parfm: Parametric Frailty Models
In parfm: Parametric Frailty Models
parfm R Documentation
Parametric Frailty Models
Description
Fits parametric Frailty Models by maximizing the marginal likelihood.
Usage
parfm(formula, cluster = NULL, strata = NULL, data,
inip = NULL, iniFpar = NULL,
dist = c("weibull", "inweibull", "frechet", "exponential",
"gompertz", "loglogistic", "lognormal", "logskewnormal"),
frailty = c("none", "gamma", "ingau", "possta",
"lognormal", "loglogistic"),
method = "nlminb",
maxit = 500, Fparscale = 1, showtime = FALSE, correct = 0)
Arguments
formula
A formula object, with the response on the left of a ~
operator,
and the terms on the right.
The response must be a survival object as returned by the Surv()
function.
The status indicator 'event' in the Surv object must be 0=alive, 1=dead.
cluster
The name of a cluster variable in data.
strata
The name of a strata variable in data.
data
A data.frame in which to interpret the variables named in
the formula.
inip
The vector of initial values.
First components are for the baseline hazard parameters according to
the order
given in 'details';
Other components are for the regression parameters according to the
order given
in 'formula'.
iniFpar
The initial value of the frailty parameter.
dist
The baseline hazard distribution.
One of weibull, inweibull, frechet, exponential,
gompertz, lognormal, loglogistic, or logskewnormal.
frailty
The Frailty distribution.
One of: none, gamma, ingau, possta or lognormal.
method
The optimisation method from the function optimx().
maxit
Maximum number of iterations (see optimx()).
Fparscale
the scaling value for the frailty parameter in optimx().
Optimisation is performed on Fpar/Fparscale.
showtime
Show the execution time?
correct
A correction factor that does not change the marginal
log-likelihood except for an additive constant given by
#clusters * correct * log(10). It may be useful in order to get finite
log-likelihood values in case of many events per cluster
with Positive Stable frailties. Note that the value of the
log-likelihood in the output is the re-adjusted value.
Details
Baseline hazards
The Weibull hazard (dist="weibull") is
h(t; rho, lambda) = rho * lambda * t^(rho - 1)
with rho, lambda > 0.
The inverse Weibull (or Fréchet) hazard (dist="inweibull" or dist="frechet") is
h(t; rho, lambda) = rho * lambda * t^(-rho - 1) / (exp(lambda t^-rho) - 1)
with rho, lambda > 0.
The exponential hazard (dist="exponential") is
h(t; lambda) = lambda
with lambda > 0.
The Gompertz hazard (dist="gompertz") is
h(t; gamma, lambda) = lambda * exp(gamma * t)
with gamma, lambda > 0.
The lognormal hazard (dist="lognormal") is
h(t; mu, sigma) = phi ( (log(t) - mu) / sigma ) / {sigma * t * [1 - Phi( (log(t)- mu) / sigma ) )]}
with mu in R, sigma > 0, and where
phi and Phi are the density and distribution functions of a standard Normal random variable.
The loglogistic hazard (dist="loglogistic") is
h(t; alpha, kappa) = [exp(alpha) * kappa * t^(kappa - 1)] / [1 + exp(alpha) * t^(kappa)]
with alpha in R, kappa > 0.
The log-skew-normal hazard (dist="logskewnormal") is
obtained as the ratio between the density and the cumulative distribution function
of a log-skew normal random variable (Azzalini, 1985),
which has density
f(t; ξ, ω, α) = \frac{2}{ω t}
φ≤ft(\frac{\log(t) - ξ}{ω}\right)
Φ≤ft(α\frac{\log(t)-ξ}{ω}\right)
with xi in R, omega > 0, alpha in R, and where
phi and Phi are the density and distribution functions of a standard Normal random variable.
Of note, if alpha=0 then the log-skew-normal boils down
to the log-normal distribution, with mu=xi and sigma=omega.
Frailty distributions
The gamma frailty distribution (frailty="gamma") is
f(u) = [theta^(-1 / theta) * u^(1 / theta - 1) * exp(-u / theta)] / [Gamma(1 / theta)]
with theta > 0 and where Gamma(.) is the gamma function.
The inverse Gaussian frailty distribution (frailty="ingau") is
f(u) = 1 / sqrt(2 * pi * theta) * u^(-3 / 2) * exp[- (u - 1)^2 / (2 * theta * u)]
with theta > 0.
The positive stable frailty distribution (frailty="poosta") is
f(u) = - 1 / (pi * u) * sum_(k=1, 2, ..., infiny) [Gamma(k * (1 - nu) + 1) / k! * (-u^(nu - 1))^k * sin((1 - nu) * k * pi)]
with 0 < nu < 1.
The lognormal frailty distribution (frailty="lognormal") is
f(u) = 1 / sqrt(2 * pi * theta) * u^(-1) * exp[- (log u)^2 / (2 * theta)]
with theta > 0.
As the Laplace tranform of the lognormal frailties does not exist in closed form,
the saddlepoint approximation is used (Goutis and Casella, 1999).
Value
An object of class parfm.
Author(s)
Federico Rotolo [aut, cre],
Marco Munda [aut],
Andrea Callegaro [ctb]
References
Munda M, Rotolo F, Legrand C (2012).
parfm: Parametric Frailty Models in R.
Journal of Statistical Software, 51(11), 1-20.
DOI <doi: 10.18637/jss.v051.i11>
Duchateau L, Janssen P (2008). The frailty model. Springer.
Wienke A (2010).
Frailty Models in Survival Analysis.
Chapman & Hall/CRC biostatistics series. Taylor and Francis.
Goutis C, Casella G (1999).
Explaining the Saddlepoint Approximation.
The American Statistician 53(3), 216-224.
DOI <doi: 10.1080/00031305.1999.10474463>
Azzalini A (1985).
A class of distributions which includes the normal ones.
Scandinavian Journal of Statistics, 12(2):171-178.
See Also
select.parfm,
ci.parfm,
predict.parfm
Examples
#------Kidney dataset------
data(kidney)
# type 'help(kidney)' for a description of the data set
kidney$sex <- kidney$sex - 1
parfm(Surv(time,status) ~ sex + age, cluster = "id",
data = kidney, dist = "exponential", frailty = "gamma")
parfm(Surv(time,status) ~ sex + age, cluster = "id",
data = kidney, dist = "exponential", frailty = "lognormal")
parfm(Surv(time,status) ~ sex + age, cluster = "id",
data = kidney, dist = "weibull", frailty = "ingau")
parfm(Surv(time,status) ~ sex + age, cluster = "id",
data = kidney, dist="gompertz", frailty="possta", method="CG")
parfm(Surv(time,status) ~ sex + age, cluster = "id",
data = kidney, dist="logskewnormal", frailty="possta", method = 'BFGS')
#--------------------------
#------Asthma dataset------
data(asthma)
head(asthma)
# type 'help(asthma)' for a description of the data set
asthma$time <- asthma$End - asthma$Begin
parfm(Surv(time, Status) ~ Drug, cluster = "Patid", data = asthma,
dist = "weibull", frailty = "gamma")
parfm(Surv(time, Status) ~ Drug, cluster = "Patid", data = asthma,
dist = "weibull", frailty = "lognormal")
parfm(Surv(Begin, End, Status) ~ Drug, cluster = "Patid",
data = asthma[asthma$Fevent == 0, ],
dist = "weibull", frailty = "lognormal", method = "Nelder-Mead")
#--------------------------
parfm documentation built on Jan. 18, 2023, 1:08 a.m.
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| parfm | R Documentation |
Parametric Frailty Models
Description
Fits parametric Frailty Models by maximizing the marginal likelihood.
Usage
parfm(formula, cluster = NULL, strata = NULL, data,
inip = NULL, iniFpar = NULL,
dist = c("weibull", "inweibull", "frechet", "exponential",
"gompertz", "loglogistic", "lognormal", "logskewnormal"),
frailty = c("none", "gamma", "ingau", "possta",
"lognormal", "loglogistic"),
method = "nlminb",
maxit = 500, Fparscale = 1, showtime = FALSE, correct = 0)
Arguments
formula |
A |
cluster |
The name of a cluster variable in data. |
strata |
The name of a strata variable in data. |
data |
A |
inip |
The vector of initial values. First components are for the baseline hazard parameters according to the order given in 'details'; Other components are for the regression parameters according to the order given in 'formula'. |
iniFpar |
The initial value of the frailty parameter. |
dist |
The baseline hazard distribution.
One of |
frailty |
The Frailty distribution.
One of: |
method |
The optimisation method from the function |
maxit |
Maximum number of iterations (see |
Fparscale |
the scaling value for the frailty parameter in |
showtime |
Show the execution time? |
correct |
A correction factor that does not change the marginal log-likelihood except for an additive constant given by #clusters * correct * log(10). It may be useful in order to get finite log-likelihood values in case of many events per cluster with Positive Stable frailties. Note that the value of the log-likelihood in the output is the re-adjusted value. |
Details
Baseline hazards
The Weibull hazard (dist="weibull") is
h(t; rho, lambda) = rho * lambda * t^(rho - 1)
with rho, lambda > 0.
The inverse Weibull (or Fréchet) hazard (dist="inweibull" or dist="frechet") is
h(t; rho, lambda) = rho * lambda * t^(-rho - 1) / (exp(lambda t^-rho) - 1)
with rho, lambda > 0.
The exponential hazard (dist="exponential") is
h(t; lambda) = lambda
with lambda > 0.
The Gompertz hazard (dist="gompertz") is
h(t; gamma, lambda) = lambda * exp(gamma * t)
with gamma, lambda > 0.
The lognormal hazard (dist="lognormal") is
h(t; mu, sigma) = phi ( (log(t) - mu) / sigma ) / {sigma * t * [1 - Phi( (log(t)- mu) / sigma ) )]}
with mu in R, sigma > 0, and where phi and Phi are the density and distribution functions of a standard Normal random variable.
The loglogistic hazard (dist="loglogistic") is
h(t; alpha, kappa) = [exp(alpha) * kappa * t^(kappa - 1)] / [1 + exp(alpha) * t^(kappa)]
with alpha in R, kappa > 0.
The log-skew-normal hazard (dist="logskewnormal") is
obtained as the ratio between the density and the cumulative distribution function
of a log-skew normal random variable (Azzalini, 1985),
which has density
f(t; ξ, ω, α) = \frac{2}{ω t} φ≤ft(\frac{\log(t) - ξ}{ω}\right) Φ≤ft(α\frac{\log(t)-ξ}{ω}\right)
with xi in R, omega > 0, alpha in R, and where phi and Phi are the density and distribution functions of a standard Normal random variable. Of note, if alpha=0 then the log-skew-normal boils down to the log-normal distribution, with mu=xi and sigma=omega.
Frailty distributions
The gamma frailty distribution (frailty="gamma") is
f(u) = [theta^(-1 / theta) * u^(1 / theta - 1) * exp(-u / theta)] / [Gamma(1 / theta)]
with theta > 0 and where Gamma(.) is the gamma function.
The inverse Gaussian frailty distribution (frailty="ingau") is
f(u) = 1 / sqrt(2 * pi * theta) * u^(-3 / 2) * exp[- (u - 1)^2 / (2 * theta * u)]
with theta > 0.
The positive stable frailty distribution (frailty="poosta") is
f(u) = - 1 / (pi * u) * sum_(k=1, 2, ..., infiny) [Gamma(k * (1 - nu) + 1) / k! * (-u^(nu - 1))^k * sin((1 - nu) * k * pi)]
with 0 < nu < 1.
The lognormal frailty distribution (frailty="lognormal") is
f(u) = 1 / sqrt(2 * pi * theta) * u^(-1) * exp[- (log u)^2 / (2 * theta)]
with theta > 0. As the Laplace tranform of the lognormal frailties does not exist in closed form, the saddlepoint approximation is used (Goutis and Casella, 1999).
Value
An object of class parfm.
Author(s)
Federico Rotolo [aut, cre], Marco Munda [aut], Andrea Callegaro [ctb]
References
Munda M, Rotolo F, Legrand C (2012). parfm: Parametric Frailty Models in R. Journal of Statistical Software, 51(11), 1-20. DOI <doi: 10.18637/jss.v051.i11>
Duchateau L, Janssen P (2008). The frailty model. Springer.
Wienke A (2010). Frailty Models in Survival Analysis. Chapman & Hall/CRC biostatistics series. Taylor and Francis.
Goutis C, Casella G (1999). Explaining the Saddlepoint Approximation. The American Statistician 53(3), 216-224. DOI <doi: 10.1080/00031305.1999.10474463>
Azzalini A (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12(2):171-178.
See Also
select.parfm,
ci.parfm,
predict.parfm
Examples
#------Kidney dataset------
data(kidney)
# type 'help(kidney)' for a description of the data set
kidney$sex <- kidney$sex - 1
parfm(Surv(time,status) ~ sex + age, cluster = "id",
data = kidney, dist = "exponential", frailty = "gamma")
parfm(Surv(time,status) ~ sex + age, cluster = "id",
data = kidney, dist = "exponential", frailty = "lognormal")
parfm(Surv(time,status) ~ sex + age, cluster = "id",
data = kidney, dist = "weibull", frailty = "ingau")
parfm(Surv(time,status) ~ sex + age, cluster = "id",
data = kidney, dist="gompertz", frailty="possta", method="CG")
parfm(Surv(time,status) ~ sex + age, cluster = "id",
data = kidney, dist="logskewnormal", frailty="possta", method = 'BFGS')
#--------------------------
#------Asthma dataset------
data(asthma)
head(asthma)
# type 'help(asthma)' for a description of the data set
asthma$time <- asthma$End - asthma$Begin
parfm(Surv(time, Status) ~ Drug, cluster = "Patid", data = asthma,
dist = "weibull", frailty = "gamma")
parfm(Surv(time, Status) ~ Drug, cluster = "Patid", data = asthma,
dist = "weibull", frailty = "lognormal")
parfm(Surv(Begin, End, Status) ~ Drug, cluster = "Patid",
data = asthma[asthma$Fevent == 0, ],
dist = "weibull", frailty = "lognormal", method = "Nelder-Mead")
#--------------------------
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