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.
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
)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.
A vector of calendar times for data cut.
Rate ratio under the null hypothesis.
Allocation ratio for the active treatment versus control. Defaults to 1 for equal randomization.
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).
A vector of accrual intensities. One for each accrual time interval.
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.
A vector of stratum fractions that sum to 1. Defaults to 1 for no stratification.
The dispersion parameter (reciprocal of the shape parameter of the gamma mixing distribution) for the active treatment group by stratum.
The dispersion parameter (reciprocal of the shape parameter of the gamma mixing distribution) for the control group by stratum.
The rate parameter of the negative binomial distribution for the active treatment group by stratum.
The rate parameter of the negative binomial distribution for the control group by stratum.
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.
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.
Duration of the enrollment period.
Follow-up time for the last enrolled subject.
Whether a fixed follow-up design is used. Defaults to 0 for variable follow-up.
Whether to calculate the variance for log rate ratio under the null hypothesis.
Kaifeng Lu, kaifenglu@gmail.com
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 log-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 $$s_1 = n r E\left\{\lambda_1 t_{1j} - \left(\lambda_1t_{1j} + \frac{1}{\kappa_1}\right) \frac{\kappa_1 e^{\tilde{\beta}_2 + \Delta}t_{1j}}{1 + \kappa_1 e^{\tilde{\beta}_2 +\Delta}t_{1j}}\right\},$$ 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 \(E\left(\frac{\partial l}{\partial \beta_2}\right) = 0.\)
# 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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