breaks) {
# o: censored lifetimes min(T_i,C_i)
# cc: censoring times C_i
# d: censoring indicators 1(T_i <= C_i)
# breaks: histogram breakpoints (vector or 'breakpts' object)
#
breaks <- as.breakpts(breaks)
bval <- breaks$val
# compile histograms (breakpoints may not span data)
obs <- whist( o, breaks=bval)
nco <- whist( o[d], breaks=bval)
cen <- whist( cc, breaks=bval)
ncc <- whist( cc[d], breaks=bval)
# number of observations exceeding largest breakpoint
upperobs <- attr(obs, "high")
uppercen <- attr(cen, "high")
# go
km <- kaplan.meier(obs, nco, breaks, upperobs=upperobs)
rs <- reduced.sample(nco, cen, ncc, uppercen=uppercen)
#
return(list(rs=rs, km=km$km, hazard=km$lambda,
r=breaks$r, breaks=bval))
}
"km.rs.opt" <-
function(o, cc, d, breaks, KM=TRUE, RS=TRUE) {
# o: censored lifetimes min(T_i,C_i)
# cc: censoring times C_i
# d: censoring indicators 1(T_i <= C_i)
# breaks: histogram breakpoints (vector or 'breakpts' object)
#
breaks <- as.breakpts(breaks)
bval <- breaks$val
out <- list(r=breaks$r, breaks=bval)
if(KM || RS)
nco <- whist( o[d], breaks=bval)
if(KM) {
obs <- whist( o, breaks=bval)
upperobs <- attr(obs, "high")
km <- kaplan.meier(obs, nco, breaks, upperobs=upperobs)
out <- append(list(km=km$km, hazard=km$lambda), out)
}
if(RS) {
cen <- whist( cc, breaks=bval)
ncc <- whist( cc[d], breaks=bval)
uppercen <- attr(cen, "high")
rs <- reduced.sample(nco, cen, ncc, uppercen=uppercen)
out <- append(list(rs=rs), out)
}
return(out)
}
censtimeCDFest <- function(o, cc, d, breaks, ...,
KM=TRUE, RS=TRUE, HAN=TRUE, RAW=TRUE,
han.denom=NULL, tt=NULL, pmax=0.9,
fname="CDF", fexpr=quote(CDF(r))) {
# Histogram-based estimation of cumulative distribution function
# of lifetimes subject to censoring.
# o: censored lifetimes min(T_i,C_i)
# cc: censoring times C_i
# d: censoring indicators 1(T_i <= C_i)
# breaks: histogram breakpoints (vector or 'breakpts' object)
# han.denom: denominator (eroded area) for each value of r
# tt: uncensored lifetimes T_i, if known
breaks <- as.breakpts(breaks)
bval <- breaks$val
rval <- breaks$r
rmax <- breaks$max
# Kaplan-Meier and/or Reduced Sample
out <- km.rs.opt(o, cc, d, breaks, KM=KM, RS=RS)
# convert to data frame
out$breaks <- NULL
df <- as.data.frame(out)
# Raw ecdf of observed lifetimes if available
if(RAW && !is.null(tt)) {
h <- whist(tt[tt <= rmax], breaks=bval)
df <- cbind(df, data.frame(raw=cumsum(h)/length(tt)))
}
# Hanisch
if(HAN) {
if(is.null(han.denom))
stop("Internal error: missing denominator for Hanisch estimator")
if(length(han.denom) != length(rval))
stop(paste("Internal error:",
"length(han.denom) =", length(han.denom),
"!=", length(rval), "= length(rvals)"))
# uncensored distances
x <- o[d]
# calculate Hanisch estimator
h <- whist(x[x <= rmax], breaks=bval)
H <- cumsum(h/han.denom)
df <- cbind(df, data.frame(han=H/max(H[is.finite(H)])))
}
# determine appropriate plotting range
bestest <- if(KM) "km" else if(HAN) "han" else if(RS) "rs" else "raw"
alim <- range(df$r[df[[bestest]] <= pmax])
# convert to fv object
nama <- c("r", "km", "hazard", "han", "rs", "raw")
avail <- c(TRUE, KM, KM, HAN, RS, RAW)
iscdf <- c(FALSE, TRUE, FALSE, TRUE, TRUE, TRUE)
labl <- c("r",
makefvlabel(NULL, "hat", fname, "km"),
"hat(lambda)(r)",
makefvlabel(NULL, "hat", fname, "han"),
makefvlabel(NULL, "hat", fname, "bord"),
makefvlabel(NULL, "hat", fname, "raw")
)[avail]
desc <- c("distance argument r",
"Kaplan-Meier estimate of %s",
"Kaplan-Meier estimate of hazard function lambda(r)",
"Hanisch estimate of %s",
"border corrected estimate of %s",
"uncorrected estimate of %s")[avail]
df <- df[, nama[avail]]
Z <- fv(df, "r", fexpr, bestest, . ~ r, alim, labl, desc,
fname=fname)
fvnames(Z, ".") <- nama[iscdf & avail]
return(Z)
}
# simple interface for students and code development
compileCDF <- function(D, B, r, ..., han.denom=NULL, check=TRUE) {
han <- !is.null(han.denom)
breaks <- breakpts.from.r(r)
if(check) {
stopifnot(length(D) == length(B) && all(D >= 0) && all(B >= 0))
if(han)
stopifnot(length(han.denom) == length(r))
}
D <- as.vector(D)
B <- as.vector(B)
# observed (censored) lifetimes
o <- pmin.int(D, B)
# censoring indicators
d <- (D <= B)
# go
result <- censtimeCDFest(o, B, d, breaks,
HAN=han,
han.denom=han.denom,
RAW=TRUE, tt=D)
result <- rebadge.fv(result, new.fname="compileCDF")
}
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