Nothing
R/LikelihoodNonMixGamma.R
In CP: Conditional Power Calculations
Defines functions
LikelihoodNonMixGamma
Documented in
LikelihoodNonMixGamma
LikelihoodNonMixGamma <- function(data1, data2, data,
a1.0, b1.0, c1.0,
a2.0, b2.0, c2.0,
a.0, b.0, c.0) {
## Calculates the maximum likelihood estimators
## of the parameters a, b and c in the non-mixture model
## with Gamma type survival, i. e.
## S(t) = c^Gamma^(0)(a, b * t), a > 0, b > 0, 0 < c < 1, t >= 0,
## with Gamma^(0) being the regularized incomplete Gamma function
## of the upper bound.
##
## Args:
## data1: Data frame which consists of at least three columns with the group
## in the first (all of group 1),
## status (1 = event, 0 = censored) in the second
## and event time in the third column.
## data2: Data frame which consists of at least three columns with the group
## in the first (all of group 2),
## status (1 = event, 0 = censored) in the second
## and event time in the third column.
## data: Data frame which consists of at least three columns with the group
## in the first (all of the same group),
## status (1 = event, 0 = censored) in the second
## and event time in the third column.
## a1.0: Initial value for the estimate of the parameter a in group 1.
## b1.0: Initial value for the estimate of the parameter b in group 1.
## c1.0: Initial value for the estimate of the parameter c in group 1.
## a2.0: Initial value for the estimate of the parameter a in group 2.
## b2.0: Initial value for the estimate of the parameter b in group 2.
## c2.0: Initial value for the estimate of the parameter c in group 2.
## a.0: Initial value for the estimate of the parameter a for all data.
## b.0: Initial value for the estimate of the parameter b for all data.
## c.0: Initial value for the estimate of the parameter c for all data.
##
## Returns:
## Returns the maximum likelihood estimates for the parameters
## a, b and c, i. e. estimates within the ordinary model
## and estimates under the proportional hazard assumption.
# GROUP 1 (data1) -> a1.hat, b1.hat, c1.hat
# log-likelihood function of parameters a, b and c for data of group 1
l1 <- function(x) {
(x[1] * log(x[2]) * sum(data1[, 2])
- lgamma(x = x[1]) * sum(data1[, 2])
+ (x[1] - 1) * sum(data1[, 2] * log(data1[, 3]))
- x[2] * sum(data1[, 2] * data1[, 3])
+ log(- log(x[3])) * sum(data1[, 2])
+ log(x[3]) * sum(stats::pgamma(q = data1[, 3],
shape = x[1],
rate = x[2])))
}
# auxiliary function
Integrand1 <- function(x, par) {
log(x) * x^(par - 1) * exp(-x)
}
# gradient of log-likelihood function
l1Grad <- function(x) {
c(log(x[2]) * sum(data1[, 2])
- digamma(x = x[1]) * sum(data1[, 2])
+ sum(data1[, 2] * log(data1[, 3]))
+ log(x[3]) * sum((gamma(x = x[1])
* sapply(data1[, 3], function(t) {
stats::integrate(f = Integrand1,
lower = 0,
upper = x[2] * t,
par = x[1])$value
})
- gamma(x = x[1]) * stats::pgamma(q = data1[, 3],
shape = x[1],
rate = x[2])
* stats::integrate(f = Integrand1,
lower = 0,
upper = Inf,
par = x[1])$value)
/ gamma(x = x[1])^2),
x[1] / x[2] * sum(data1[, 2])
- sum(data1[, 2] * data1[, 3])
+ log(x[3]) * sum((x[2] * data1[, 3])^(x[1] - 1) * exp(- x[2] * data1[, 3]) * data1[, 3]
/ gamma(x = x[1])),
1 / (x[3] * log(x[3])) * sum(data1[, 2])
+ 1 / x[3] * sum(stats::pgamma(q = data1[, 3],
shape = x[1],
rate = x[2])))
}
# constraints for parameters a, b and c for data of group 1
eps <- 1e-7
A1 <- matrix(data = c(1, 0, 0,
0, 1, 0,
0, 0, 1,
0, 0, - 1),
nrow = 4,
ncol = 3,
byrow = TRUE)
b1 <- c(eps, eps, eps, - (1 - eps))
# maximum likelihood estimators for parameters a, b and c for data of group 1
res1 <- stats::constrOptim(theta = c(a1.0, b1.0, c1.0),
f = l1,
grad = l1Grad,
ui = A1,
ci = b1,
control = list(fnscale = -1))
# GROUP 2 (data2) -> a2.hat, b2.hat, c2.hat
# log-likelihood function of parameters a, b and c for data of group 2
l2 <- function(x) {
(x[1] * log(x[2]) * sum(data2[, 2])
- lgamma(x = x[1]) * sum(data2[, 2])
+ (x[1] - 1) * sum(data2[, 2] * log(data2[, 3]))
- x[2] * sum(data2[, 2] * data2[, 3])
+ log(- log(x[3])) * sum(data2[, 2])
+ log(x[3]) * sum(stats::pgamma(q = data2[, 3],
shape = x[1],
rate = x[2])))
}
# auxiliary function
Integrand2 <- function(x, par) {
log(x) * x^(par - 1) * exp(-x)
}
# gradient of log-likelihood function
l2Grad <- function(x) {
c(log(x[2]) * sum(data2[, 2])
- digamma(x = x[1]) * sum(data2[, 2])
+ sum(data2[, 2] * log(data2[, 3]))
+ log(x[3]) * sum((gamma(x = x[1])
* sapply(data2[, 3], function(t) {
stats::integrate(f = Integrand2,
lower = 0,
upper = x[2] * t,
par = x[1])$value
})
- gamma(x = x[1]) * stats::pgamma(q = data2[, 3],
shape = x[1],
rate = x[2])
* stats::integrate(f = Integrand2,
lower = 0,
upper = Inf,
par = x[1])$value)
/ gamma(x = x[1])^2),
x[1] / x[2] * sum(data2[, 2])
- sum(data2[, 2] * data2[, 3])
+ log(x[3]) * sum((x[2] * data2[, 3])^(x[1] - 1) * exp(- x[2] * data2[, 3]) * data2[, 3]
/ gamma(x = x[1])),
1 / (x[3] * log(x[3])) * sum(data2[, 2])
+ 1 / x[3] * sum(stats::pgamma(q = data2[, 3],
shape = x[1],
rate = x[2])))
}
# constraints for parameters a, b and c for data of group 2
eps <- 1e-7
A2 <- matrix(data = c(1, 0, 0,
0, 1, 0,
0, 0, 1,
0, 0, - 1),
nrow = 4,
ncol = 3,
byrow = TRUE)
b2 <- c(eps, eps, eps, - (1 - eps))
# maximum likelihood estimators for parameters a, b and c for data of group 2
res2 <- stats::constrOptim(theta = c(a2.0, b2.0, c2.0),
f = l2,
grad = l2Grad,
ui = A2,
ci = b2,
control = list(fnscale = -1))
# ALL DATA (data) -> a.hat, b.hat
# log-likelihood function of parameters a, b and c for all data
lAll <- function(x) {
(x[1] * log(x[2]) * sum(data[, 2])
- lgamma(x = x[1]) * sum(data[, 2])
+ (x[1] - 1) * sum(data[, 2] * log(data[, 3]))
- x[2] * sum(data[, 2] * data[, 3])
+ log(- log(x[3])) * sum(data[, 2])
+ log(x[3]) * sum(stats::pgamma(q = data[, 3],
shape = x[1],
rate = x[2])))
}
# auxiliary function
IntegrandAll <- function(x, par) {
log(x) * x^(par - 1) * exp(-x)
}
# gradient of log-likelihood function
lAllGrad <- function(x) {
c(log(x[2]) * sum(data[, 2])
- digamma(x = x[1]) * sum(data[, 2])
+ sum(data[, 2] * log(data[, 3]))
+ log(x[3]) * sum((gamma(x = x[1])
* sapply(data[, 3], function(t) {
stats::integrate(f = IntegrandAll,
lower = 0,
upper = x[2] * t,
par = x[1])$value
})
- gamma(x = x[1]) * stats::pgamma(q = data[, 3],
shape = x[1],
rate = x[2])
* stats::integrate(f = IntegrandAll,
lower = 0,
upper = Inf,
par = x[1])$value)
/ gamma(x = x[1])^2),
x[1] / x[2] * sum(data[, 2])
- sum(data[, 2] * data[, 3])
+ log(x[3]) * sum((x[2] * data[, 3])^(x[1] - 1) * exp(- x[2] * data[, 3]) * data[, 3]
/ gamma(x = x[1])),
1 / (x[3] * log(x[3])) * sum(data[, 2])
+ 1 / x[3] * sum(stats::pgamma(q = data[, 3],
shape = x[1],
rate = x[2])))
}
# constraints for parameters a, b and c for all data
eps <- 1e-7
AAll <- matrix(data = c(1, 0, 0,
0, 1, 0,
0, 0, 1,
0, 0, - 1),
nrow = 4,
ncol = 3,
byrow = TRUE)
bAll <- c(eps, eps, eps, - (1 - eps))
# maximum likelihood estimators for parameters a, b and c for all data
resAll <- stats::constrOptim(theta = c(a.0, b.0, c.0),
f = lAll,
grad = lAllGrad,
ui = AAll,
ci = bAll,
control = list(fnscale = -1))
a.hat <- resAll$par[1]
b.hat <- resAll$par[2]
# GROUP 1 (data1) with fixed a and b -> c1.cond.hat
# log-likelihood function of parameter c for data of group 1
l1.cond <- function(x) {
(a.hat * log(b.hat) * sum(data1[, 2])
- lgamma(x = a.hat) * sum(data1[, 2])
+ (a.hat - 1) * sum(data1[, 2] * log(data1[, 3]))
- b.hat * sum(data1[, 2] * data1[, 3])
+ log(- log(x)) * sum(data1[, 2])
+ log(x) * sum(stats::pgamma(q = data1[, 3],
shape = a.hat,
rate = b.hat)))
}
# auxiliary function
Integrand1.cond <- function(x, par) {
log(x) * x^(par - 1) * exp(-x)
}
# gradient of log-likelihood function
l1Grad.cond <- function(x) {
(1 / (x * log(x)) * sum(data1[, 2])
+ 1 / x * sum(stats::pgamma(q = data1[, 3],
shape = a.hat,
rate = b.hat)))
}
# constraints for parameter c for data of group 1
eps <- 1e-7
A1.cond <- matrix(data = c(1,
- 1),
nrow = 2,
ncol = 1,
byrow = TRUE)
b1.cond <- c(eps, - (1 - eps))
# maximum likelihood estimator for parameter c for data of group 1
res1.cond <- stats::constrOptim(theta = c1.0,
f = l1.cond,
grad = l1Grad.cond,
ui = A1.cond,
ci = b1.cond,
control = list(fnscale = -1))
# GROUP 2 (data2) with fixed a and b -> c2.cond.hat
# log-likelihood function of parameter c for data of group 2
l2.cond <- function(x) {
(a.hat * log(b.hat) * sum(data2[, 2])
- lgamma(x = a.hat) * sum(data2[, 2])
+ (a.hat - 1) * sum(data2[, 2] * log(data2[, 3]))
- b.hat * sum(data2[, 2] * data2[, 3])
+ log(- log(x)) * sum(data2[, 2])
+ log(x) * sum(stats::pgamma(q = data2[, 3],
shape = a.hat,
rate = b.hat)))
}
# auxiliary function
Integrand2.cond <- function(x, par) {
log(x) * x^(par - 1) * exp(-x)
}
# gradient of log-likelihood function
l2Grad.cond <- function(x) {
(1 / (x * log(x)) * sum(data2[, 2])
+ 1 / x * sum(stats::pgamma(q = data2[, 3],
shape = a.hat,
rate = b.hat)))
}
# constraints for parameter c for data of group 2
eps <- 1e-7
A2.cond <- matrix(data = c(1,
- 1),
nrow = 2,
ncol = 1,
byrow = TRUE)
b2.cond <- c(eps, - (1 - eps))
# maximum likelihood estimator for parameter c for data of group 2
res2.cond <- stats::constrOptim(theta = c2.0,
f = l2.cond,
grad = l2Grad.cond,
ui = A2.cond,
ci = b2.cond,
control = list(fnscale = -1))
return(c(res1$par[1], res1$par[2], res1$par[3],
res2$par[1], res2$par[2], res2$par[3],
resAll$par[1], resAll$par[2],
res1.cond$par,
res2.cond$par,
res1.cond$value,
res2.cond$value))
}
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Defines functions LikelihoodNonMixGamma
Documented in LikelihoodNonMixGamma
LikelihoodNonMixGamma <- function(data1, data2, data,
a1.0, b1.0, c1.0,
a2.0, b2.0, c2.0,
a.0, b.0, c.0) {
## Calculates the maximum likelihood estimators
## of the parameters a, b and c in the non-mixture model
## with Gamma type survival, i. e.
## S(t) = c^Gamma^(0)(a, b * t), a > 0, b > 0, 0 < c < 1, t >= 0,
## with Gamma^(0) being the regularized incomplete Gamma function
## of the upper bound.
##
## Args:
## data1: Data frame which consists of at least three columns with the group
## in the first (all of group 1),
## status (1 = event, 0 = censored) in the second
## and event time in the third column.
## data2: Data frame which consists of at least three columns with the group
## in the first (all of group 2),
## status (1 = event, 0 = censored) in the second
## and event time in the third column.
## data: Data frame which consists of at least three columns with the group
## in the first (all of the same group),
## status (1 = event, 0 = censored) in the second
## and event time in the third column.
## a1.0: Initial value for the estimate of the parameter a in group 1.
## b1.0: Initial value for the estimate of the parameter b in group 1.
## c1.0: Initial value for the estimate of the parameter c in group 1.
## a2.0: Initial value for the estimate of the parameter a in group 2.
## b2.0: Initial value for the estimate of the parameter b in group 2.
## c2.0: Initial value for the estimate of the parameter c in group 2.
## a.0: Initial value for the estimate of the parameter a for all data.
## b.0: Initial value for the estimate of the parameter b for all data.
## c.0: Initial value for the estimate of the parameter c for all data.
##
## Returns:
## Returns the maximum likelihood estimates for the parameters
## a, b and c, i. e. estimates within the ordinary model
## and estimates under the proportional hazard assumption.
# GROUP 1 (data1) -> a1.hat, b1.hat, c1.hat
# log-likelihood function of parameters a, b and c for data of group 1
l1 <- function(x) {
(x[1] * log(x[2]) * sum(data1[, 2])
- lgamma(x = x[1]) * sum(data1[, 2])
+ (x[1] - 1) * sum(data1[, 2] * log(data1[, 3]))
- x[2] * sum(data1[, 2] * data1[, 3])
+ log(- log(x[3])) * sum(data1[, 2])
+ log(x[3]) * sum(stats::pgamma(q = data1[, 3],
shape = x[1],
rate = x[2])))
}
# auxiliary function
Integrand1 <- function(x, par) {
log(x) * x^(par - 1) * exp(-x)
}
# gradient of log-likelihood function
l1Grad <- function(x) {
c(log(x[2]) * sum(data1[, 2])
- digamma(x = x[1]) * sum(data1[, 2])
+ sum(data1[, 2] * log(data1[, 3]))
+ log(x[3]) * sum((gamma(x = x[1])
* sapply(data1[, 3], function(t) {
stats::integrate(f = Integrand1,
lower = 0,
upper = x[2] * t,
par = x[1])$value
})
- gamma(x = x[1]) * stats::pgamma(q = data1[, 3],
shape = x[1],
rate = x[2])
* stats::integrate(f = Integrand1,
lower = 0,
upper = Inf,
par = x[1])$value)
/ gamma(x = x[1])^2),
x[1] / x[2] * sum(data1[, 2])
- sum(data1[, 2] * data1[, 3])
+ log(x[3]) * sum((x[2] * data1[, 3])^(x[1] - 1) * exp(- x[2] * data1[, 3]) * data1[, 3]
/ gamma(x = x[1])),
1 / (x[3] * log(x[3])) * sum(data1[, 2])
+ 1 / x[3] * sum(stats::pgamma(q = data1[, 3],
shape = x[1],
rate = x[2])))
}
# constraints for parameters a, b and c for data of group 1
eps <- 1e-7
A1 <- matrix(data = c(1, 0, 0,
0, 1, 0,
0, 0, 1,
0, 0, - 1),
nrow = 4,
ncol = 3,
byrow = TRUE)
b1 <- c(eps, eps, eps, - (1 - eps))
# maximum likelihood estimators for parameters a, b and c for data of group 1
res1 <- stats::constrOptim(theta = c(a1.0, b1.0, c1.0),
f = l1,
grad = l1Grad,
ui = A1,
ci = b1,
control = list(fnscale = -1))
# GROUP 2 (data2) -> a2.hat, b2.hat, c2.hat
# log-likelihood function of parameters a, b and c for data of group 2
l2 <- function(x) {
(x[1] * log(x[2]) * sum(data2[, 2])
- lgamma(x = x[1]) * sum(data2[, 2])
+ (x[1] - 1) * sum(data2[, 2] * log(data2[, 3]))
- x[2] * sum(data2[, 2] * data2[, 3])
+ log(- log(x[3])) * sum(data2[, 2])
+ log(x[3]) * sum(stats::pgamma(q = data2[, 3],
shape = x[1],
rate = x[2])))
}
# auxiliary function
Integrand2 <- function(x, par) {
log(x) * x^(par - 1) * exp(-x)
}
# gradient of log-likelihood function
l2Grad <- function(x) {
c(log(x[2]) * sum(data2[, 2])
- digamma(x = x[1]) * sum(data2[, 2])
+ sum(data2[, 2] * log(data2[, 3]))
+ log(x[3]) * sum((gamma(x = x[1])
* sapply(data2[, 3], function(t) {
stats::integrate(f = Integrand2,
lower = 0,
upper = x[2] * t,
par = x[1])$value
})
- gamma(x = x[1]) * stats::pgamma(q = data2[, 3],
shape = x[1],
rate = x[2])
* stats::integrate(f = Integrand2,
lower = 0,
upper = Inf,
par = x[1])$value)
/ gamma(x = x[1])^2),
x[1] / x[2] * sum(data2[, 2])
- sum(data2[, 2] * data2[, 3])
+ log(x[3]) * sum((x[2] * data2[, 3])^(x[1] - 1) * exp(- x[2] * data2[, 3]) * data2[, 3]
/ gamma(x = x[1])),
1 / (x[3] * log(x[3])) * sum(data2[, 2])
+ 1 / x[3] * sum(stats::pgamma(q = data2[, 3],
shape = x[1],
rate = x[2])))
}
# constraints for parameters a, b and c for data of group 2
eps <- 1e-7
A2 <- matrix(data = c(1, 0, 0,
0, 1, 0,
0, 0, 1,
0, 0, - 1),
nrow = 4,
ncol = 3,
byrow = TRUE)
b2 <- c(eps, eps, eps, - (1 - eps))
# maximum likelihood estimators for parameters a, b and c for data of group 2
res2 <- stats::constrOptim(theta = c(a2.0, b2.0, c2.0),
f = l2,
grad = l2Grad,
ui = A2,
ci = b2,
control = list(fnscale = -1))
# ALL DATA (data) -> a.hat, b.hat
# log-likelihood function of parameters a, b and c for all data
lAll <- function(x) {
(x[1] * log(x[2]) * sum(data[, 2])
- lgamma(x = x[1]) * sum(data[, 2])
+ (x[1] - 1) * sum(data[, 2] * log(data[, 3]))
- x[2] * sum(data[, 2] * data[, 3])
+ log(- log(x[3])) * sum(data[, 2])
+ log(x[3]) * sum(stats::pgamma(q = data[, 3],
shape = x[1],
rate = x[2])))
}
# auxiliary function
IntegrandAll <- function(x, par) {
log(x) * x^(par - 1) * exp(-x)
}
# gradient of log-likelihood function
lAllGrad <- function(x) {
c(log(x[2]) * sum(data[, 2])
- digamma(x = x[1]) * sum(data[, 2])
+ sum(data[, 2] * log(data[, 3]))
+ log(x[3]) * sum((gamma(x = x[1])
* sapply(data[, 3], function(t) {
stats::integrate(f = IntegrandAll,
lower = 0,
upper = x[2] * t,
par = x[1])$value
})
- gamma(x = x[1]) * stats::pgamma(q = data[, 3],
shape = x[1],
rate = x[2])
* stats::integrate(f = IntegrandAll,
lower = 0,
upper = Inf,
par = x[1])$value)
/ gamma(x = x[1])^2),
x[1] / x[2] * sum(data[, 2])
- sum(data[, 2] * data[, 3])
+ log(x[3]) * sum((x[2] * data[, 3])^(x[1] - 1) * exp(- x[2] * data[, 3]) * data[, 3]
/ gamma(x = x[1])),
1 / (x[3] * log(x[3])) * sum(data[, 2])
+ 1 / x[3] * sum(stats::pgamma(q = data[, 3],
shape = x[1],
rate = x[2])))
}
# constraints for parameters a, b and c for all data
eps <- 1e-7
AAll <- matrix(data = c(1, 0, 0,
0, 1, 0,
0, 0, 1,
0, 0, - 1),
nrow = 4,
ncol = 3,
byrow = TRUE)
bAll <- c(eps, eps, eps, - (1 - eps))
# maximum likelihood estimators for parameters a, b and c for all data
resAll <- stats::constrOptim(theta = c(a.0, b.0, c.0),
f = lAll,
grad = lAllGrad,
ui = AAll,
ci = bAll,
control = list(fnscale = -1))
a.hat <- resAll$par[1]
b.hat <- resAll$par[2]
# GROUP 1 (data1) with fixed a and b -> c1.cond.hat
# log-likelihood function of parameter c for data of group 1
l1.cond <- function(x) {
(a.hat * log(b.hat) * sum(data1[, 2])
- lgamma(x = a.hat) * sum(data1[, 2])
+ (a.hat - 1) * sum(data1[, 2] * log(data1[, 3]))
- b.hat * sum(data1[, 2] * data1[, 3])
+ log(- log(x)) * sum(data1[, 2])
+ log(x) * sum(stats::pgamma(q = data1[, 3],
shape = a.hat,
rate = b.hat)))
}
# auxiliary function
Integrand1.cond <- function(x, par) {
log(x) * x^(par - 1) * exp(-x)
}
# gradient of log-likelihood function
l1Grad.cond <- function(x) {
(1 / (x * log(x)) * sum(data1[, 2])
+ 1 / x * sum(stats::pgamma(q = data1[, 3],
shape = a.hat,
rate = b.hat)))
}
# constraints for parameter c for data of group 1
eps <- 1e-7
A1.cond <- matrix(data = c(1,
- 1),
nrow = 2,
ncol = 1,
byrow = TRUE)
b1.cond <- c(eps, - (1 - eps))
# maximum likelihood estimator for parameter c for data of group 1
res1.cond <- stats::constrOptim(theta = c1.0,
f = l1.cond,
grad = l1Grad.cond,
ui = A1.cond,
ci = b1.cond,
control = list(fnscale = -1))
# GROUP 2 (data2) with fixed a and b -> c2.cond.hat
# log-likelihood function of parameter c for data of group 2
l2.cond <- function(x) {
(a.hat * log(b.hat) * sum(data2[, 2])
- lgamma(x = a.hat) * sum(data2[, 2])
+ (a.hat - 1) * sum(data2[, 2] * log(data2[, 3]))
- b.hat * sum(data2[, 2] * data2[, 3])
+ log(- log(x)) * sum(data2[, 2])
+ log(x) * sum(stats::pgamma(q = data2[, 3],
shape = a.hat,
rate = b.hat)))
}
# auxiliary function
Integrand2.cond <- function(x, par) {
log(x) * x^(par - 1) * exp(-x)
}
# gradient of log-likelihood function
l2Grad.cond <- function(x) {
(1 / (x * log(x)) * sum(data2[, 2])
+ 1 / x * sum(stats::pgamma(q = data2[, 3],
shape = a.hat,
rate = b.hat)))
}
# constraints for parameter c for data of group 2
eps <- 1e-7
A2.cond <- matrix(data = c(1,
- 1),
nrow = 2,
ncol = 1,
byrow = TRUE)
b2.cond <- c(eps, - (1 - eps))
# maximum likelihood estimator for parameter c for data of group 2
res2.cond <- stats::constrOptim(theta = c2.0,
f = l2.cond,
grad = l2Grad.cond,
ui = A2.cond,
ci = b2.cond,
control = list(fnscale = -1))
return(c(res1$par[1], res1$par[2], res1$par[3],
res2$par[1], res2$par[2], res2$par[3],
resAll$par[1], resAll$par[2],
res1.cond$par,
res2.cond$par,
res1.cond$value,
res2.cond$value))
}
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