R: Coxian phase-type distribution with two phases

2phase {msm} | R Documentation

## Coxian phase-type distribution with two phases

### Description

Density, distribution, quantile functions and other utilities for the Coxian phase-type distribution with two phases.

### Usage

      d2phase(x, l1, mu1, mu2, log=FALSE)
      p2phase(q, l1, mu1, mu2, lower.tail=TRUE, log.p=FALSE)
      q2phase(p, l1, mu1, mu2, lower.tail=TRUE, log.p=FALSE)
      r2phase(n, l1, mu1, mu2)
      h2phase(x, l1, mu1, mu2, log=FALSE)

### Arguments

`x,q` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations. If `length(n) > 1`, the length is taken to be the number required.
`l1` | Intensity for transition between phase 1 and phase 2.
`mu1` | Intensity for transition from phase 1 to exit.
`mu2` | Intensity for transition from phase 2 to exit.
`log` | logical; if TRUE, return log density or log hazard.
`log.p` | logical; if TRUE, probabilities p are given as log(p).
`lower.tail` | logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

### Details

This is the distribution of the time to reach state 3 in a continuous-time Markov model with three states and transitions permitted from state 1 to state 2 (with intensity lambda1) state 1 to state 3 (intensity mu1) and state 2 to state 3 (intensity mu2). States 1 and 2 are the two "phases" and state 3 is the "exit" state.

The density is

f(t | l1, mu1) = exp(-(l1+mu1)*t)*(mu1 + (l1+mu1)*l1*t)

if l1 + mu1 = mu2, and

f(t | l1, mu1, mu2) = ((l1+mu1)*exp(-(l1+mu1)*t)*(mu2-mu1) + mu2*l1*exp(-mu2*t))/(l1+mu1-mu2)

otherwise. The distribution function is

F(t | l1, mu1) = 1 - exp(-(l1+mu1)*t)*(1 + l1*t)

if l1 + mu1 = mu2, and

F(t | l1, mu1, mu2) = 1 - (exp(-(l1+mu1)*t)*(-mu1+mu2) + l1*exp(-mu2*t))/(l1+mu1-mu2)

otherwise. Quantiles are calculated by numerically inverting the distribution function.

The mean is (1 + l1/mu2) / (l1 + mu1).

The variance is (2 + 2*l1*(l1+mu1+ mu2)/mu2^2 - (1 + l1/mu2)^2)/(l1+mu1)^2.

If mu1=mu2 it reduces to an exponential distribution with rate mu1, and the parameter l1 is redundant. Or also if l1=0.

The hazard at x=0 is μ_1, and smoothly increasing if mu1<mu2. If l1 + mu1 >= mu2 it increases to an asymptote of mu2, and if l1 + mu1 <= mu2 it increases to an asymptote of l1 + mu1. The hazard is decreasing if mu1>mu2, to an asymptote of mu2.

### Value

`d2phase` gives the density, `p2phase` gives the distribution function, `q2phase` gives the quantile function, `r2phase` generates random deviates, and `h2phase` gives the hazard.

### Alternative parameterisation

An individual following this distribution can be seen as coming from a mixture of two populations:

1) "short stayers" whose mean sojourn time is M1 = 1/(l1+mu1) and sojourn distribution is exponential with rate l1+mu1.

2) "long stayers" whose mean sojourn time 1/(l1+mu1) + 1/mu2 and sojourn distribution is the sum of two exponentials with rate l1+mu1 and mu2 respectively. The individual is a "long stayer" with probability p=λ_1/(λ_1 + μ_1).

Thus a two-phase distribution can be more intuitively parameterised by the short and long stay means M_1 < M_2 and the long stay probability p. Given these parameters, the transition intensities are l1=p/M1, mu1=(1-p)/M1, and mu2 = 1/(M2 - M1). This can be useful for choosing intuitively reasonable initial values for procedures to fit these models to data.

The hazard is increasing at least if M2 < 2M1, and also only if (M2 - 2M1)/(M2 - M1) < p.

For increasing hazards with l1 + mu1 <= mu2, the maximum hazard ratio between any time t and time 0 is 1/(1-p).

For increasing hazards with l1 + mu1 >= mu2, the maximum hazard ratio is M1/((1-p)(M2 - M1)). This is the minimum hazard ratio for decreasing hazards.

### General phase-type distributions

This is a special case of the n-phase Coxian phase-type distribution, which in turn is a special case of the (general) phase-type distribution. The actuar R package implements a general n-phase distribution defined by the time to absorption of a general continuous-time Markov chain with a single absorbing state, where the process starts in one of the transient states with a given probability.

### Author(s)

C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk

### References

C. Dutang, V. Goulet and M. Pigeon (2008). actuar: An R Package for Actuarial Science. Journal of Statistical Software, vol. 25, no. 7, 1-37. URL http://www.jstatsoft.org/v25/i07

[Package msm version 1.6.9 Index]