twophase: 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)

Value

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

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.

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.

Author

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

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 $\mu_2$.

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