dmcr: Density Function for Mixture Cure Rate Distribution

In phe9480/rgs: What the Package Does (One Line, Title Case)

Description

Consider the survival function of mixture cure rate (MCR) distribution: S(t) = p + (1-p)S0(t), where S0(t) is a survival distribution for susceptible subject, i.e., S0(0)=1 and S0(t) -> 0 as t -> Infinity. S0(t) can be any proper survival function. For generality of S0(t), consider the generalized modified Weibull (GMW) distribution with parameters (alpha, beta, gamma, lambda) Martinez et al(2013).

Usage

dmcr(
  t,
  p = 0.3,
  alpha = log(2)/12,
  beta = 1,
  gamma = 1,
  lambda = 0,
  tau = 0,
  psi = 1
)

Arguments

p	Cure rate parameter. When p = 0, it reduces to GMW(alpha, beta, gamma, lambda) distribution.
alpha	Generalized modified Weibull (GMW) distribution parameters. alpha > 0
beta	Generalized modified Weibull (GMW) distribution parameters. beta > 0
gamma	Generalized modified Weibull (GMW) distribution parameters. gamma >= 0 but gamma and lambda cannot be both 0.
lambda	Generalized modified Weibull (GMW) distribution parameters. lambda >= 0 but gamma and lambda cannot be both 0.
tau	Threshold for delayed effect period. tau = 0 reduces no delayed effect.
psi	Hazard ratio after delayed effect. psi = 1 reduces to the survival function without proportional hazards after delayed period (tau).

Details

alpha: scale parameter beta and gamma: shape parameters lambda: acceleration parameter

S0(t) = 1-(1-exp(-alphat^gammaexp(lambda*t)))^beta

Special cases: (1) Weibull dist: lambda = 0, beta = 1. Beware of the parameterization difference. (2) Exponential dist: lambda = 0, beta = 1, gamma = 1. The shape parameter (hazard rate) is alpha. (3) Rayleigh dist: lambda = 0, beta = 1, gamma = 2. (4) Exponentiated Weibull dist (EW): lambda = 0 (5) Exponentiated exponential dist (EE): lambda = 0 and gamma = 1 (6) Generalized Rayleigh dist (GR): lambda = 0, gamma = 2 (7) Modified Weibull dist (MW): beta = 1

Let T be the survival time according to survival function S1(t) below. Denote T's distribution as MCR(p, alpha, beta, gamma, lambda, tau, psi), where tau is the delayed effect and psi is the hazard ratio after delayed effect, ie. proportional hazards to S(t) after delay tau. S(t) = p + (1-p)*S0(t) S1(t) = S(t)I(t<tau) + S(tau)^(1-psi)*S(t)^psi In brief, S0(t) is the proper GMW distribution (alpha, beta, gamma, lambda); S(t) is MCR with additional cure rate parameter p; In reference to S(t), S1(t) is a delayed effect distribution and proportional hazards after delay.

MCR(p, alpha, beta, gamma, lambda, tau, psi=1) reduces to S(t) MCR(p, alpha, beta, gamma, lambda, tau=0, psi) reduces to S(t)^psi, i.e. proportional hazards. MCR(p=0, alpha, beta, gamma, lambda, tau=0, psi=1) reduces to S0(t) MCR(p=0, alpha, beta=1, gamma=1, lambda=0, tau=0, psi=1) reduces to exponential dist.

Value

An object with m: Median; and x: samples

Examples

#(1) Exponential distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/12
x = dmcr(t=12, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0)

t = seq(0,50)
d = sapply(t, dmcr, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0)
plot(d)

#(2) Weibull distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/12
x = dmcr(t=12, p=0.15, alpha = log(2)/12, beta=1, gamma=0.9, lambda=0)

d = sapply(t, dmcr, p=0.15, alpha = log(2)/12, beta=1, gamma=0.9, lambda=0)
plot(d)

#(3) Rayleigh distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/100
x = dmcr(t=12, p=0.15, alpha = alpha, beta=1, gamma=2, lambda=0)

#(4) GMW distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/12
x = dmcr(t=12, p=0.15, alpha = alpha, beta=1, gamma=1, lambda=0.2)

phe9480/rgs documentation built on March 1, 2022, 12:26 a.m.