Skip to contents

Survival Distributions with piece-wise constant hazards and multiple states

Source: R/multistate.R
dmstate.Rd

Densitiy, distribution function, hazard function, cumulative hazard function and survival function of multi-state survival functions.

Usage

dmstate(x, t, Q, pi, abs)

pmstate(q, t, Q, pi, abs)

hmstate(x, t, Q, pi, abs)

chmstate(x, t, Q, pi, abs)

smstate(q, t, Q, pi, abs)

Arguments

x

vector of quantiles

t

vector of left interval borders

Q

Q-matrices of the process, see details

pi

initial distribution

abs

indicator vector of absorbing states, see details

q

vector of quantiles

Value

dmstate gives the density evaluated at x.

pmstate gives the distribution function evaluated at q.

hmstate gives the hazard function evaluated at x.

chmstate gives the cumulative hazard function evaluated at x.

smstate gives the survival function evaluated at q.

Details

Q is an array of dimensions N x N x M where M is the number of time intervals and N is the number of states. Every slice of Q along the third dimension is an N x N Q-matrix. Each row of the Q-matrix contains the hazard-rates for transitioning from the respective state to each other state in the off-diagonal elements. The diagonal element is minus the sum of the other elements (such that the row sums are 0 for each row). (See Norris (1997) Part 2, Continuous-time Markov chains I, for the definition of Q-matrices and the theory of continuous time markov chains.)

abs is a vector that is one for each absorbing state that corresponds to an event of interest and zero everywhere else. With this different events of interest can be encoded for the same model. For example overall survival and progression free survival can be encoded by setting abs to one in the "death" state or the "death" and the "progressed disease" state and leaving Q and pi the same.

The initial distribution pi can be used to set the probabilities of starting in different stages. The starting distribution in combination with Q can be used to model sub-populations. The corresponding values of pi are then the prevalence of the sub-populations in the initial state.

The densities, distribution functions, ... now correspond to the event of entering one of the absorbing states when the initial distribution in the states is pi.

Functions

  • dmstate(): density of survival distributions for a piece-wise exponential multi-state model

  • pmstate(): distribution function of survival distributions for a piece-wise exponential multi-state model

  • hmstate(): hazard of survival distributions for a piece-wise exponential multi-state model

  • chmstate(): cumulative hazard of survival distributions for a piece-wise exponential multi-state model

  • smstate(): survival function of survival distributions for a piece-wise exponential multi-state model

References

Norris, J. R. (1997) Markov Chains Cambridge University Press

Examples

# Example 1: Proportional Hazards
Tint <- 0
Q <- matrix(
  c(
    -0.1, 0.1,
    0  , 0
  ), 2, 2, byrow = TRUE
)
dim(Q) <- c(2,2,1)
pi <- c(1,0)
abs <- c(0,1)

t <- 0:100

par(mfrow=c(3,2))
plot.new()
text(0.5,0.5,"example 1 proportional hazards")
plot(t, pmstate(t, Tint, Q, pi, abs), type="l")
plot(t, smstate(t, Tint, Q, pi, abs), type="l")
plot(t, dmstate(t, Tint, Q, pi, abs), type="l")
plot(t, hmstate(t, Tint, Q, pi, abs), type="l", ylim=c(0,1))
plot(t, chmstate(t, Tint, Q, pi, abs), type="l")


# Example 2: Disease Progression
Tint <- 0
Q <- matrix(
  c(
    -0.3, 0.2, 0.1,
    0  ,-0.4, 0.4,
    0  ,   0,   0
  ), 3, 3, byrow = TRUE
)
dim(Q) <- c(3,3,1)
pi <- c(1,0,0)
abs_os  <- c(0,0,1)
abs_pfs <- c(0,1,1)

t <- seq(0,20, by=0.1)

par(mfrow=c(3,2))
plot.new()
text(0.5,0.5,"example 2a disease progression\noverall survival")
plot(t, pmstate(t, Tint, Q, pi, abs_os), type="l")
plot(t, smstate(t, Tint, Q, pi, abs_os), type="l")
plot(t, dmstate(t, Tint, Q, pi, abs_os), type="l")
plot(t, hmstate(t, Tint, Q, pi, abs_os), type="l", ylim=c(0,1))
plot(t, chmstate(t, Tint, Q, pi, abs_os), type="l")


par(mfrow=c(3,2))
plot.new()
text(0.5,0.5,"example 2b disease progression\nprogression-free survival")
plot(t, pmstate(t, Tint, Q, pi, abs_pfs), type="l")
plot(t, smstate(t, Tint, Q, pi, abs_pfs), type="l")
plot(t, dmstate(t, Tint, Q, pi, abs_pfs), type="l")
plot(t, hmstate(t, Tint, Q, pi, abs_pfs), type="l", ylim=c(0,1))
plot(t, chmstate(t, Tint, Q, pi, abs_pfs), type="l")


# Example 3: Sub-Populations
Tint <- 0
Q <- matrix(
  c(
    -0.4, 0  , 0.4,
    0  ,-0.1, 0.1,
    0  ,   0,   0
  ), 3, 3, byrow = TRUE
)
dim(Q) <- c(3,3,1)
pi <- c(0.5,0.5,0)
abs <- c(0,0,1)

t <- seq(0, 40, by=0.1)

par(mfrow=c(3,2))
plot.new()
text(0.5,0.5,"example 3 sub-populations")
plot(t, pmstate(t, Tint, Q, pi, abs), type="l")
plot(t, smstate(t, Tint, Q, pi, abs), type="l")
plot(t, dmstate(t, Tint, Q, pi, abs), type="l")
plot(t, hmstate(t, Tint, Q, pi, abs), type="l", ylim=c(0,1))
plot(t, chmstate(t, Tint, Q, pi, abs), type="l")



# Example 4: Delayed Effect in one group and immediate effect in the other group
Tint <- c(0,20)
Q <- array(NA_real_, dim=c(3,3,2))
Q[,,1] <- matrix(
  c(
    -0.2, 0   , 0.2 ,
    0  ,-0.05, 0.05,
    0  ,    0, 0
  ), 3, 3, byrow = TRUE
)
Q[,,2] <- matrix(
  c(
    -0.05, 0   , 0.05 ,
    0  ,-0.05, 0.05,
    0  ,    0, 0
  ), 3, 3, byrow = TRUE
)

pi <- c(0.75,0.25,0)
abs <- c(0,0,1)

t <- seq(0, 100, by=0.1)

par(mfrow=c(3,2))
plot.new()
text(0.5,0.5,"example 4\ndelayed effect in one group\nimmediate effect in the other")
plot(t, pmstate(t, Tint, Q, pi, abs), type="l")
plot(t, smstate(t, Tint, Q, pi, abs), type="l")
plot(t, dmstate(t, Tint, Q, pi, abs), type="l")
plot(t, hmstate(t, Tint, Q, pi, abs), type="l", ylim=c(0,0.2))
plot(t, chmstate(t, Tint, Q, pi, abs), type="l")