smcr: Survival Function for Mixture Cure Rate Distribution
In phe9480/rgs: What the Package Does (One Line, Title Case)

Description Usage Arguments Details Value Examples

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).

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

`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).

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.

An object with m: Median; and x: samples

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

t=seq(1,100)
c = sapply(t, smcr, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0)
plot(t, c, type="l", ylab="", main="Survival Function")

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

#(3) Rayleigh distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/100
x = smcr(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 = smcr(t = 12, p = 0.15, alpha = alpha, beta=1, gamma=1, lambda=0.2)