# tests/mstate.R In survival: Survival Analysis

```#
# 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))
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))
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))
all.equal(mfit\$time, c(2, 3, 4, 5, 8, 9, 10, 11))

# 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])
aeq(mfit2\$p0, p0(w1)[swap])

mfit2b <- survfit(Surv(t1, t2, st) ~ 1, mtest2, id=id, istate=i0)#scrambled
aeq(mfit2b\$pstate, dopstate(w1)[,swap])
aeq(mfit2b\$p0, p0(w1)[swap])

mfit2b\$call <- mfit2\$call <- NULL
all.equal(mfit2b, mfit2)
aeq(mfit2\$transitions, c(2,0,1,0, 0,2,0,0, 1,1,1,0, 0,0,0,2))

# 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])
aeq(mfit3\$pstate, dopstate(1:10)[,swap])

# 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])
aeq(mfit3\$influence[,2,], dp2(w1)[,swap])
aeq(mfit3\$influence[,3,], dp3(w1)[,swap])
aeq(mfit3\$influence[,4,], dp4(w1)[,swap])
aeq(mfit3\$influence[,5,], dp5(w1)[,swap])
aeq(mfit3\$influence[,6,], dp8(w1)[,swap])
aeq(mfit3\$influence[,7,], dp9(w1)[,swap])
aeq(mfit3\$influence[,8,], dp10(w1)[,swap])
aeq(mfit3\$influence[,9,], dp10(w1)[,swap]) # no changes at time 11

# 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='.'))

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])
aeq(hazard(mfit3, 2), aj3(w1)[swap, swap])
aeq(hazard(mfit3, 3), aj4(w1)[swap, swap])
aeq(hazard(mfit3, 4), aj5(w1)[swap, swap])
aeq(hazard(mfit3, 5), aj8(w1)[swap, swap])
aeq(hazard(mfit3, 6), aj9(w1)[swap, swap])
```

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survival documentation built on Aug. 24, 2021, 5:06 p.m.