R version 4.4.0 RC (2024-04-16 r86458 ucrt) -- "Puppy Cup"
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> library(survival)
> aeq <- function(x, y, ...) all.equal(as.vector(x), as.vector(y), ...)
> 
> # Check that a multi-state model, correctly set up, gives the same
> # solution as a time-dependent covariate.
> # This is a stronger test than mstrata: there the covariate which was mapped
> #  into a state was constant, here it is time-dependent.
> #
> # First build the TD data set from pbcseq, with a categorical bilirubin
> pbc1 <- pbcseq
> pbc1$bili4 <- cut(pbc1$bili, c(0,1, 2,4, 100), 
+                   c("normal", "1-2x", "2-4x", ">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", "1-2x", "2-4x", ">4", "death"))
> check1 <- survcheck(Surv(tstart, tstop, bstat) ~ 1, istate= bili4,
+                     id = id, data=pbc2)
> check1$transitions
        to
from     normal 1-2x 2-4x  >4 death (censored)
  normal      0   81   10   3     9         77
  1-2x       61    0   68  15     9         36
  2-4x        2   33    0  94    12         24
  >4          1    3   28   0   110         35
  death       0    0    0   0     0          0
> all.equal(as.character(pbc2$bili4), as.character(check1$istate))
[1] TRUE
> # the above verifies that I created the data set correctly
> 
> # Standard coxph fit with a time dependent bili4 variable.
> fit1 <- coxph(Surv(tstart, tstop, death) ~ age + bili4, pbc2)
> 
> # An additive multi-state fit, where bili4 is a state
> #  The three forms below should all give identical models
> fit2 <- coxph(list(Surv(tstart, tstop, bstat) ~ 1,
+                    c(1:4):5 ~ age / common + shared), id= id, istate=bili4,
+               data=pbc2)
> fit2b <- coxph(list(Surv(tstart, tstop, bstat) ~ 1,
+                    1:5 + 2:5 + 3:5 + 4:5 ~ age / common + shared), 
+               id= id, istate=bili4, data=pbc2)
> fit2c <- coxph(list(Surv(tstart, tstop, bstat) ~ 1,
+                    0:5 ~ age / common + shared), 
+               id= id, istate=bili4, data=pbc2)
> 
> # Make sure the names are correct and the coefficients match
> aeq(coef(fit1), coef(fit2))
[1] TRUE
> aeq(names(coef(fit2)), c("age", "ph(2:5/1:5)", "ph(3:5/1:5)", "ph(4:5/1:5)"))
[1] TRUE
> all.equal(coef(fit2), coef(fit2b))
[1] TRUE
> all.equal(coef(fit2), coef(fit2c))
[1] TRUE
> 
> # Now a model with a separate age effect for each bilirubin group
> fit3  <- coxph(Surv(tstart, tstop, death) ~ age*bili4, pbc2)
> fit3b <- coxph(Surv(tstart, tstop, death) ~ bili4/age, pbc2)
> fit4 <-  coxph(list(Surv(tstart, tstop, bstat) ~ 1,
+                    c(1:4):5 ~ age / shared), id= id, istate=bili4,
+               data=pbc2)
> all.equal(fit3$loglik, fit3b$loglik)
[1] TRUE
> all.equal(fit3$loglik, fit4$loglik)
[1] TRUE
> 
> # The coefficients are quite different due to different codings for dummy vars
> # Unpack the interaction, first 4 coefs will be the age effect within each
> #  bilirubin group
> temp <- c(coef(fit3)[1] + c(0, coef(fit3)[5:7]), coef(fit3)[2:4])
> names(temp)[1:4] <- c("age1", "age2", "age3", "age4")
> aeq(temp, coef(fit3b)[c(4:7, 1:3)])
[1] TRUE
> aeq(temp, coef(fit4))
[1] TRUE
> 
> # Third, a model with separate baseline hazards for each bili group
> fit5 <- coxph(Surv(tstart, tstop, death) ~ strata(bili4)/age, pbc2,
+               cluster=id)
> fit6 <- coxph(list(Surv(tstart, tstop, bstat) ~ 1, 0:5 ~ age),
+                    id=id, istate=bili4, pbc2)
> aeq(coef(fit5), coef(fit6))
[1] TRUE
> aeq(fit5$var, fit6$var)
[1] TRUE
> aeq(fit5$naive.var, fit6$naive.var)
[1] TRUE
> 
> proc.time()
   user  system elapsed 
   1.23    0.12    1.29