
R Under development (unstable) (2024-04-01 r86255) -- "Unsuffered Consequences"
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Platform: x86_64-pc-linux-gnu

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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, time0=FALSE)
> 
> # 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
> 
> aeq(mfit$n.risk,  matrix(c(4,4,3,2,1,1,0,0,
+                                  0,1,1,2,2,1,0,0,
+                                  0,0,1,1,1,1,2,1,
+                                  0,0,0,0,0,1,0,0), ncol=4))
[1] TRUE
> aeq(mfit$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
> aeq(mfit$n.transition, matrix(c(1,0,1,0,0,0,0,0,
+                            0,0,0,0,0,0,1,0,
+                            0,1,0,1,0,0,0,0,
+                            0,0,0,0,0,1,0,0,
+                            0,0,0,0,0,1,0,0,
+                            0,0,0,0,1,0,0,0), ncol=6))
[1] TRUE
> all.equal(mfit$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", "a", "b", "c"))
> tdata$i0 <- factor(tdata$i0, c(4, 1:3),
+                     labels=c("entry", "a", "b", "c"))
> if (FALSE) {
+     #useful picture     
+     temp <- survcheck(Surv(t1, t2, st) ~1, tdata, id=id, istate=i0)
+     plot(c(0,11), c(1,5.5), type='n', xlab="Time", ylab= "Subject")
+     with(tdata, segments(t1+.1, id, t2, id, col=as.numeric(temp$istate)))
+     with(subset(tdata, st!= "censor"),
+           text(t2, id+.15, as.character(st)))
+     with(tdata, text((t1+t2)/2, id+.25, wt))
+     with(subset(tdata, !duplicated(id)),
+            text(t1, id+.15, as.character(i0)))
+     #abline(v=c(2:5, 8:11), lty=3, col='gray')
+ }
> 
> 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
> 
> # When the functions below were written, the entry state was listed last.
> # Currently the entry state is first, so "[swap]" was added to the aj routines
> #  rather than rearranging the formulas
> swap <- c(4,1,2,3)
> p0 <- function(w) c( w[1]+ w[6], w[4], w[9], 0)/ (w[1]+ w[4] + w[6] + w[9])
> 
> #  aj2 = Aalen-Johansen H matrix at time 2, etc.
> aj2 <- function(w) {
+     #subject 4 moves from entry to 'a'  
+     rbind(c(1, 0, 0, 0),  
+           c(0, 1, 0, 0),
+           c(0, 0, 1, 0),
+           c(w[6], 0, 0, w[1])/(w[1] + w[6]))[swap, swap] 
+ }
> 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))[swap,swap]
> aj4 <- function(w) {
+     # subject 1 moves from entry to a
+     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])) [swap, swap]
+ }
> aj5 <- function(w) {
+     # subject 2 from a to b
+     rbind(c(w[2]+w[7], w[4], 0, 0)/(w[2]+ w[4] + w[7]), 
+                          c(0, 1, 0, 0),  
+                          c(0, 0, 1, 0),
+                          c(0, 0, 0, 1))[swap, swap]
+ }
> aj8 <- function(w) rbind(c(w[2], 0, w[7], 0)/(w[2]+ w[7]), # 4  a to c
+                          c(0, 1, 0, 0),  
+                          c(0, 0, 1, 0),
+                          c(0, 0, 0, 1))[swap, swap]
> aj9 <- function(w) rbind(c(0, 1, 0, 0), # 1  a to b
+                          c(0, 1, 0, 0),  
+                          c(0, 0, 1, 0),
+                          c(0, 0, 1 ,0)) [swap, swap] # 3 entry to c
> aj10 <- function(w)rbind(c(1, 0, 0, 0),
+                          c(1, 0, 0, 0),  #1 b to a
+                          c(0, 0, 1, 0),
+                          c(0, 0, 0, 1))[swap, swap]
> 
> #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, 
+                  time0=FALSE) # ordered
> aeq(mfit2$pstate, dopstate(w1))
[1] TRUE
> aeq(mfit2$p0, p0(w1))
[1] TRUE
> 
> mfit2b <- survfit(Surv(t1, t2, st) ~ 1, mtest2, id=id, istate=i0, time0=FALSE)
> aeq(mfit2b$pstate, dopstate(w1))
[1] TRUE
> aeq(mfit2b$p0, p0(w1))
[1] TRUE
> 
> mfit2b$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, time0=FALSE)
> aeq(mfit3$p0, p0(1:10))
[1] TRUE
> aeq(mfit3$pstate, dopstate(1:10))
[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[1]+ w[6], w[4], w[9], 0)/ (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) {   # influence just before the first event
+   p <- p0(w)
+   w0 <- w[c(1,4,6,9)]  # the 4 obs at the start, subjects 1, 2, 4, 5
+   rbind(c(1,0, 0, 0) - p,   # subject 1 affects p[entry]
+         c(0,1, 0, 0) - p,   # subject 2 affects p[a]
+         0,                   # subject 3 affects none
+         c(1, 0, 0, 0) - p,   # subject 4 affect p[entry]
+         c(0, 0, 1, 0) - p)/   # subject 5 affects p[b]
+       sum(w0)
+ }
>   
> 
> dp2 <- function(w) {
+     h2 <- aj2(w)   # H matrix at time 2
+     part1 <- dp0(w) %*% h2
+ 
+     # 1 and 4 in entry, obs 4 moves from entry to a
+     mult  <- p0(w)[1]/(w[1] + w[6])  #p(t-) / weights in state
+     part2 <- rbind((c(1,0,0,0)- h2[1,]) * mult,
+                    0,
+                    0,
+                    (c(0,1,0,0) - h2[1,]) * 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 entry (obs 1 and 5) 1 moves to a
+     mult <- dopstate(w)[2,1]/ (w[1] + w[5])   # p_1(time 4-0) / wt
+     part2 <- rbind((c(0,1,0,0)- h4[1,]) * mult,
+                    0,
+                    (c(1,0,0,0)- h4[1,]) * mult,
+                    0,
+                    0)
+     part1 + part2
+ }
> dp5 <- function(w) {
+     h5 <- aj5(w)   # H matrix at time 5
+     part1 <- dp4(w) %*% h5
+ 
+     # subjects 124 in state a (obs 2,4,7), 2 goes to b
+     mult <- dopstate(w)[3,2]/ (w[2] + w[4] + w[7]) 
+     part2 <- rbind((c(0,1,0,0)- h5[2,]) * mult,
+                    (c(0,0,1,0)- h5[2,]) * mult,
+                    0,
+                    (c(0,1,0,0)- h5[2,]) * mult,
+                    0)
+     part1 + part2
+ }
> dp8 <- function(w) {
+     h8 <- aj8(w)   # H matrix at time 8
+     part1 <- dp5(w) %*% h8
+ 
+     # subjects 14 in state a (obs 2 &7), 4 goes to c
+     mult <- dopstate(w)[4, 2]/ (w[2] + w[7]) 
+     part2 <- rbind((c(0,1,0,0)- h8[2,]) * mult,
+                    0,
+                    0,
+                    (c(0,0,0,1)- h8[2,]) * mult,
+                    0)
+     part1 + part2
+ }
> dp9 <- function(w) dp8(w) %*% aj9(w)
> dp10<- function(w) dp9(w) %*% aj10(w)
> 
> #
> # Feb 4 2024: discovered that the variance computation above is incorrect.
> # Let U = influence for phat, with one row per observation in the data
> # The weighted per subject influence is Z(t)= BDU(t) where 
> #  B= rbind(c(1,1,1,0,0,0,0,0,0,0), 
> #           c(0,0,0,1,0,0,0,0,0,0),
> #           c(0,0,0,0,1,0,0,0,0,0),
> #           c(0,0,0,0,0,1,1,1,0,0),
> #           c(0,0,0,0,0,0,0,0,1,1))
> #  and D= diag(1:10)
> # which can be summarized as "weight each row, then add over subjects". 
> # The variance at time t is the column sums of Z^2(t) (elementwise squares)
> #
> # The code above for dp0, dp2, etc returns BU, which matches the computation of
> #  the influence in survfitci.c. If the weight for a given subject is constant 
> #  over time, then BDU= WBU where W is the diagonal matrix of per-subject 
> #  weights: survfitci.c implicitly made this assumption, and was correct
> #  in this case. It returned U as the influence, which matches dp0 etc.
> #
> # survfitci.c has been replaced by survfitaj.c, which uses the careful
> # derivations in the methods vignette, and returns BDU.
> # The checks below have been changed to a case with constant weights per
> #  subject.  R code to test for general weights is in mstate2.R
> #
> w1 <- tdata$id 
> mfit4 <- survfit(Surv(t1, t2, st) ~1, tdata, id=id, weights=id, istate=i0,
+                  influence=TRUE, time0= FALSE)
> aeq(mfit4$influence[,1,], 1:5*dp2(w1))  #time 2
[1] TRUE
> aeq(mfit4$influence[,2,], 1:5*dp3(w1))
[1] TRUE
> aeq(mfit4$influence[,3,], 1:5*dp4(w1))
[1] TRUE
> aeq(mfit4$influence[,4,], 1:5*dp5(w1))
[1] TRUE
> aeq(mfit4$influence[,5,], 1:5*dp8(w1)) # time 8
[1] TRUE
> aeq(mfit4$influence[,6,], 1:5* dp9(w1))
[1] TRUE
> aeq(mfit4$influence[,7,], 1:5* dp10(w1))
[1] TRUE
> aeq(mfit4$influence[,8,], 1:5* dp10(w1)) # no changes at time 11
[1] TRUE
> 
> ssq <- function(x) sqrt(sum(x^2))
> temp2 <- apply(mfit4$influence.pstate, 2:3, ssq)
> aeq(temp2, mfit4$std.err)  
[1] TRUE
> 
> if (FALSE) { # old test, survfitci returned the time 0 influence as well
+     w1 <- 1:10
+     aeq(mfit3$influence[,1,], dp0(w1))
+     aeq(mfit3$influence[,2,], dp2(w1))
+     aeq(mfit3$influence[,3,], dp3(w1))
+     aeq(mfit3$influence[,4,], dp4(w1))
+     aeq(mfit3$influence[,5,], dp5(w1))
+     aeq(mfit3$influence[,6,], dp8(w1))
+     aeq(mfit3$influence[,7,], dp9(w1))
+     aeq(mfit3$influence[,8,], dp10(w1))
+     aeq(mfit3$influence[,9,], dp10(w1)) # no changes at time 11
+ } # end of if (FALSE)
> 
> # The cumulative hazard at each time point is remapped from a matrix
> #  into a vector (in survfit)
> # First check out the names
> nstate <- length(mfit4$states)
> temp <- matrix(0, nstate, nstate)
> indx1 <- match(rownames(mfit4$transitions), mfit4$states)
> indx2 <- match(colnames(mfit4$transitions), mfit4$states, nomatch=0)
> temp[indx1, indx2] <- mfit4$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(mfit4$cumhaz), paste(from, to, sep='.'))
[1] TRUE
> 
> # check the cumulative hazard
> temp <- mfit4$n.risk[,from]
> hazard <- mfit4$n.transition/ifelse(temp==0, 1, temp)
> aeq(apply(hazard, 2, cumsum), mfit4$cumhaz)
[1] TRUE
> 
> 
> proc.time()
   user  system elapsed 
  0.958   0.105   1.055 
