R Under development (unstable) (2023-08-12 r84939 ucrt) -- "Unsuffered Consequences"
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> #
> # A tiny multi-state example
> #
> library(survival)
> aeq <- function(x,y) all.equal(as.vector(x), as.vector(y))
> mtest <- data.frame(id= c(1, 1, 1,  2,  3,  4, 4, 4,  5, 5),
+                     t1= c(0, 4, 9,  0,  2,  0, 2, 8,  1, 3),
+                     t2= c(4, 9, 10, 5,  9,  2, 8, 9,  3, 11),
+                     st= c(1, 2,  1, 2,  3,  1, 3, 0,  2,  0))
> 
> mtest$state <- factor(mtest$st, 0:3, c("censor", "a", "b", "c"))
> 
> if (FALSE) {
+     # this graph is very useful when debugging
+     temp <- survcheck(Surv(t1, t2, state) ~1, mtest, id=id)
+     plot(c(0,11), c(1,5.1), type='n', xlab="Time", ylab= "Subject")
+     with(mtest, segments(t1+.1, id, t2, id, col=as.numeric(temp$istate)))
+     event <- subset(mtest, state!='censor')
+     text(event$t2, event$id+.2, as.character(event$state))
+ }
> 
> mtest <- mtest[c(1,3,2,4,5,7,6,10, 9, 8),]  #not in time order
> 
> mfit <- survfit(Surv(t1, t2, state) ~ 1, mtest, id=id)
> 
> # True results
> #
> #time       state                    probabilities
> #         entry  a   b  c         entry  a    b     c
> #
> #0        124                      1     0    0     0
> #1+       1245
> #2+       1235   4                3/4   1/4   0     0    4 -> a, add 3
> #3+       123    4   5            9/16  1/4  3/16   0    5 -> b
> #4+        23    14  5            6/16  7/16 3/16   0    1 -> a
> #5+        3     14  5            3/16  7/16 6/16   0    2 -> b, exits
> #8+        3     1   5  4         3/16  7/32 6/16  7/32  4 -> c
> #9+                  15            0     0  19/32 13/32  1->b, 3->c & exit
> # 10+            1   5                19/64 19/64 13/32  1->a
> 
> # In mfit, the "entry" state is first in the matrices, when this function was
> #  first created it was the last.
> swap <- c(4,1,2,3)  # at one time it was last
> all.equal(mfit$n.risk,  matrix(c(4,4,3,2,1,1,0,0,
+                                  0,1,1,2,2,1,0,0,
+                                  0,0,1,1,1,1,2,1,
+                                  0,0,0,0,0,1,0,0), ncol=4))
[1] TRUE
> all.equal(mfit$pstate,  matrix(c(24, 18, 12,  6, 6, 0, 0,  0,
+                                   8,  8, 14, 14, 7, 0,  9.5, 9.5, 
+                                   0,  6,  6, 12, 12,19,9.5, 9.5, 
+                                   0,  0,  0,  0, 7, 13, 13, 13)/32, ncol=4))
[1] TRUE
> all.equal(mfit$n.event, matrix(c(0,0,0,0,0,0,0,0,
+                                  1,0,1,0,0,0,1,0,
+                                  0,1,0,1,0,1,0,0,
+                                  0,0,0,0,1,1,0,0), ncol=4))
[1] TRUE
> all.equal(mfit$time, c(2, 3, 4, 5, 8, 9, 10, 11))
[1] TRUE
> 
> 
> # Somewhat more complex.
> #  Scramble the input data
> #  Not everyone starts at the same time or in the same state
> #  Case weights
> #
> tdata <- data.frame(id= c(1, 1, 1,  2,  3,  4, 4, 4,  5,  5),
+                     t1= c(0, 4, 9,  1,  2,  0, 2, 8,  1,  3),
+                     t2= c(4, 9, 10, 5,  9,  2, 8, 9,  3, 11),
+                     st= c(1, 2,  1, 2,  3,  1, 3, 0,  3,  0),
+                     i0= c(4, 1,  2, 1,  4,  4, 1, 3,  2,  3),
+                     wt= 1:10)
> 
> tdata$st <- factor(tdata$st, c(0:3),
+                     labels=c("censor", "1", "2", "3"))
> tdata$i0 <- factor(tdata$i0, 1:4,
+                     labels=c("1", "2", "3", "entry"))
> 
> tfun <- function(data=tdata) {
+     reorder <- c(10, 9, 1, 2, 5, 4, 3, 7, 8, 6)
+     new <- data[reorder,]
+     new
+ }
> 
> # These weight vectors are in the order of tdata
> # w[9] is the weight for subject 5 at time 1.5, for instance
> # p0 is defined as all those at risk just before the first event, which in
> #  this data set is entry:a at time 2 for id=4; id 1,2,4,5 at risk
> p0 <- function(w) c(w[4], w[9], 0, w[1]+ w[6])/ (w[1]+ w[4] + w[6] + w[9])
> 
> #  aj2 = Aalen-Johansen H matrix at time 2, etc.
> aj2 <- function(w) {
+     rbind(c(1, 0, 0, 0),    # state a (1) stays put
+           c(0, 1, 0, 0),
+           c(0, 0, 1, 0),
+           c(w[6], 0, 0, w[1])/(w[1] + w[6]))  #subject 4 moves to 'a'
+ }
> aj3 <- function(w) rbind(c(1, 0, 0, 0),   
+                          c(0, 0, 1, 0),  # 5 moves from b to c
+                          c(0, 0, 1, 0),
+                          c(0, 0, 0, 1))
> aj4 <- function(w) rbind(c(1, 0, 0, 0),
+                          c(0, 1, 0, 0),  
+                          c(0, 0, 1, 0),
+                          c(w[1], 0, 0, w[5])/(w[1] + w[5])) #1 moves from 4 to a
> aj5 <- function(w) rbind(c(w[2]+w[7], w[4], 0, 0)/(w[2]+ w[4] + w[7]), #2 to b
+                          c(0, 1, 0, 0),  
+                          c(0, 0, 1, 0),
+                          c(0, 0, 0, 1))
> aj8 <- function(w) rbind(c(w[2], 0, w[7], 0)/(w[2]+ w[7]), # 4  to c
+                          c(0, 1, 0, 0),  
+                          c(0, 0, 1, 0),
+                          c(0, 0, 0, 1))
> aj9 <- function(w) rbind(c(0, 1, 0, 0), # 1  to b
+                          c(0, 1, 0, 0),  
+                          c(0, 0, 1, 0),
+                          c(0, 0, 1 ,0)) # 3 to c
> aj10 <- function(w)rbind(c(1, 0, 0, 0),
+                          c(1, 0, 0, 0),  #1 back to a
+                          c(0, 0, 1, 0),
+                          c(0, 0, 0, 1))
> 
> #time       state               
> #         a   b  c  entry
> #
> #1        2   5     14       initial distribution
> #2        24  5     1        4 -> a, add 3
> #3        24     5  13       5 from b to c
> #4       124     5   3       1 -> a
> #5        14     5   3       2 -> b, exits
> #8        1      45  3       4 -> c
> #9            1  45          1->b, 3->c & exit
> #10       1      45          1->a
> 
> # P is a product of matrices
> dopstate <- function(w) {
+     p1 <- p0(w)
+     p2 <- p1 %*% aj2(w)
+     p3 <- p2 %*% aj3(w)
+     p4 <- p3 %*% aj4(w)
+     p5 <- p4 %*% aj5(w)
+     p8 <- p5 %*% aj8(w)
+     p9 <- p8 %*% aj9(w)
+     p10<- p9 %*% aj10(w)
+     rbind(p2, p3, p4, p5, p8, p9, p10, p10)
+ }
> 
> # Check the pstate estimate
> w1 <- rep(1,10)
> mtest2 <- tfun(tdata)  # scrambled order
> mfit2 <- survfit(Surv(t1, t2, st) ~ 1, tdata, id=id, istate=i0) # ordered
> aeq(mfit2$pstate, dopstate(w1)[,swap])
[1] TRUE
> aeq(mfit2$p0, p0(w1)[swap])
[1] TRUE
> 
> mfit2b <- survfit(Surv(t1, t2, st) ~ 1, mtest2, id=id, istate=i0)#scrambled
> aeq(mfit2b$pstate, dopstate(w1)[,swap])
[1] TRUE
> aeq(mfit2b$p0, p0(w1)[swap])
[1] TRUE
> 
> mfit2b$call <- mfit2$call <- NULL
> all.equal(mfit2b, mfit2) 
[1] TRUE
> aeq(mfit2$transitions, c(2,0,1,0, 0,2,0,0, 1,1,1,0, 0,0,0,2))
[1] TRUE
> 
> # Now the harder one, where subjects change weights
> mfit3  <- survfit(Surv(t1, t2, st) ~ 1, tdata, id=id, istate=i0,
+                   weights=wt, influence=TRUE)
> aeq(mfit3$p0, p0(1:10)[swap])
[1] TRUE
> aeq(mfit3$pstate, dopstate(1:10)[,swap])
[1] TRUE
>     
> 
> # The derivative of a matrix product AB is (dA)B + A(dB) where dA is the
> #  elementwise derivative of A and etc for B.
> # dp0 creates the derivatives of p0 with respect to each subject, a 5 by 4
> #  matrix
> # All the functions below are hand coded for a weight vector that is in
> #  exactly the same order as the rows of mtest.
> # Since p0 = (w[4], w[9], 0, w[1]+ w[6])/ (w[1]+ w[4] + w[6] + w[9])
> #      and subject id is 1,1,1, 2, 3, 4,4,4, 5,5
> #   we get the derivative below
> # 
> dp0 <- function(w) {
+   p <- p0(w)
+   w0 <- w[c(1,4,6,9)]  # the 4 obs at the start, subjects 1, 2, 4, 5
+   rbind(c(0, 0, 0, 1) - p,   # subject 1 affects p[4]
+         c(1, 0, 0, 0) - p,   # subject 2 affects p[1]
+         0,                   # subject 3 affects none
+         c(0, 0, 0, 1) - p,   # subject 4 affect p[4]
+         c(0, 1, 0, 0) - p)/   # subject 5 affects p[2]
+       sum(w0)
+ }
>   
> 
> dp2 <- function(w) {
+     h2 <- aj2(w)   # H matrix at time 2
+     part1 <- dp0(w) %*% h2
+ 
+     # 1 and 4 in state 4, obs 4 moves from entry to a
+     mult  <- p0(w)[4]/(w[1] + w[6])  #p(t-) / weights in state
+     part2 <- rbind((c(0,0,0,1)- h2[4,]) * mult,
+                    0,
+                    0,
+                    (c(1,0,0,0) - h2[4,]) * mult,
+                    0)
+     part1 + part2
+ }
> 
> dp3 <- function(w) {
+     dp2(w) %*% aj3(w)
+ }
> 
> dp4 <- function(w) {
+     h4 <- aj4(w)   # H matrix at time 4
+     part1 <- dp3(w) %*% h4
+ 
+     # subjects 1 and 3 in state 4, obs 1 and 5, 1 moves to a
+     mult <- dopstate(w)[2,4]/ (w[1] + w[5])   # p_4(time 4-0) / wt
+     part2 <- rbind((c(1,0,0,0)- h4[4,]) * mult,
+                    0,
+                    (c(0,0,0,1)- h4[4,]) * mult,
+                    0,
+                    0)
+     part1 + part2
+ }
> dp5 <- function(w) {
+     h5 <- aj5(w)   # H matrix at time 5
+     part1 <- dp4(w) %*% h5
+ 
+     # subjects 124 in state 1, obs 2,4,7, 2 goes to 2
+     mult <- dopstate(w)[3,1]/ (denom <- w[2] + w[4] + w[7]) 
+     part2 <- rbind((c(1,0,0,0)- h5[1,]) * mult,
+                    (c(0,1,0,0)- h5[1,]) * mult,
+                    0,
+                    (c(1,0,0,0)- h5[1,]) * mult,
+                    0)
+     part1 + part2
+ }
> dp8 <- function(w) {
+     h8 <- aj8(w)   # H matrix at time 8
+     part1 <- dp5(w) %*% h8
+ 
+     # subjects 14 in state 1, obs 2 &7, 4 goes to c
+     mult <- dopstate(w)[4, 1]/ (w[2] + w[7]) 
+     part2 <- rbind((c(1,0,0,0)- h8[1,]) * mult,
+                    0,
+                    0,
+                    (c(0,0,1,0)- h8[1,]) * mult,
+                    0)
+     part1 + part2
+ }
> dp9 <- function(w) dp8(w) %*% aj9(w)
> dp10<- function(w) dp9(w) %*% aj10(w)
> 
> w1 <- 1:10
> aeq(mfit3$influence[,1,], dp0(w1)[,swap])
[1] TRUE
> aeq(mfit3$influence[,2,], dp2(w1)[,swap])
[1] TRUE
> aeq(mfit3$influence[,3,], dp3(w1)[,swap])
[1] TRUE
> aeq(mfit3$influence[,4,], dp4(w1)[,swap])
[1] TRUE
> aeq(mfit3$influence[,5,], dp5(w1)[,swap])
[1] TRUE
> aeq(mfit3$influence[,6,], dp8(w1)[,swap])
[1] TRUE
> aeq(mfit3$influence[,7,], dp9(w1)[,swap])
[1] TRUE
> aeq(mfit3$influence[,8,], dp10(w1)[,swap])
[1] TRUE
> aeq(mfit3$influence[,9,], dp10(w1)[,swap]) # no changes at time 11
[1] TRUE
> 
> 
> # The cumulative hazard at each time point is remapped from a matrix
> #  into a vector (in survfit)
> # First check out the names
> nstate <- length(mfit3$states)
> temp <- matrix(0, nstate, nstate)
> indx1 <- match(rownames(mfit3$transitions), mfit3$states)
> indx2 <- match(colnames(mfit3$transitions), mfit3$states, nomatch=0)
> temp[indx1, indx2] <- mfit3$transitions[, indx2>0]
> # temp is an nstate by nstate version of the transitions matrix
> from <- row(temp)[temp>0]
> to   <- col(temp)[temp>0]
>  
> all.equal(colnames(mfit3$cumhaz), paste(from, to, sep='.'))
[1] TRUE
> 
> 
> hazard <- function(fit, i, indx=which(temp>0)) {
+     nstate <- length(fit$states)
+     cmat <- matrix(0, nstate, nstate)
+     if (i==1) cmat[indx] <- fit$cumhaz[i,]
+     else cmat[indx] <- fit$cumhaz[i,] - fit$cumhaz[i-1,]
+ 
+     diag(cmat) <- 1- rowSums(cmat)
+     cmat
+ }
> 
> aeq(hazard(mfit3, 1), aj2(w1)[swap, swap])
[1] TRUE
> aeq(hazard(mfit3, 2), aj3(w1)[swap, swap])
[1] TRUE
> aeq(hazard(mfit3, 3), aj4(w1)[swap, swap])
[1] TRUE
> aeq(hazard(mfit3, 4), aj5(w1)[swap, swap])
[1] TRUE
> aeq(hazard(mfit3, 5), aj8(w1)[swap, swap])
[1] TRUE
> aeq(hazard(mfit3, 6), aj9(w1)[swap, swap])
[1] TRUE
> 
> proc.time()
   user  system elapsed 
   1.01    0.09    1.09