2phase {msm} | R Documentation |
Density, distribution, quantile functions and other utilities for the Coxian phase-type distribution with two phases.
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)
x,q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
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]. |
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
.
d2phase
gives the density, p2phase
gives the distribution
function, q2phase
gives the quantile function, r2phase
generates random deviates, and h2phase
gives the hazard.
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))
. This is the minimum hazard ratio for
decreasing hazards.
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.
C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk
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