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Type 'q()' to quit R. > library(survival) > aeq <- function(x, y, ...) all.equal(as.vector(x), as.vector(y), ...) > > # Test the survival curve for a fit with shared hazards. > # Use the pbcseq data set, and turn bilirubin into a time-dependent state with > # 4 levels, and a shared baseline hazard for the 4 transitions to death. > # The subtlety is that coefficients for a shared (proportional) baseline hazard > # are attached to a state, not to an observation. > # (A bilirubin value of <1 is normal.) > pbc1 <- pbcseq > pbc1$bili4 <- cut(pbc1$bili, c(0,1, 2,4, 100), + c("normal", "1-2", "2-4", ">4")) > ptemp <- subset(pbc1, !duplicated(id)) # first row of each > > pbc2 <- tmerge(ptemp[, c("id", "age", "sex")], ptemp, id, + death= event(futime, status==2)) > > pbc2 <- tmerge(pbc2, pbc1, id=id, bili = tdc(day, bili), + bili4 = tdc(day, bili4), bstat = event(day, as.numeric(bili4))) > btemp <- with(pbc2, ifelse(death, 5, bstat)) > > # a row with the same starting and ending bili4 level is not an event > b2 <- ifelse(((as.numeric(pbc2$bili4)) == btemp), 0, btemp) > pbc2$bstat <- factor(b2, 0:5, + c("censor", "normal", "1-2", "2-4", ">4", "death")) > check1 <- survcheck(Surv(tstart, tstop, bstat) ~ 1, istate= bili4, + id = id, data=pbc2) > check1$transitions to from normal 1-2 2-4 >4 death (censored) normal 0 81 10 3 9 77 1-2 61 0 68 15 9 36 2-4 2 33 0 94 12 24 >4 1 3 28 0 110 35 death 0 0 0 0 0 0 > > fit2 <- coxph(list(Surv(tstart, tstop, bstat) ~ 1, + c(1:4):5 ~ age / common + shared), id= id, istate=bili4, + data=pbc2) > > # Before we tackle fit2, start small with just 9 subjects, coefs fixed to > # simple values to make hand computation easier. There are no transitions > # from state 3 to death in this subset. Since it is a shared hazard the > # subjects in state 3 ARE at risk and so are found in the denominator of the > # hazard, but since none of the progress the MLE for that ph coef is -infinity. > # We set it to -1. > pbc3 <- subset(pbc2, id < 10) > pbc3$age <- round(pbc3$age) # easier to do "by hand" sums > fit3 <- coxph(list(Surv(tstart, tstop, bstat) ~ 1, + c(1:4):5 ~ age / common + shared), x=TRUE, + id= id, istate=bili4, data=pbc3, init= c(.05, .6, -1, 1.1), iter=0) > # a mixed p0 gives a stronger test than our usual (1, 0,0,0,0) > surv3 <- survfit(fit3, newdata=list(age=50), p0=c(.4, .3, .2, .1, 0)) > > etime <- sort(unique(pbc3$tstop[pbc3$bstat != "censor"])) > # At event time 1 (182), all 9 are at risk, (3,3,2,1) in initial states 1-4 > atrisk <- pbc3$tstart < etime[1] & pbc3$tstop >= etime[1] # all 9 at risk > table(pbc3$bili4[atrisk]) normal 1-2 2-4 >4 3 3 2 1 > > # One event occurs at 182, a 2:1 transition (1-2 to normal) > # Risk scores for the non-death transitions are all exp(0) =1, > # so the hazard matrix H will have second row of (1/3, -1/3, 0,0,0) and all > # other rows are 0. > with(subset(pbc3, tstop== 182), table(istate= bili4, state=bstat)) state istate censor normal 1-2 2-4 >4 death normal 0 0 0 0 0 0 1-2 0 1 0 0 0 0 2-4 0 0 0 0 0 0 >4 0 0 0 0 0 0 > > # The next four events are from 3:4, 3:2, 2:3, and 1:2, so also have > # simple transtions, i.e., no covariates so all risk scores are exp(0) =1 > # > hmat <- array(0, dim=c(5,5,6)) # first 6 hazard matrices, start with 3,3,2,1 > hmat[2,1,1] <- 1/3; hmat[2,2,1] <- -1/3 # new count= 4,2,2,1 > hmat[3,4,2] <- 1/2; hmat[3,3,2] <- -1/2 # new count= 4,2,1,2 > hmat[3,2,3] <- 1 ; hmat[3,3,3] <- -1 # new count= 4,3,0,2 > hmat[2,3,4] <- 1/3; hmat[2,2,4] <- -1/3 # new count= 4,2,1,2 > hmat[1,2,5] <- 1/4; hmat[1,1,5] <- -1/4 # new count= 3,3,1,2 > > # Event 6 is a transition from state 4 to death, at day 400 > # For the shared hazard, the denominator is all those in states 1,2,3, or 4. > atrisk <- with(pbc3, tstart < etime[6] & tstop >= etime[6]) > table(pbc3$bili4[atrisk]) # current states just before time 6 normal 1-2 2-4 >4 3 3 1 2 > > adata <- subset(pbc3, atrisk) > eta <- with(adata, .05*(age-50) + .6*(bili4=="1-2") + 1.1*(bili4 == ">4") - + 1*(bili4=="2-4")) > cbind(adata[,c('id', 'age', 'tstop', 'bili4', 'bstat')], eta, risk=exp(eta)) id age tstop bili4 bstat eta risk 2 1 59 400 >4 death 1.55 4.7114702 5 2 56 768 normal 1-2 0.30 1.3498588 14 3 70 743 1-2 censor 1.60 4.9530324 18 4 55 729 1-2 2-4 0.85 2.3396469 25 5 38 769 2-4 >4 -1.60 0.2018965 30 6 66 737 normal censor 0.80 2.2255409 36 7 56 545 1-2 normal 0.90 2.4596031 44 8 53 795 normal censor 0.15 1.1618342 52 9 43 723 >4 censor 0.75 2.1170000 > basehaz <- 1/sum(exp(eta)) > hmat[1,5,6] <- basehaz; hmat[1,1,6] <- -basehaz > hmat[2,5,6] <- basehaz * exp(.6); hmat[2,2,6] <- -basehaz*exp(.6) > hmat[3,5,6] <- basehaz * exp(-1); hmat[3,3,6] <- -basehaz*exp(-1) > hmat[4,5,6] <- basehaz * exp(1.1); hmat[4,4,6] <- -basehaz*exp(1.1) > # double check: sum of per-subject hazards at this time point = number of > # events at this time point > sum(basehaz * exp(eta)) ==1 [1] TRUE > > tmat <- array(0., dim= dim(hmat)) # transition matrices > pstate <- matrix((4:0)/10, nrow=1) > for (i in 1:6) { + tmat[,,i] <- as.matrix(Matrix::expm(hmat[,,i])) + pstate <- rbind(pstate, pstate[i,]%*% tmat[,,i]) + } > > dtime <- which(surv3$time %in% etime) # skip censored rows > aeq(surv3$pstate[dtime[1:6],1,], pstate[-1,]) [1] TRUE > > # > # A function to do the above "by hand" calculations, over all time points > # It is verified for the particular fit we did, but written for > # more generality. > # fit: a multi-state fit, with shared baselines > # istate: the inital state for each row of data > # p0: starting dist for compuation > # x0: curve for this set of covariates > # > mysurv <- function(fit, istate, p0, x0, debug=0) { + if (!inherits(fit, 'coxphms')) stop("invalid fit") + smap <- fit$smap + from <- as.numeric(sub(":.*$", "", colnames(smap))) + to <- as.numeric(sub("^.*:", "", colnames(smap))) + shared <- duplicated(smap[1,]) + nshare <- sum(shared) + bcoef <- rep(1, ncol(smap)) # coefficients for shared baseline + beta <- coef(fit, matrix=TRUE) + if (nshare >0) { + # coefficients for shared baseline will be the last nshare of them + i <- seq(length=nshare, to=length(fit$coefficients)) + bcoef[shared] <- exp(fit$coefficients[i]) + # remove shared coef rows from beta + phrow <- apply(fit$cmap, 1, function(x) any(x %in% i)) + beta <- beta[!phrow,, drop=FALSE] + } + + # Make the values for istate and state match the 1:2, etc of the fit, + # i.e., the order of fit$states + # istate and state are used in tables, using factors makes sure the result + # is always the right size + nstate <- length(fit$states) + state <- factor(fit$y[,3], 1:nstate) # endpoint of a transition + if (length(istate) != nrow(fit$y)) stop ("mismatched istate") + istate <- factor(as.character(istate), fit$states) + + # set up output + ntran <- ncol(smap) # number of transitions + utime <- sort(unique(fit$y[!is.na(state), 2])) # unique event times + ntime <- length(utime) + tmat <- matrix(0, nstate, nstate) # transtion matrix at this time point + pmat <- diag(nstate) # product of transitions + nrisk <- matrix(0., ntime, nstate) #number at risk + wtrisk<- matrix(0., ntime, ntran) # weighted number per transtion + nevent <- matrix(0L, ntime, nstate) # number of events of each type + pstate <- matrix(0L, ntime, nstate) # probability in state + hmat <- matrix(0., nstate, nstate) # working matrix of hazards + + # eta is a matrix of (x for subject - x0) %*% coef, one row per subject, + # one column per transition + eta <- (fit$x - rep(x0, each= nrow(fit$y))) %*% beta + rwt <- exp(eta) # the risk weight for each obs + + t1 <- fit$y[,1] + t2 <- fit$y[,2] + for (i in 1:ntime) { + atrisk <- (t1 < utime[i] & utime[i] <= t2) # risk set at this time + event <- which(utime[i] == t2) # potential events, at this time + nrisk[i,] <- c(table(istate[atrisk])) # number at risk in each state + nevent[i,] <- c(table(state[event])) + # The linear predictor and hence the number at risk is different for + # every transition. Also, some will not be at risk for the transition. + # + for (k in 1:ntran) { + atrisk2 <- (atrisk & (as.numeric(istate) == from[k])) + wtrisk[i,k] <- sum(rwt[atrisk2,k]) + } + dtemp <- table(istate[event], state[event]) #censors don't count + + # fill in hmat, one hazard at a time + hmat <- 0*hmat + for (j in unique(smap)) { + # for each baseline hazard + k <- which(smap == j) # transitons that share this hazard + deaths <- sum(dtemp[cbind(from[k], to[k])]) # total events + if (deaths==0) hmat[cbind(from[k], to[k])] <- 0 # avoid 0/0 + else { + hazard <- deaths/ sum(wtrisk[i, k] * bcoef[k]) #shared baseline + hmat[cbind(from[k], to[k])] <- hazard * bcoef[k] # PH + } + } + diag(hmat) <- diag(hmat) - rowSums(hmat) # rows sum to zero + tmat <- as.matrix(Matrix::expm(hmat)) # transtion matrix + # if (i >= debug) browser() + pmat <- pmat %*% tmat + pstate[i,] <- drop(p0 %*% pmat) + } + list(time=utime, nrisk=nrisk, nevent=nevent, pstate=pstate, + wtrisk= wtrisk, P=pmat) + } > > test3 <- mysurv(fit3, pbc3$bili4, p0= 4:0/10, x0 =50) > aeq(test3$pstate, surv3$pstate[match(test3$time, surv3$time),1,]) [1] TRUE > > # Now with the full data set > fit2 <- coxph(list(Surv(tstart, tstop, bstat) ~ 1, + c(1:4):5 ~ age / common + shared), id= id, istate=bili4, + data=pbc2, ties='breslow', x=TRUE) > surv2 <- survfit(fit2, newdata=list(age=50), p0=c(.4, .3, .2, .1, 0)) > test2 <- mysurv(fit2, pbc2$bili4, p0= 4:0/10, fit2, x0 =50) > aeq(test2$pstate, surv2$pstate[match(test2$time, surv2$time),1,]) [1] TRUE > > > if (FALSE){ + # for testing, make a plot + xfun <- function(i) { + j <- match(test2$time[i], surv2$time) + all.equal(test2$pstate[i,], surv2$pstate[j,1,]) + } + plot(surv2, col=1:5, lwd=2) + matpoints(test2$time, test2$pstate, col=1:5, pch='o') + } > > > proc.time() user system elapsed 1.71 0.21 1.92