survival: Survival Analysis

library(survival)
options(na.action=na.exclude) # preserve missings
options(contrasts=c('contr.treatment', 'contr.poly')) #ensure constrast type
aeq <- function(x,y) all.equal(as.vector(x), as.vector(y))

#
# Tests from the appendix of Therneau and Grambsch
#  a. Data set 1 and Breslow estimate
#  The data below is not in time order, to also test sorting, and has 1 NA
#
test1 <- data.frame(time=  c(9, 3,1,1,6,6,8),
                    status=c(1,NA,1,0,1,1,0),
                    x=     c(0, 2,1,1,1,0,0))

# Nelson-Aalen influence
s1 <- survfit(Surv(time, status) ~1, test1, id=1:7, influence=TRUE)
inf1 <- matrix(c(10, rep(-2,5), 10, -2, 7,7, -11, -11)/72,
               ncol=2)
indx <- order(test1$time[!is.na(test1$status)])
aeq(s1$influence.chaz[indx,], inf1[,c(1,2,2,2)])

# KM influence
inf2 <- matrix(c(-20, rep(4,5), -10, 2, -13, -13, 17, 17,
                 rep(0,6))/144, ncol=3)
aeq(s1$influence.surv[indx,], inf2[, c(1,2,2,3)])

# Fleming-Harrington influence
s2 <- survfit(Surv(time, status) ~ 1, test1, id=1:7, ctype=2, influence=2)
inf3 <- matrix(c( rep(c(5, -1), c(1, 5))/36, c(5,-1)/36, 
                 c(21,21,-29, -29)/144), ncol=2)
aeq(s2$influence.chaz[indx,], inf3[,c(1,2,2,2)])


# Breslow estimate
byhand1 <- function(beta, newx=0) {
    r <- exp(beta)
    loglik <- 2*beta - (log(3*r+3) + 2*log(r+3))
    u <- (6 + 3*r - r^2) / ((r+1)*(r+3))
    imat <- r/(r+1)^2 + 6*r/(r+3)^2

    x <- c(1,1,1,0,0,0)
    status <- c(1,0,1,1,0,1)
    xbar <- c(r/(r+1), r/(r+3), 0, 0)  # at times 1, 6, 8 and 9
    haz <- c(1/(3*r+3), 2/(r+3), 0, 1 )
    ties <- c(1,1,2,2,3,4)
    wt <- c(r,r,r,1,1,1)
    mart <- c(1,0,1,1,0,1) -  wt* (cumsum(haz))[ties]  #martingale residual

    a <- 3*(r+1)^2; b<- (r+3)^2
    score <- c((2*r+3)/a, -r/a, -r/a + 3*(3-r)/b,  r/a - r*(r+1)/b,
               r/a + 2*r/b, r/a + 2*r/b)

    # Schoenfeld residual
    scho <- c(1/(r+1), 1- (r/(3+r)), 0-(r/(3+r)) , 0)

    surv  <- exp(-cumsum(haz)* exp(beta*newx))
    varhaz.g <- cumsum(c(1/(3*r+3)^2, 2/(r+3)^2, 0, 1 ))

    varhaz.d <- cumsum((newx-xbar) * haz)

    varhaz <- (varhaz.g + varhaz.d^2/ imat) * exp(2*beta*newx)

    names(xbar) <- names(haz) <- 1:4
    names(surv) <- names(varhaz) <- 1:4
    list(loglik=loglik, u=u, imat=imat, xbar=xbar, haz=haz,
	     mart=mart,  score=score,
		scho=scho, surv=surv, var=varhaz,
		varhaz.g=varhaz.g, varhaz.d=varhaz.d)
    }



fit0 <-coxph(Surv(time, status) ~x, test1, iter=0, method='breslow')
truth0 <- byhand1(0,0)
aeq(truth0$loglik, fit0$loglik[1])
aeq(1/truth0$imat, fit0$var)
aeq(truth0$mart, fit0$resid[c(2:6,1)])
aeq(truth0$scho, resid(fit0, 'schoen'))
aeq(truth0$score, resid(fit0, 'score')[c(3:7,1)])
sfit <- survfit(fit0, list(x=0))
aeq(sfit$cumhaz, cumsum(truth0$haz))
aeq(sfit$surv, exp(-cumsum(truth0$haz)))
aeq(sfit$std.err^2, c(7/180, 2/9, 2/9, 11/9))
aeq(resid(fit0, 'score'), c(5/24, NA, 5/12, -1/12, 7/24, -1/24, 5/24))

fit1 <- coxph(Surv(time, status) ~x, test1, iter=1, method='breslow')
aeq(fit1$coef, 8/5)

# This next gives an ignorable warning message
fit2 <- coxph(Surv(time, status) ~x, test1, method='breslow', iter=2)
aeq(round(fit2$coef, 6), 1.472724)

fit <- coxph(Surv(time, status) ~x, test1, method='breslow', eps=1e-8,
             nocenter=NULL)
aeq(fit$coef, log(1.5 + sqrt(33)/2))  # the true solution
truth <- byhand1(fit$coef, 0)
aeq(truth$loglik, fit$loglik[2])
aeq(1/truth$imat, fit$var)
aeq(truth$mart, fit$resid[c(2:6,1)])
aeq(truth$scho, resid(fit, 'schoen'))
aeq(truth$score, resid(fit, 'score')[c(3:7,1)])
expect <- predict(fit, type='expected', newdata=test1) #force recalc
aeq(test1$status[-2] -fit$resid, expect[-2]) #tests the predict function

sfit <- survfit(fit, list(x=0), censor=FALSE)
aeq(sfit$std.err^2, truth$var[c(1,2,4)]) # sfit skips time 8 (no events there)
aeq(-log(sfit$surv), (cumsum(truth$haz))[c(1,2,4)])
sfit <- survfit(fit, list(x=0), censor=TRUE)
aeq(sfit$std.err^2, truth$var) 
aeq(-log(sfit$surv), (cumsum(truth$haz)))

# 
# Done with the formal test, now print out lots of bits
#
resid(fit)
resid(fit, 'scor')
resid(fit, 'scho')

predict(fit, type='lp', se.fit=T)
predict(fit, type='risk', se.fit=T)
predict(fit, type='expected', se.fit=T)
predict(fit, type='terms', se.fit=T)

summary(survfit(fit, list(x=2)))