nbstat: Negative Binomial Rate Ratio
In lrstat: Power and Sample Size Calculation for Non-Proportional Hazards and Beyond
nbstat R Documentation
Negative Binomial Rate Ratio
Description
Obtains the number of subjects accrued, number of events,
number of dropouts, number of subjects reaching the maximum
follow-up, total exposure, and variance for log rate in each group,
rate ratio, variance, and Wald test statistic of
log rate ratio at given calendar times.
Usage
nbstat(
time = NA_real_,
rateRatioH0 = 1,
allocationRatioPlanned = 1,
accrualTime = 0L,
accrualIntensity = NA_real_,
piecewiseSurvivalTime = 0L,
stratumFraction = 1L,
kappa1 = NA_real_,
kappa2 = NA_real_,
lambda1 = NA_real_,
lambda2 = NA_real_,
gamma1 = 0L,
gamma2 = 0L,
accrualDuration = NA_real_,
followupTime = NA_real_,
fixedFollowup = 0L,
nullVariance = 0L
)
Arguments
time
A vector of calendar times for data cut.
rateRatioH0
Rate ratio under the null hypothesis.
allocationRatioPlanned
Allocation ratio for the active treatment
versus control. Defaults to 1 for equal randomization.
accrualTime
A vector that specifies the starting time of
piecewise Poisson enrollment time intervals. Must start with 0, e.g.,
c(0, 3) breaks the time axis into 2 accrual intervals:
[0, 3) and [3, Inf).
accrualIntensity
A vector of accrual intensities. One for
each accrual time interval.
piecewiseSurvivalTime
A vector that specifies the starting time of
piecewise exponential survival time intervals. Must start with 0, e.g.,
c(0, 6) breaks the time axis into 2 event intervals:
[0, 6) and [6, Inf).
Defaults to 0 for exponential distribution.
stratumFraction
A vector of stratum fractions that sum to 1.
Defaults to 1 for no stratification.
kappa1
The dispersion parameter (reciprocal of the shape
parameter of the gamma mixing distribution) for the active treatment
group by stratum.
kappa2
The dispersion parameter (reciprocal of the shape
parameter of the gamma mixing distribution) for the control group by
stratum.
lambda1
The rate parameter of the negative binomial distribution
for the active treatment group by stratum.
lambda2
The rate parameter of the negative binomial distribution
for the control group by stratum.
gamma1
The hazard rate for exponential dropout, a vector of hazard
rates for piecewise exponential dropout applicable for all strata, or a
vector of hazard rates for dropout in each analysis time interval
by stratum for the active treatment group.
gamma2
The hazard rate for exponential dropout, a vector of hazard
rates for piecewise exponential dropout applicable for all strata, or a
vector of hazard rates for dropout in each analysis time interval
by stratum for the control group.
accrualDuration
Duration of the enrollment period.
followupTime
Follow-up time for the last enrolled subject.
fixedFollowup
Whether a fixed follow-up design is used.
Defaults to 0 for variable follow-up.
nullVariance
Whether to calculate the variance for log rate ratio
under the null hypothesis.
Details
The probability mass function for a negative binomial distribution with
dispersion parameter \kappa_i and rate parameter \lambda_i
is given by
P(Y_{ij} = y) = \frac{\Gamma(y+1/\kappa_i)}{\Gamma(1/\kappa_i) y!}
\left(\frac{1}{1 + \kappa_i \lambda_i t_{ij}}\right)^{1/\kappa_i}
\left(\frac{\kappa_i \lambda_i t_{ij}}
{1 + \kappa_i \lambda_i t_{ij}}\right)^{y},
where Y_{ij} is the event count for subject j in
treatment group i, and t_{ij} is the exposure time for
the subject. If \kappa_i=0, the negative binomial distribution
reduces to the Poisson distribution.
For treatment group i, let \beta_i = \log(\lambda_i).
The likelihood for \{(\kappa_i, \beta_i):i=1,2\} can be written as
l = \sum_{i=1}^{2}\sum_{j=1}^{n_{i}}
\{\log \Gamma(y_{ij} + 1/\kappa_i) - \log \Gamma(1/\kappa_i) + y_{ij}
(\log(\kappa_i) + \beta_i) - (y_{ij} + 1/\kappa_i)
\log(1+ \kappa_i \exp(\beta_i) t_{ij})\}.
It follows that
\frac{\partial l}{\partial \beta_i} = \sum_{j=1}^{n_i}
\left\{y_{ij} - (y_{ij} + 1/\kappa_i)
\frac{\kappa_i \exp(\beta_i) t_{ij}}
{1 + \kappa_i \exp(\beta_i)t_{ij}}\right\},
and
-\frac{\partial^2 l}{\partial \beta_i^2} =
\sum_{j=1}^{n_i} (y_{ij} + 1/\kappa_i) \frac{\kappa_i \lambda_i t_{ij}}
{(1 + \kappa_i \lambda_i t_{ij})^2}.
The Fisher information for \beta_i is
E\left(-\frac{\partial^2 l}{\partial \beta_i^2}\right)
= n_i E\left(\frac{\lambda_i t_{ij}}
{1 + \kappa_i \lambda_i t_{ij}}\right).
In addition, we can show that
E\left(-\frac{\partial^2 l}
{\partial \beta_i \partial \kappa_i}\right) = 0.
Therefore, the variance of \hat{\beta}_i is
Var(\hat{\beta}_i) = \frac{1}{n_i} \left\{
E\left(\frac{\lambda_i t_{ij}}{1 + \kappa_i \lambda_i t_{ij}}\right)
\right\}^{-1}.
To evaluate the integral, we need to obtain the distribution of the
exposure time,
t_{ij} = \min(\tau - W_{ij}, C_{ij}, T_{fmax}),
where \tau denotes the calendar time since trial start,
W_{ij} denotes the enrollment time for subject j
in treatment group i, C_{ij} denotes the time to dropout
after enrollment for subject j in treatment group i, and
T_{fmax} denotes the maximum follow-up time for
all subjects. Therefore,
P(t_{ij} \geq t) = P(W_{ij} \leq \tau - t)P(C_{ij} \geq t)
I(t\leq T_{fmax}).
Let H denote the distribution function of the enrollment time,
and G_i denote the survival function of the dropout time for
treatment group i. By the change of variables, we have
E\left(\frac{\lambda_i t_{ij}}{1 + \kappa_i \lambda_i t_{ij}}
\right) = \int_{0}^{\tau \wedge T_{fmax}}
\frac{\lambda_i}{(1 + \kappa_i \lambda_i t)^2} H(\tau - t) G_i(t) dt.
A numerical integration algorithm for a univariate function can be
used to evaluate the above integral.
For the restricted maximum likelihood (reml) estimate of
(\beta_1,\beta_2) subject to the
constraint that \beta_1 - \beta_2 = \Delta, we express the
log-likelihood in terms of (\beta_2,\Delta,\kappa_1,\kappa_2),
and takes the derivative of the log-likelihood function with respect
to \beta_2. The resulting score equation has asymptotic limit
E\left(\frac{\partial l}{\partial \beta_2}\right) = s_1 + s_2,
where
s_1 = n r E\left\{\lambda1_1 t_{1j} - \left(\lambda_1t_{1j}
+ \frac{1}{\kappa_1}\right) \frac{\kappa_1 e^{\tilde{\beta}_2 +
\Delta}t_{1j}}{1 + \kappa_1 e^{\tilde{\beta}_2 +\Delta}t_{1j}}\right\},
and
s_2 = n (1-r) E\left\{\lambda_2 t_{2j} -
\left(\lambda_2 t_{2j} + \frac{1}{\kappa_2}\right)
\frac{\kappa_2 e^{\tilde{\beta}_2} t_{2j}}
{1 + \kappa_2 e^{\tilde{\beta}_2}t_{2j}}\right\}.
Here r is the randomization probability for the active
treatment group. The asymptotic limit of the reml of \beta_2
is the solution \tilde{\beta}_2 to
E\left(\frac{\partial l}{\partial \beta_2}\right) = 0.
Value
A list with two components:
-
resultsUnderH1: A data frame containing the following variables:
-
time: The analysis time since trial start.
-
subjects: The number of enrolled subjects.
-
nevents: The total number of events.
-
nevents1: The number of events in the active treatment
group.
-
nevents2: The number of events in the control group.
-
ndropouts: The total number of dropouts.
-
ndropouts1: The number of dropouts in the active treatment
group.
-
ndropouts2: The number of dropouts in the control group.
-
nfmax: The total number of subjects reaching maximum
follow-up.
-
nfmax1: The number of subjects reaching maximum follow-up
in the active treatment group.
-
nfmax2: The number of subjects reaching maximum follow-up
in the control group.
-
exposure: The total exposure time.
-
exposure1: The exposure time for the active treatment group.
-
exposure2: The exposure time for the control group.
-
rateRatio: The rate ratio of the active treatment group
versus the control group.
-
vlogRate1: The variance for the log rate
parameter for the active treatment group.
-
vlogRate2: The variance for the log rate
parameter for the control group.
-
vlogRR: The variance of log rate ratio.
-
information: The information of log rate ratio.
-
zlogRR: The Z-statistic for log rate ratio.
-
resultsUnderH0 when nullVariance = TRUE: A data frame
with the following variables:
-
time: The analysis time since trial start.
-
lambda1H0: The restricted maximum likelihood estimate
of the event rate for the active treatment group.
-
lambda2H0: The restricted maximum likelihood estimate
of the event rate for the control group.
-
rateRatioH0: The rate ratio under H0.
-
vlogRate1H0: The variance for the log rate
parameter for the active treatment group under H0.
-
vlogRate2H0: The variance for the log rate
parameter for the control group under H0.
-
vlogRRH0: The variance of log rate ratio under H0.
-
informationH0: The information of log rate ratio under H0.
-
zlogRRH0: The Z-statistic for log rate ratio with variance
evaluated under H0.
-
varianceRatio: The ratio of the variance under H0 versus
the variance under H1.
-
lambda1: The true event rate for the active treatment group.
-
lambda2: The true event rate for the control group.
-
rateRatio: The true rate ratio.
-
resultsUnderH0 when nullVariance = FALSE: A data frame
with the following variables:
-
time: The analysis time since trial start.
-
rateRatioH0: The rate ratio under H0.
-
varianceRatio: Equal to 1.
-
lambda1: The true event rate for the active treatment group.
-
lambda2: The true event rate for the control group.
-
rateRatio: The true rate ratio.
Author(s)
Kaifeng Lu, kaifenglu@gmail.com
Examples
# Example 1: Variable follow-up design
nbstat(time = c(1, 1.25, 2, 3, 4),
accrualIntensity = 1956/1.25,
kappa1 = 5,
kappa2 = 5,
lambda1 = 0.7*0.125,
lambda2 = 0.125,
gamma1 = 0,
gamma2 = 0,
accrualDuration = 1.25,
followupTime = 2.75)
# Example 2: Fixed follow-up design
nbstat(time = c(0.5, 1, 1.5, 2),
accrualIntensity = 220/1.5,
stratumFraction = c(0.2, 0.8),
kappa1 = 3,
kappa2 = 3,
lambda1 = c(0.5*8.4, 0.6*10.5),
lambda2 = c(8.4, 10.5),
gamma1 = 0,
gamma2 = 0,
accrualDuration = 1.5,
followupTime = 0.5,
fixedFollowup = 1,
nullVariance = 1)
lrstat documentation built on Oct. 18, 2024, 9:06 a.m.
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| nbstat | R Documentation |
Negative Binomial Rate Ratio
Description
Obtains the number of subjects accrued, number of events, number of dropouts, number of subjects reaching the maximum follow-up, total exposure, and variance for log rate in each group, rate ratio, variance, and Wald test statistic of log rate ratio at given calendar times.
Usage
nbstat(
time = NA_real_,
rateRatioH0 = 1,
allocationRatioPlanned = 1,
accrualTime = 0L,
accrualIntensity = NA_real_,
piecewiseSurvivalTime = 0L,
stratumFraction = 1L,
kappa1 = NA_real_,
kappa2 = NA_real_,
lambda1 = NA_real_,
lambda2 = NA_real_,
gamma1 = 0L,
gamma2 = 0L,
accrualDuration = NA_real_,
followupTime = NA_real_,
fixedFollowup = 0L,
nullVariance = 0L
)
Arguments
time |
A vector of calendar times for data cut. |
rateRatioH0 |
Rate ratio under the null hypothesis. |
allocationRatioPlanned |
Allocation ratio for the active treatment versus control. Defaults to 1 for equal randomization. |
accrualTime |
A vector that specifies the starting time of
piecewise Poisson enrollment time intervals. Must start with 0, e.g.,
|
accrualIntensity |
A vector of accrual intensities. One for each accrual time interval. |
piecewiseSurvivalTime |
A vector that specifies the starting time of
piecewise exponential survival time intervals. Must start with 0, e.g.,
|
stratumFraction |
A vector of stratum fractions that sum to 1. Defaults to 1 for no stratification. |
kappa1 |
The dispersion parameter (reciprocal of the shape parameter of the gamma mixing distribution) for the active treatment group by stratum. |
kappa2 |
The dispersion parameter (reciprocal of the shape parameter of the gamma mixing distribution) for the control group by stratum. |
lambda1 |
The rate parameter of the negative binomial distribution for the active treatment group by stratum. |
lambda2 |
The rate parameter of the negative binomial distribution for the control group by stratum. |
gamma1 |
The hazard rate for exponential dropout, a vector of hazard rates for piecewise exponential dropout applicable for all strata, or a vector of hazard rates for dropout in each analysis time interval by stratum for the active treatment group. |
gamma2 |
The hazard rate for exponential dropout, a vector of hazard rates for piecewise exponential dropout applicable for all strata, or a vector of hazard rates for dropout in each analysis time interval by stratum for the control group. |
accrualDuration |
Duration of the enrollment period. |
followupTime |
Follow-up time for the last enrolled subject. |
fixedFollowup |
Whether a fixed follow-up design is used. Defaults to 0 for variable follow-up. |
nullVariance |
Whether to calculate the variance for log rate ratio under the null hypothesis. |
Details
The probability mass function for a negative binomial distribution with
dispersion parameter \kappa_i and rate parameter \lambda_i
is given by
P(Y_{ij} = y) = \frac{\Gamma(y+1/\kappa_i)}{\Gamma(1/\kappa_i) y!}
\left(\frac{1}{1 + \kappa_i \lambda_i t_{ij}}\right)^{1/\kappa_i}
\left(\frac{\kappa_i \lambda_i t_{ij}}
{1 + \kappa_i \lambda_i t_{ij}}\right)^{y},
where Y_{ij} is the event count for subject j in
treatment group i, and t_{ij} is the exposure time for
the subject. If \kappa_i=0, the negative binomial distribution
reduces to the Poisson distribution.
For treatment group i, let \beta_i = \log(\lambda_i).
The likelihood for \{(\kappa_i, \beta_i):i=1,2\} can be written as
l = \sum_{i=1}^{2}\sum_{j=1}^{n_{i}}
\{\log \Gamma(y_{ij} + 1/\kappa_i) - \log \Gamma(1/\kappa_i) + y_{ij}
(\log(\kappa_i) + \beta_i) - (y_{ij} + 1/\kappa_i)
\log(1+ \kappa_i \exp(\beta_i) t_{ij})\}.
It follows that
\frac{\partial l}{\partial \beta_i} = \sum_{j=1}^{n_i}
\left\{y_{ij} - (y_{ij} + 1/\kappa_i)
\frac{\kappa_i \exp(\beta_i) t_{ij}}
{1 + \kappa_i \exp(\beta_i)t_{ij}}\right\},
and
-\frac{\partial^2 l}{\partial \beta_i^2} =
\sum_{j=1}^{n_i} (y_{ij} + 1/\kappa_i) \frac{\kappa_i \lambda_i t_{ij}}
{(1 + \kappa_i \lambda_i t_{ij})^2}.
The Fisher information for \beta_i is
E\left(-\frac{\partial^2 l}{\partial \beta_i^2}\right)
= n_i E\left(\frac{\lambda_i t_{ij}}
{1 + \kappa_i \lambda_i t_{ij}}\right).
In addition, we can show that
E\left(-\frac{\partial^2 l}
{\partial \beta_i \partial \kappa_i}\right) = 0.
Therefore, the variance of \hat{\beta}_i is
Var(\hat{\beta}_i) = \frac{1}{n_i} \left\{
E\left(\frac{\lambda_i t_{ij}}{1 + \kappa_i \lambda_i t_{ij}}\right)
\right\}^{-1}.
To evaluate the integral, we need to obtain the distribution of the exposure time,
t_{ij} = \min(\tau - W_{ij}, C_{ij}, T_{fmax}),
where \tau denotes the calendar time since trial start,
W_{ij} denotes the enrollment time for subject j
in treatment group i, C_{ij} denotes the time to dropout
after enrollment for subject j in treatment group i, and
T_{fmax} denotes the maximum follow-up time for
all subjects. Therefore,
P(t_{ij} \geq t) = P(W_{ij} \leq \tau - t)P(C_{ij} \geq t)
I(t\leq T_{fmax}).
Let H denote the distribution function of the enrollment time,
and G_i denote the survival function of the dropout time for
treatment group i. By the change of variables, we have
E\left(\frac{\lambda_i t_{ij}}{1 + \kappa_i \lambda_i t_{ij}}
\right) = \int_{0}^{\tau \wedge T_{fmax}}
\frac{\lambda_i}{(1 + \kappa_i \lambda_i t)^2} H(\tau - t) G_i(t) dt.
A numerical integration algorithm for a univariate function can be used to evaluate the above integral.
For the restricted maximum likelihood (reml) estimate of
(\beta_1,\beta_2) subject to the
constraint that \beta_1 - \beta_2 = \Delta, we express the
log-likelihood in terms of (\beta_2,\Delta,\kappa_1,\kappa_2),
and takes the derivative of the log-likelihood function with respect
to \beta_2. The resulting score equation has asymptotic limit
E\left(\frac{\partial l}{\partial \beta_2}\right) = s_1 + s_2,
where
s_1 = n r E\left\{\lambda1_1 t_{1j} - \left(\lambda_1t_{1j}
+ \frac{1}{\kappa_1}\right) \frac{\kappa_1 e^{\tilde{\beta}_2 +
\Delta}t_{1j}}{1 + \kappa_1 e^{\tilde{\beta}_2 +\Delta}t_{1j}}\right\},
and
s_2 = n (1-r) E\left\{\lambda_2 t_{2j} -
\left(\lambda_2 t_{2j} + \frac{1}{\kappa_2}\right)
\frac{\kappa_2 e^{\tilde{\beta}_2} t_{2j}}
{1 + \kappa_2 e^{\tilde{\beta}_2}t_{2j}}\right\}.
Here r is the randomization probability for the active
treatment group. The asymptotic limit of the reml of \beta_2
is the solution \tilde{\beta}_2 to
E\left(\frac{\partial l}{\partial \beta_2}\right) = 0.
Value
A list with two components:
-
resultsUnderH1: A data frame containing the following variables:-
time: The analysis time since trial start. -
subjects: The number of enrolled subjects. -
nevents: The total number of events. -
nevents1: The number of events in the active treatment group. -
nevents2: The number of events in the control group. -
ndropouts: The total number of dropouts. -
ndropouts1: The number of dropouts in the active treatment group. -
ndropouts2: The number of dropouts in the control group. -
nfmax: The total number of subjects reaching maximum follow-up. -
nfmax1: The number of subjects reaching maximum follow-up in the active treatment group. -
nfmax2: The number of subjects reaching maximum follow-up in the control group. -
exposure: The total exposure time. -
exposure1: The exposure time for the active treatment group. -
exposure2: The exposure time for the control group. -
rateRatio: The rate ratio of the active treatment group versus the control group. -
vlogRate1: The variance for the log rate parameter for the active treatment group. -
vlogRate2: The variance for the log rate parameter for the control group. -
vlogRR: The variance of log rate ratio. -
information: The information of log rate ratio. -
zlogRR: The Z-statistic for log rate ratio.
-
-
resultsUnderH0whennullVariance = TRUE: A data frame with the following variables:-
time: The analysis time since trial start. -
lambda1H0: The restricted maximum likelihood estimate of the event rate for the active treatment group. -
lambda2H0: The restricted maximum likelihood estimate of the event rate for the control group. -
rateRatioH0: The rate ratio under H0. -
vlogRate1H0: The variance for the log rate parameter for the active treatment group under H0. -
vlogRate2H0: The variance for the log rate parameter for the control group under H0. -
vlogRRH0: The variance of log rate ratio under H0. -
informationH0: The information of log rate ratio under H0. -
zlogRRH0: The Z-statistic for log rate ratio with variance evaluated under H0. -
varianceRatio: The ratio of the variance under H0 versus the variance under H1. -
lambda1: The true event rate for the active treatment group. -
lambda2: The true event rate for the control group. -
rateRatio: The true rate ratio.
-
-
resultsUnderH0whennullVariance = FALSE: A data frame with the following variables:-
time: The analysis time since trial start. -
rateRatioH0: The rate ratio under H0. -
varianceRatio: Equal to 1. -
lambda1: The true event rate for the active treatment group. -
lambda2: The true event rate for the control group. -
rateRatio: The true rate ratio.
-
Author(s)
Kaifeng Lu, kaifenglu@gmail.com
Examples
# Example 1: Variable follow-up design
nbstat(time = c(1, 1.25, 2, 3, 4),
accrualIntensity = 1956/1.25,
kappa1 = 5,
kappa2 = 5,
lambda1 = 0.7*0.125,
lambda2 = 0.125,
gamma1 = 0,
gamma2 = 0,
accrualDuration = 1.25,
followupTime = 2.75)
# Example 2: Fixed follow-up design
nbstat(time = c(0.5, 1, 1.5, 2),
accrualIntensity = 220/1.5,
stratumFraction = c(0.2, 0.8),
kappa1 = 3,
kappa2 = 3,
lambda1 = c(0.5*8.4, 0.6*10.5),
lambda2 = c(8.4, 10.5),
gamma1 = 0,
gamma2 = 0,
accrualDuration = 1.5,
followupTime = 0.5,
fixedFollowup = 1,
nullVariance = 1)
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