Coxian phase-type distribution with two phases

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.
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.
lower.tail: logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
log.p: logical; if TRUE, probabilities p are given as log(p).
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.

Value

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

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 $\lambda_1$) state 1 to state 3 (intensity $\mu_1$) and state 2 to state 3 (intensity $\mu_2$). States 1 and 2 are the two "phases" and state 3 is the "exit" state.

The density is

$$f(t | \lambda_1, \mu_1) = e^{-(\lambda_1+\mu_1)t}(\mu_1 + (\lambda_1+\mu_1)\lambda_1 t)$$

if $\lambda_1 + \mu_1 = \mu_2$, and

$$f(t | \lambda_1, \mu_1, \mu_2) = \frac{(\lambda_1+\mu_1)e^{-(\lambda_1+\mu_1)t}(\mu_2-\mu_1) + \mu_2\lambda_1e^{-\mu_2t}}{\lambda_1+\mu_1-\mu_2}$$

otherwise. The distribution function is

$$F(t | \lambda_1, \mu_1) = 1 - e^{-(\lambda_1+\mu_1) t} (1 + \lambda_1 t)$$

if $\lambda_1 + \mu_1 = \mu_2$, and

$$F(t | \lambda_1, \mu_1, \mu_2) = 1 - \frac{e^{-(\lambda_1 + \mu_1)t} (\mu_2 - \mu_1) + \lambda_1 e^{-\mu_2 t}}{ \lambda_1 + \mu_1 - \mu_2}$$

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

The mean is $(1 + \lambda_1/\mu_2) / (\lambda_1 + \mu_1)$.

The variance is $(2 + 2\lambda_1(\lambda_1+\mu_1+ \mu_2)/\mu_2^2 - (1 + \lambda_1/\mu_2)^2)/(\lambda_1+\mu_1)^2$.

If $\mu_1=\mu_2$ it reduces to an exponential distribution with rate $\mu_1$, and the parameter $\lambda_1$ is redundant. Or also if $\lambda_1=0$.

The hazard at $x=0$ is $\mu_1$, and smoothly increasing if $\mu_1<\mu_2$. If $\lambda_1 + \mu_1 \geq \mu_2$ it increases to an asymptote of $\mu_2$, and if $\lambda_1 + \mu_1 \leq \mu_2$ it increases to an asymptote of $\lambda_1 + \mu_1$. The hazard is decreasing if $\mu_1>\mu_2$, to an asymptote of $\mu_2$.

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 $M_1 = $$ 1/(\lambda_1+\mu_1)$ and sojourn distribution is exponential with rate $\lambda_1 + \mu_1$.

2) "long stayers" whose mean sojourn time $M_2 = $$ 1/(\lambda_1+\mu_1) + 1/\mu_2$ and sojourn distribution is the sum of two exponentials with rate $\lambda_1 + $$ \mu_1$ and $\mu_2$ respectively. The individual is a "long stayer" with probability $p=\lambda_1/(\lambda_1 + \mu_1)$.

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 $\lambda_1=p/M_1$, $\mu_1=(1-p)/M_1$, and $\mu_2=1/(M_2-M_1)$. 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 $M_2 < 2M_1$, and also only if $(M_2 - 2M_1)/(M_2 - M_1) < p$.

For increasing hazards with $\lambda_1 + \mu_1 \leq \mu_2$, the maximum hazard ratio between any time $t$ and time 0 is $1/(1-p)$.

For increasing hazards with $\lambda_1 + \mu_1 \geq \mu_2$, the maximum hazard ratio is $M_1/((1-p)(M_2 - M_1))$$ M_1))$. This is the minimum hazard ratio for decreasing hazards.

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

Author

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