115 lines
3.8 KiB
R
115 lines
3.8 KiB
R
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library(survival)
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options(na.action=na.exclude) # preserve missings
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options(contrasts=c('contr.treatment', 'contr.poly')) #ensure constrast type
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# Tests of the weighted Cox model
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# This is section 1.3 of my appendix -- no yet found in any of the
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# printings though, it awaits the next edition
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#
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# Efron approximation
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#
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aeq <- function(x,y) all.equal(as.vector(x), as.vector(y))
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testw1 <- data.frame(time= c(1,1,2,2,2,2,3,4,5),
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status= c(1,0,1,1,1,0,0,1,0),
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x= c(2,0,1,1,0,1,0,1,0),
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wt = c(1,2,3,4,3,2,1,2,1))
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xx <- testw1$wt
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# Efron estimate
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byhand <- function(beta, newx=0) {
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r <- exp(beta)
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a <- 7*r +3; b<- 4*r+2
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loglik <- 11*beta - (log(r^2 + 11*r +7) + 10*log(11*r +5)/3 +
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10*log(a*2/3 +b)/3 + 10*log(a/3 +b)/3 +2*log(2*r+1))
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hazard <- c(1/(r^2 + 11*r +7),
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10/(3*c(11*r +5, a*2/3 +b, a/3+b)), 2/(2*r+1))
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temp <- c(hazard[1], hazard[1]+hazard[2] + hazard[3]*2/3 + hazard[4]/3,
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cumsum(hazard)[4:5])
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risk <- c(r^2, 1,r,r,1,r,1,r,1)
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expected <- risk* temp[c(1,1,2,2,2,3,3,4,4)]
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# The matrix of weights, one row per obs, one col per death
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# deaths at 1,2,2,2, and 4
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riskmat <- matrix(c(1,1,1,1,1,1,1,1,1,
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0,0,1,1,1,1,1,1,1,
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0,0,2/3,2/3,2/3,1,1,1,1,
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0,0,1/3,1/3,1/3,1,1,1,1,
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0,0,0,0,0,0,0,1,1), ncol=5)
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wtmat <- diag(c(r^2, 2, 3*r, 4*r, 3, 2*r, 1, 2*r, 1)) %*% riskmat
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x <- c(2,0,1,1,0,1,0,1,0)
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xbar <- colSums(x*wtmat)/ colSums(wtmat)
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imat <- (4*r^2 + 11*r)*hazard[1] - xbar[1]^2 +
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10* mean(xbar[2:4] - xbar[2:4]^2) + 2*(xbar[5] - xbar[5]^2)
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status <- c(1,0,1,1,1,0,0,1,0)
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wt <- c(1,2,3,4,3,2,1,2,1)
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# Table of sums for score resids
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hazmat <- riskmat %*% diag(c(1,10/3,10/3, 10/3,2)/colSums(wtmat))
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dM <- -risk*hazmat #Expected part
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dM[1,1] <- dM[1,1] +1 # deaths at time 1
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for (i in 2:4) dM[3:5, i] <- dM[3:5,i] + 1/3
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dM[8,5] <- dM[8,5] +1
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mart <- rowSums(dM)
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resid <-dM * outer(x, xbar ,'-')
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# Increments to the variance of the hazard
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var.g <- cumsum(hazard^2* c(1,3/10, 3/10, 3/10, 1/2))
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var.d <- cumsum((xbar-newx)*hazard)
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sxbar <- c(xbar[1], mean(xbar[2:4]), xbar[5]) #xbar for Schoen
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list(loglik=loglik, imat=imat, hazard=hazard, xbar=xbar,
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mart=status-expected, expected=expected,
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score=rowSums(resid), schoen=c(2,1,1,0,1) - sxbar[c(1,2,2,2,3)],
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varhaz=((var.g + var.d^2/imat)* exp(2*beta*newx))[c(1,4,5)])
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}
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# Verify
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temp <- byhand(0,0)
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aeq(temp$xbar, c(13/19, 11/16, 26/38, 19/28, 2/3))
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aeq(temp$hazard, c(1/19, 5/24, 5/19, 5/14, 2/3))
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fit0 <- coxph(Surv(time, status) ~x, testw1, weights=wt, iter=0)
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fit <- coxph(Surv(time, status) ~x, testw1, weights=wt)
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truth0 <- byhand(0,pi)
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aeq(fit0$loglik[1], truth0$loglik)
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aeq(1/truth0$imat, fit0$var)
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aeq(truth0$mart, fit0$residuals)
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aeq(truth0$schoen, resid(fit0, 'schoen'))
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aeq(truth0$score, resid(fit0, 'score'))
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sfit <- survfit(fit0, list(x=pi), censor=FALSE)
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aeq(sfit$std.err^2, truth0$varhaz)
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aeq(-log(sfit$surv), cumsum(truth0$hazard)[c(1,4,5)])
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truth <- byhand(fit$coefficients, .3)
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aeq(truth$loglik, fit$loglik[2])
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aeq(1/truth$imat, fit$var)
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aeq(truth$mart, fit$residuals)
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aeq(truth$schoen, resid(fit, 'schoen'))
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aeq(truth$score, resid(fit, 'score'))
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sfit <- survfit(fit, list(x=.3), censor=FALSE)
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aeq(sfit$std.err^2, truth$varhaz)
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aeq(-log(sfit$surv), (cumsum(truth$hazard)* exp(fit$coefficients*.3))[c(1,4,5)])
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fit0
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summary(fit)
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resid(fit0, type='score')
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resid(fit0, type='scho')
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resid(fit, type='score')
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resid(fit, type='scho')
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rr1 <- resid(fit, type='mart')
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rr2 <- resid(fit, type='mart', weighted=T)
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aeq(rr2/rr1, testw1$wt)
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rr1 <- resid(fit, type='score')
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rr2 <- resid(fit, type='score', weighted=T)
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aeq(rr2/rr1, testw1$wt)
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