rmcr: Random Number Generator for Mixture Cure Rate Distribution
In phe9480/rgs: What the Package Does (One Line, Title Case)
Description
Usage
Arguments
Details
Examples
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
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Arguments
n
Number of samples following the MCR distribution
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.
Draw u from U(0, 1);
if u >= 1 - S(tau)^(1-psi)*p^psi, let t = Infinity;
otherwise, let v = ((1-u1)^(1/psi) / (S(tau)^(1-psi)) - p) / (1-p),
and t = inv.S0(v), where inv.S0 is the inverse function of S0(t).
Repeat N times.
Then the N samples t_1, ..., t_N follow the MCR dist above.
Examples
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set.seed(2022)
#(1) Exponential distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/12
x = rmcr(n=10000, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0)
m = qmcr(pct=0.5, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0)
median(x)
km<-survival::survfit(survival::Surv(x,rep(1,length(x)))~rep(1,length(x)))
plot(km, ylab="Survival")
abline(h = c(p, 0.5), col="gray80", lty=2)
#(2) Weibull distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/12
x = rmcr(n=10000, p=0.15, alpha = log(2)/12, beta=1, gamma=0.9, lambda=0)
m = qmcr(pct=0.5, p=0.15, alpha = log(2)/12, beta=1, gamma=0.9, lambda=0)
median(x)
#(3) Rayleigh distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/100
x = rmcr(n=10000, p=0.15, alpha = alpha, beta=1, gamma=2, lambda=0)
m = qmcr(pct=0.5, p=0.15, alpha = alpha, beta=1, gamma=2, lambda=0)
median(x)
#(4) GMW distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/12
x = rmcr(n=10000, p=0.15, alpha = alpha, beta=1, gamma=1, lambda=0.2)
m = qmcr(pct=0.5, p=0.15, alpha = alpha, beta=1, gamma=1, lambda=0.2)
median(x)
#(5) No cure rate
x = rmcr(n=100000, p=0, alpha = log(2)/12, beta=1, gamma=1, lambda=0)
median(x)
#(6) Delayed effect 6 months, proportional hazards HR = 0.6
set.seed(2022)
#control arm:
x0 = rmcr(n=1000, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0, tau=0, psi=1)
median(x0)
m0 = qmcr(pct=0.5, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0, tau=0, psi=1)
m0
#Experimental arm:
x1 = rmcr(n=1000, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0, tau=6, psi=0.5)
median(x1)
m1 = qmcr(pct=0.5, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0, tau=6, psi=0.5)
m1
x = c(x0, x1); group = c(rep(0,length(x0)),rep(1,length(x1)))
event = rep(1, length(x)) #all events
km<-survival::survfit(survival::Surv(x,event)~group)
plot(km, ylab="Survival")
abline(h = c(p, 0.5), col="gray80", lty=2)
abline(v=c(6, m1, m0), col="gray80", lty=2)
phe9480/rgs documentation built on March 1, 2022, 12:26 a.m.
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Description Usage Arguments Details Examples
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
1 2 3 4 5 6 7 8 9 10 |
Arguments
n |
Number of samples following the MCR distribution |
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.
Draw u from U(0, 1); if u >= 1 - S(tau)^(1-psi)*p^psi, let t = Infinity; otherwise, let v = ((1-u1)^(1/psi) / (S(tau)^(1-psi)) - p) / (1-p), and t = inv.S0(v), where inv.S0 is the inverse function of S0(t). Repeat N times.
Then the N samples t_1, ..., t_N follow the MCR dist above.
Examples
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set.seed(2022)
#(1) Exponential distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/12
x = rmcr(n=10000, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0)
m = qmcr(pct=0.5, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0)
median(x)
km<-survival::survfit(survival::Surv(x,rep(1,length(x)))~rep(1,length(x)))
plot(km, ylab="Survival")
abline(h = c(p, 0.5), col="gray80", lty=2)
#(2) Weibull distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/12
x = rmcr(n=10000, p=0.15, alpha = log(2)/12, beta=1, gamma=0.9, lambda=0)
m = qmcr(pct=0.5, p=0.15, alpha = log(2)/12, beta=1, gamma=0.9, lambda=0)
median(x)
#(3) Rayleigh distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/100
x = rmcr(n=10000, p=0.15, alpha = alpha, beta=1, gamma=2, lambda=0)
m = qmcr(pct=0.5, p=0.15, alpha = alpha, beta=1, gamma=2, lambda=0)
median(x)
#(4) GMW distribution mixture with cure rate 0.15
p = 0.15; alpha = log(2)/12
x = rmcr(n=10000, p=0.15, alpha = alpha, beta=1, gamma=1, lambda=0.2)
m = qmcr(pct=0.5, p=0.15, alpha = alpha, beta=1, gamma=1, lambda=0.2)
median(x)
#(5) No cure rate
x = rmcr(n=100000, p=0, alpha = log(2)/12, beta=1, gamma=1, lambda=0)
median(x)
#(6) Delayed effect 6 months, proportional hazards HR = 0.6
set.seed(2022)
#control arm:
x0 = rmcr(n=1000, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0, tau=0, psi=1)
median(x0)
m0 = qmcr(pct=0.5, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0, tau=0, psi=1)
m0
#Experimental arm:
x1 = rmcr(n=1000, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0, tau=6, psi=0.5)
median(x1)
m1 = qmcr(pct=0.5, p=0.15, alpha = log(2)/12, beta=1, gamma=1, lambda=0, tau=6, psi=0.5)
m1
x = c(x0, x1); group = c(rep(0,length(x0)),rep(1,length(x1)))
event = rep(1, length(x)) #all events
km<-survival::survfit(survival::Surv(x,event)~group)
plot(km, ylab="Survival")
abline(h = c(p, 0.5), col="gray80", lty=2)
abline(v=c(6, m1, m0), col="gray80", lty=2)
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