Blinded Sample Size Recalculation for a One Subgroup Design

Description

Given data from an Internal Pilot Study (IPS), bssr.1subgroup reestimates the nuisance parameters, i.e. variances and prevalence, and recalculates the required sample size for proving a desired alternative when testing for an effect in the full or subpopulation. See 'Details' for more information.

Usage

bssr.1subgroup(
  data,
  alpha,
  beta,
  delta,
  eps = 0.001,
  approx = c("conservative.t", "liberal.t", "normal"),
  df = c("n", "n1"),
  adjust = c("YES", "NO"),
  k = 1,
  nmax = 1000
)

Arguments

Details

This function performs blinded nuisance parameter reestimation in a design with a subgroup within a full population where we want to test for treatment effects between a control and a treatment group. Then the required sample size for the control and treatment group to prove an existing alternative delta with a specified power 1-beta when testing the global null hypothesis H_0: \Delta_F=\Delta_S=0 to level alpha is calculated.

The data matrix data should have three columns: The first column has to be a binary variable (0=treatment group, 1=control group). The second column should also contain a binary variable giving the full population/subgroup differentiation (0=full population, 1=subpopulation). The last column contains the observations.

For sample sizes n_C and n_T of the control and treatment group, respectively, the argument k is the sample size allocation factor, i.e. k = n_T/n_C.

The parameter df provides a difference to the standard sample size calculation procedure implemented in n.1subgroup. When applying a multivariate t distribution approximation to approximate the joint distribution of the standardized test statistics it gives the opportunity to use degrees of freedom depending on the number of subjects in the IPS instead of degrees of freedom depending on the projected final sample size. Note that this leads to better performance when dealing with extremely small subgroup sample sizes but significantly increases the calculated final sample size.

Value

bssr.1subgroup returns a list containing the recalculated required sample size within the control group and treatment group along with all relevant parameters. Use summary.bssrest for a structured overview.

Source

bssr.1subgroup uses code contributed by Marius Placzek.

See Also

n.1subgroup for sample size calculation prior to the trial.

Examples

#Given data from the Internal Pilot Study, reestimate the nuisance parameters and
#recalculate the required sample size to correctly reject with
#80% probability when testing the global Nullhypothesis H_0: Delta_F=Delta_S = 0
#assuming the true effect Delta_S=1 is in the subgroup (no effect outside of the subgroup).

random<-r.1subgroup(n=50, delta=c(0,1), sigma=c(1,1.2), tau=0.4, fix.tau="YES", k=2)
reestimate<-bssr.1subgroup(data=random,alpha=0.05,beta=0.1,delta=c(0,1),eps=0.001,
approx="conservative.t",df="n1",k=2,adjust="NO")
summary(reestimate)

Blinded Sample Size Recalculation for longitudinal data in a One Subgroup Design

Description

Given re-estimations from an Internal Pilot Study (IPS), bssr.GEE.1subgroup re-estimates required sample size given the re-estimated nuisance parameters are given. bssr.gee.1subgroup is a wrapper for n.gee.1subgroup where the re-estimation of the variances can be highly dependable on the user and should be supplied separately. see "detail" for more information.

Usage

bssr.gee.1subgroup(
  alpha,
  tail = "both",
  beta = NULL,
  delta,
  estsigma,
  tau = 0.5,
  k = 1
)

Arguments

Details

This function provides a simple warped for n.gee.1subgroup where instead of initial assumptions, reestimated nuisance parameter are used. For more information see n.gee.1subgroup. Required samplesize to test alternative delta with specified power 1-beta when testing the global null hypothesis H_0: \beta_3^F=\beta_3^S=0 to level alpha is estimated. When testing outcomes have variance estsigma.

For sample sizes n_C and n_T of the control and treatment group respectively, the argument k is the sample size allocation factor, i.e. k = n_T/n_C and tau represents the ratio of the sub-population.

Value

bssr.gee.1subgroup returns a list containing the recalculated sample sizes along with all relevant parameters. Use summary.bssrest for a structured overview.

Source

bssr.gee.1subgroup uses code contributed by Roland Gerard Gera.

See Also

n.gee.1subgroup for sample size calculation prior to a trial and estimcov how the re-estimate nuisance parameters. See sim.gee for a working example for an initial sample size estimation and a re-estimation mid trial.

Examples

estimate<-bssr.gee.1subgroup(alpha=0.05,beta=0.2,delta=c(0.1,0.1),estsigma=c(0.8,0.4),tau=0.4, k=1)
summary(estimate)

Blinded Sample Size Reestimation for Longitudinal Count Data with marginal Negative Binomial Distribution and underlying Gamma Frailty with Autoregressive Correlation Structure of Order One

Description

bssr.nb.gf fits blinded observations and recalculates the sample size required for sustaining power at desired alternative when testing for trend parameters in a Gamma frailty models. See 'Details' for more information.

Usage

bssr.nb.gf(
  data,
  alpha = 0.025,
  power = 0.8,
  delta,
  h0 = 0,
  tp,
  k,
  trend = c("constant", "exponential", "custom"),
  approx = 20
)

Arguments

Details

The function recalculates a sample size for testing in constant and exponential trends.

Under a constant trend, the means in control and experiment group are equal to \lambda_1 and \lambda_1 + \lambda_2, respectively. The treatment effect delta is therefore equal to \lambda_2.

Under an exponential trend, the means in control and experiment group are equal to exp(\lambda_1+t \cdot \lambda_2) and \lambda_1 + t\cdot \lambda_2 + t\cdot \lambda_3, respectively. The treatment effect delta is therefore equal to \lambda_3.

bssr.nb.gf returns the required sample size for the control and treatment group required to prove an existing alternative delta with a specified power power when testing the null hypothesis H_0: \delta \ge h_0 at level alpha. Nuisance parameters are estimated through the blinded observations data, thus not further required. For sample sizes n_C and n_T of the control and treatment group, respectively, the argument k is the desired sample size allocation factor at the end of the study, i.e. k = n_T/n_C.

Value

bssr.nb.gf returns the required sample size within the control group and treatment group.

Source

bssr.nb.gf uses code contributed by Thomas Asendorf.

See Also

rnbinom.gf for information on the Gamma Frailty model, n.nb.gf for calculating initial sample size required when performing inference, fit.nb.gf for calculating initial parameters required when performing sample size estimation.

Examples

##The example is commented as it may take longer than 10 seconds to run.
##Please uncomment prior to execution.

##Example for constant rates
#set.seed(12)
#h<-function(lambda.eta){
#   lambda.eta[2]
#}
#hgrad<-function(lambda.eta){
#   c(0, 1, 0)
#}

##Calculate initial sample size
#estimate<-n.nb.gf(lambda=c(0,-0.3), size=1, rho=0.5, tp=6, k=1, h=h, hgrad=hgrad,
#   h0=0, trend="constant", approx=20)

##Generate and permutate data with different nuisance parameters
#random<-get.groups(n=round(estimate$n/2), size=c(0.8, 0.8), lambda=c(0.5, -0.3),
#   rho=c(0.4, 0.4), tp=6, trend="constant")
#random<-random[sample(1:nrow(random), nrow(random)), ]

##Recalculate sample size with data
#reestimate<-bssr.nb.gf(data=random, alpha=0.025, power=0.8, delta=-0.3, h0=0,
#   tp=6, k=1, trend="constant", approx = 20)

#summary(reestimate)

Blinded Sample Size Reestimation for Longitudinal Count Data using the NB-INAR(1) Model

Description

bssr.nb.inar1 fits blinded observations and recalculates the sample size required for proving a desired alternative when testing for a rate ratio between two groups unequal to one. See 'Details' for more information.

Usage

bssr.nb.inar1(alpha, power, delta, x, n, k)

Arguments

Details

When testing for differences between rates \mu_C and \mu_T of two groups, a control and a treatment group respectively, we usually test for the ratio between the two rates, i.e. \mu_T/\mu_C = 1. The ratio of the two rates is refered to as \delta, i.e. \delta = \mu_T/\mu_C.

bssr.nb.inar1 gives back the required sample size for the control and treatment group required to prove an existing alternative theta with a specified power power when testing the null hypothesis H_0: \mu_T/\mu_C \ge 1 to level alpha. Nuisance parameters are estimated through the blinded observations x, thus not further required.

for sample sizes n_C and n_T of the control and treatment group, respectively, the argument k is the desired sample size allocation factor at the end of the study, i.e. k = n_T/n_C.

Value

rnbinom.inar1 returns the required sample size within the control group and treatment group.

Source

rnbinom.inar1 uses code contributed by Thomas Asendorf.

See Also

rnbinom.inar1 for information on the NB-INAR(1) model, n.nb.inar1 for calculating initial sample size required when performing inference, fit.nb.inar1 for calculating initial parameters required when performing sample size estimation

Examples

#Calculate required sample size to find significant difference with
#80% probability when testing the Nullhypothesis H_0: mu_T/mu_C >= 1
#assuming the true effect delta is 0.8 and rate, size and correlation
#parameter in the control group are 2, 1 and 0.5, respectively.

estimate<-n.nb.inar1(alpha=0.025, power=0.8, delta=0.8, muC=2, size=1, rho=0.5, tp=7, k=1)

#Simulate data
placebo<-rnbinom.inar1(n=50, size=1, mu=2, rho=0.5, tp=7)
treatment<-rnbinom.inar1(n=50, size=1, mu=1.6, rho=0.5, tp=7)

#Blinded sample size reestimation
blinded.data<-rbind(placebo, treatment)[sample(1:100),]
estimate<-bssr.nb.inar1(alpha=0.025, power=0.8, delta=0.8, x=blinded.data, n=c(50,50), k=1)
summary(estimate)

Estimation of simulation parameters

Description

estimcov estimates the covariance matrix and dropout rates given a dataset and observation-times

Usage

estimcov(
  data,
  Time,
  Startvalues = c(3, 0.5, 1),
  stepwidth = c(0.001, 0.001, 0.001),
  maxiter = 10000,
  lower = c(1e-04, 1e-04, 1e-04),
  upper = c(Inf, 5, 3)
)

Arguments

Details

This function is designed to estimate the variance, rho and theta and a vector with the dropout rate in the data.

Value

estimcov returns a list with two entries. In the first the parameters variance, rho and theta are returned and in the second a vector with the dropout-rate is returned.

Source

estimcov uses code contributed by Roland Gerard Gera.

Examples

# First generate a dataset with 200 patients, rho =0.25 and tau = 0.5 and
# then estimate the parameters using estimcov.

set.seed(2015)
dataset <- r.gee.1subgroup(n=200, reg=list(c(0,0,0,0.1),c(0,0,0,0.1)), sigma=c(3,2.5),
  tau=0.5, rho=0.25, theta=1, k=1.5, Time=c(0:5), OD=0)

estimations <- estimcov(data=dataset,Time=c(0:5))
estimations[[1]]
estimations[[2]]

Fitting Longitudinal Data from a Gamma Frailty Model with Frailty of Autoregressive Correlation Structure of Order One

Description

fit.nb.gf fits data using the pseudo maximum likelihood of a Gamma frailty model

Usage

fit.nb.gf(
  dataC,
  dataE,
  trend = c("constant", "exponential"),
  lower,
  upper,
  method = "L-BFGS-B",
  start,
  approx = 20,
  rho = FALSE,
  H0 = FALSE,
  h0 = 0
)

Arguments

Details

the function fit.nb.gf fits a Gamma frailty model as found in Fiocco (2009). The fitting function allows for incomplete follow up, but not for intermittent missingness.

When calculating the expected sandwich estimator required for the sample size, certain terms can not be computed analytically and have to be approximated numerically. The value approx defines how close the approximation is to the true expected sandwich estimator. High values of approx provide better approximations but are compuationally more expensive.

If parameter H0 is set to TRUE, the hessian and outer gradient are calculated under the assumption that lambda[2] \geq h0 if trend = "constant" or lambda[3] \geq h0 if trend = "exponential".

Value

fit.nb.gf returns estimates of the trend parameters lambda, dispersion parameter size, Hessian matrix hessian, outer gradient product matrix ogradient and, if inquired, correlation coefficient rho.

Source

fit.nb.gf uses code contributed by Thomas Asendorf.

References

Fiocco M, Putter H, Van Houwelingen JC, (2009), A new serially correlated gamma-frailty process for longitudinal count data Biostatistics Vol. 10, No. 2, pp. 245-257.

See Also

rnbinom.gf for information on the Gamma frailty model, n.nb.gf for calculating initial sample size required when performing inference, bssr.nb.gf for blinded sample size reestimation within a running trial, optim for more information on the used minimization algorithms.

Examples

#Generate data from the Gamma frailty model
random<-get.groups(n=c(1000,1000), size=c(0.7, 0.7), lambda=c(0.8, -0.5), rho=c(0.6, 0.6),
  tp=7, trend="constant")
fit.nb.gf(dataC=random[1001:2000,], dataE=random[1:1000,], trend="constant")

Fitting Longitudinal Data with Negative Binomial Marginal Distribution and Autoregressive Correlation Structure of Order One: NB-INAR(1)

Description

fit.nb.inar1 fits data using the maximum likelihood of a reparametrized NB-INAR(1) model.

Usage

fit.nb.inar1(
  x,
  lower = rep(10, 3)^-5,
  upper = c(10^5, 10^5, 1 - 10^-5),
  method = "L-BFGS-B",
  start
)

Arguments

Details

the function fit.nb.inar1 fits a reparametrization of the NB-INAR(1) model as found in McKenzie (1986). The reparametrized model assumes equal means and dispersion parameter between time points with an autoregressive correlation structure. The function is especially useful for estimating parameters for an initial sample size calculation using n.nb.inar1. The fitting function allows for incomplete follow up, but not for intermittent missingness.

Value

fit.nb.inar1 return estimates of the mean mu, dispersion parameter size and correlation coefficient rho.

Source

fit.nb.inar1 uses code contributed by Thomas Asendorf.

References

McKenzie Ed (1986), Autoregressive Moving-Average Processes with Negative-Binomial and Geometric Marginal Distributions. Advances in Applied Probability Vol. 18, No. 3, pp. 679-705.

See Also

rnbinom.inar1 for information on the NB-INAR(1) model, n.nb.inar1 for calculating initial sample size required when performing inference, bssr.nb.inar1 for blinded sample size reestimation within a running trial, optim for more information on the used minimization algorithms.

Examples

#Generate data from the NB-INAR(1) model
set.seed(8)
random<-rnbinom.inar1(n=1000, size=1.5, mu=2, rho=0.6, tp=7)

estimate<-fit.nb.inar1(random)
estimate

Generation of a covariance or a correlation matrix

Description

Generate a covariance or correlation matrix given parameters var, rho, theta for the covariance structure, Time for the observed timepoints and cov=TRUE if a covariance or cov=FALSE if a correlation-matrix is generated.

Usage

gen_cov_cor(var = 1, rho, theta, Time, cov = TRUE)

Arguments

Details

gen_cov_cor is used to generate either a covariance or a correlation matrix. Given vector Time and parameters var, rho and theta the following two equations are used to calculate the covariance and the correlation between two timepoints, respectively: cov(Time[i],Time[j])=var*(rho^(abs(Time[i]-Time[j])^theta)) corr(Time[i],Time[j])=rho^(abs(Time[i]-Time[j])^theta) ]]

Value

gen_cov_cor returns a covariance or correlation matrix.

Source

gen_cov_cor uses code contributed by Roland Gerard Gera

@seealso r.gee.1subgroup for information on the generated longitudinal data and n.gee.1subgroup for the calculation of initial sample sizes for longitudinal GEE-models and bssr.gee.1subgroup for blinded sample size re-estimation within a trial. See estimcov for more information on the used minimization algorithms.

Examples

#Generate a covariance-matrix with measurements at Baseline and at times c(1,1.5,2,5)

covar<-gen_cov_cor(var=3,rho=0.25,theta=1,Time=c(0,1,1.5,2,5),cov=TRUE)
covar

#Generate a correlation-matrix with the same values

corr<-gen_cov_cor(rho=0.25,theta=1,Time=c(0,1,1.5,2,5),cov=FALSE)
corr

Generate Time Series with Negative Binomial Distribution and Multivariate Gamma Frailty with Autoregressive Correlation Structure of Order One with Trend

Description

rnbinom.gf generates one or more independent time series following the Gamma frailty model. The generated data has negative binomial marginal distribution and the underlying multivariate Gamma frailty an autoregressive covariance structure.

Usage

get.groups(n, size, lambda, rho, tp, trend)

Arguments

Details

The function relies on rnbinom.gf for creating data with underlying constant or exponential trends.

Value

get.groups returns a matrix of dimension n x tp with marginal negative binomial distribution with means corresponding to trend parameters lambda, common dispersion parameter size and a correlation induce by rho, the correlation coefficient of the autoregressive multivariate Gamma frailty.

Source

rnbinom.gf computes observations from a Gamma frailty model by Fiocco et. al. 2009 using code contributed by Thomas Asendorf.

References

Fiocco M, Putter H, Van Houwelingen JC, (2009), A new serially correlated gamma-frailty process for longitudinal count data Biostatistics Vol. 10, No. 2, pp. 245-257.

See Also

rnbinom.gf for information on the Gamma frailty model.

Examples

random<-get.groups(n=c(1000,1000), size=c(0.5, 0.5), lambda=c(1, 2), rho=c(0.6, 0.6), tp=7,
  trend="constant")
head(random)

Sample Size Calculation for a One Subgroup Design

Description

n.1subgroup calculates the required sample size for proving a desired alternative when testing for an effect in the full or subpopulation. See 'Details' for more information.

Usage

n.1subgroup(
  alpha,
  beta,
  delta,
  sigma,
  tau,
  eps = 0.001,
  approx = c("conservative.t", "liberal.t", "normal"),
  k = 1,
  nmax = 1000,
  nmin = 0
)

Arguments

Details

This function performs sample size estimation in a design with a subgroup within a full population where we want to test for treatment effects between a control and a treatment group. Since patients from the subgroup might potentially benefit from the treatment more than patients not included in that subgroup, one might prefer testing hypothesis cercerning the full population and the subpopulation at the same time. Here standardized test statistics are their joined distributions are used to calculate the required sample size for the control and treatment group to prove an existing alternative delta with a specified power 1-beta when testing the global null hypothesis H_0: \Delta_F=\Delta_S=0 to level alpha.

For sample sizes n_C and n_T of the control and treatment group, respectively, the argument k is the sample size allocation factor, i.e. k = n_T/n_C.

Value

n.1subgroup returns the required sample size within the control group and treatment group.

Source

n.1subgroup uses code contributed by Marius Placzek.

See Also

#' bssr.1subgroup for blinded sample size reestimation within a running trial.

Examples

#Calculate required sample size to correctly reject with
#80% probability when testing the global Nullhypothesis H_0: Delta_F=Delta_S = 0
#assuming the true effect Delta_S=1 is in the subgroup (no effect outside of the subgroup)
#with subgroup prevalence tau=0.4.
#The variances in and outside of the subgroup are unequal, sigma=c(1,1.2).

estimate<-n.1subgroup(alpha=0.025,beta=0.1,delta=c(0,1),sigma=c(1,1.2),tau=0.4,eps=0.0001,
approx="conservative.t",k=2)
summary(estimate)

Sample Size estimation for longitudinal GEE models

Description

n.gee.1subgroup calculates the required sample size for proving a desired alternative when testing a regression coefficients in a full and/or a subpopulation. See 'Details' for more information.

Usage

n.gee.1subgroup(
  alpha,
  tail = "both",
  beta = NULL,
  delta,
  sigma,
  tau = 0.5,
  k = 1,
  npow = NULL,
  nmax = Inf
)

Arguments

Details

This function performs a sample size estimation in a design with a nested subgroup within an overall population. To calculate the required sample only the value of tested regressor needs to inserted as delta. sigma is the variance of that regressor. The power for the global null hypothesis is given by 1-beta and alpha specifies the false positve level for rejecting H_0: \Delta_F=\Delta_S=0 to level alpha.

Here argument k denotes the sample size allocation factor between treatment groups, i.e. k = n_T/n_C.

Value

n.gee.1subgroup returns the required sample size within the control group and treatment group.

Source

n.gee.1subgroup uses code contributed by Roland Gerard Gera.

See Also

bssr.1subgroup for blinded sample size re-estimation within a running trial and sandwich for estimating asymptotic covarianc mtrices in GEE models.

Examples

#Calculate required sample size to correctly reject Null with
#80% probability when testing global Nullhypothesis H_0: Delta_F=Delta_S = 0, while
#assuming the coefficient in and outside of the subgroup is Delta=c(0.1,0,1) with a
#subgroup-prevalence of tau=0.4.
#The variances of regressors in delta when variances are unequal sigma=c(0.8,0.4).

estimate<-n.gee.1subgroup(alpha=0.05,beta=0.2,delta=c(0.1,0.1),sigma=c(0.8,0.4),tau=0.4, k=1)
summary(estimate)

#Alternatively we can estimate the power our study would have
#if we know the effect in and outside our subgroup as
#well as the variance of the regressors. Here we
#estimate that only 300 Patiens total can be recruited and we are interested
#in the power that would give us.

n.gee.1subgroup(alpha=0.05,delta=c(0.1,0.1),sigma=c(0.8,0.4),tau=0.4, k=1, npow=300)

Sample Size Calculation for Comparing Two Groups when observing Longitudinal Count Data with marginal Negative Binomial Distribution and underlying Gamma Frailty with Autoregressive Correlation Structure of Order One

Description

n.nb.gf calculates required sample sizes for testing trend parameters in a Gamma frailty model

Usage

n.nb.gf(
  alpha = 0.025,
  power = 0.8,
  lambda,
  size,
  rho,
  tp,
  k = 1,
  h,
  hgrad,
  h0,
  trend = c("constant", "exponential", "custom"),
  approx = 20
)

Arguments

Details

The function calculates required samples sizes for testing trend parameters of trends in longitudinal negative binomial data. The underlying one-sided null-hypothesis is defined by H_0: h(\eta, \lambda) \geq h_0 vs. the alternative H_A: h(\eta, \lambda) < h_0. For testing these hypothesis, the program therefore requires a function h and a value h0.

n.nb.gf gives back the required sample size for the control and treatment group, to prove an existing alternative h(\eta, \lambda) - h_0 with a power of power when testing at level alpha. For sample sizes n_C and n_T of the control and treatment group, respectively, the argument k is the sample size allocation factor, i.e. k = n_T/n_C.

When calculating the expected sandwich estimator required for the sample size, certain terms can not be computed analytically and have to be approximated numerically. The value approx defines how close the approximation is to the true expected sandwich estimator. High values of approx provide better approximations but are compuationally more expensive.

Value

n.nb.gf returns the required sample size within the control group and treatment group.

Source

n.nb.gf uses code contributed by Thomas Asendorf.

See Also

rnbinom.gf for information on the Gamma frailty model, fit.nb.gf for calculating initial parameters required when performing sample size estimation, bssr.nb.gf for blinded sample size reestimation within a running trial.

Examples

##The example is commented as it may take longer than 10 seconds to run.
##Please uncomment prior to execution.

##Example for constant rates
#h<-function(lambda.eta){
#   lambda.eta[2]
#}
#hgrad<-function(lambda.eta){
#   c(0, 1, 0)
#}

##We assume the rate in the control group to be exp(lambda[1]) = exp(0) and an
##effect of lambda[2] = -0.3. The \code{size} is assumed to be 1 and the correlation
##coefficient \code{\rho} 0.5. At the end of the study, we would like to test
##the treatment effect specified in lambda[2], and therefore define function
##\code{h} and value \code{h0} accordingly.

#estimate<-n.nb.gf(lambda=c(0,-0.3), size=1, rho=1, tp=6, k=1, h=h, hgrad=hgrad,
#   h0=0.2, trend="constant", approx=20)
#summary(estimate)

##Example for exponential trend
#h<-function(lambda.eta){
#   lambda.eta[3]
#}
#hgrad<-function(lambda.eta){
#   c(0, 0, 1, 0)
#}

#estimate<-n.nb.gf(lambda=c(0, 0, -0.3/6), size=1, rho=0.5, tp=7, k=1, h=h, hgrad=hgrad,
#   h0=0, trend="exponential", approx=20)
#summary(estimate)

Sample Size Calculation for Comparing Two Groups when observing Longitudinal Count Data with marginal Negative Binomial Distribution and Autoregressive Correlation Structure of Order One: NB-INAR(1)

Description

n.nb.inar1 calculates the required sample size for proving a desired alternative when testing for a rate ratio between two groups unequal to one. Also gives back power for a specified sample size. See 'Details' for more information.

Usage

n.nb.inar1(
  alpha,
  power = NULL,
  delta,
  muC,
  size,
  rho,
  tp,
  k,
  npow = NULL,
  nmax = Inf
)

Arguments

Details

When testing for differences between rates \mu_C and \mu_T of two groups, a control and a treatment group respectively, we usually test for the ratio between the two rates, i.e. \mu_T/\mu_C = 1. The ratio of the two rates is refered to as \delta, i.e. \delta = \mu_T/\mu_C.

n.nb.inar1 gives back the required sample size for the control and treatment group required to prove an existing alternative theta with a specified power power when testing the null hypothesis H_0: \mu_T/\mu_C \ge 1 to level alpha. If power is not specified but instead npow, the power achieved with a total sample size of npow is calculated.

For sample sizes n_C and n_T of the control and treatment group, respectively, the argument k is the sample size allocation factor, i.e. k = n_T/n_C.

Value

rnbinom.inar1 returns the required sample size within the control group and treatment group.

Source

rnbinom.inar1 uses code contributed by Thomas Asendorf.

See Also

rnbinom.inar1 for information on the NB-INAR(1) model, fit.nb.inar1 for calculating initial parameters required when performing sample size estimation, bssr.nb.inar1 for blinded sample size reestimation within a running trial.

Examples

#Calculate required sample size to find significant difference with
#80% probability when testing the Nullhypothesis H_0: mu_T/mu_C >= 1
#assuming the true effect delta is 0.8 and rate, size and correlation
#parameter in the control group are 2, 1 and 0.5, respectively.

estimate<-n.nb.inar1(alpha=0.025, power=0.8, delta=0.8, muC=2, size=1, rho=0.5, tp=7, k=1)
summary(estimate)

estimate<-n.nb.inar1(alpha=0.025, npow=200, delta=0.8, muC=2, size=1, rho=0.5, tp=7, k=1)
summary(estimate)

Generate dataset of normal distributed observations in a one subgroup design

Description

r.1subgroup generates data for a design with one subgroup within a full population. Each observation is normal distributed with mean 0 in the placebo group and a potential effect in the treatment group. Whether the effect is solely in the subgroup or additionally a certain amount outside of the subgroup can be specified as well as potentially different variances within the subgroup and outside of the subgroup.

Usage

r.1subgroup(n, delta, sigma, tau, fix.tau = c("YES", "NO"), k)

Arguments

Details

For delta=(\Delta_F\S, \Delta_S)' and sigma=(\sigma_F\S, \sigma_S)' this function r.1subgroup generates data as follows:

Placebo group outside of subgroup ~N(0,\sigma^2_F\S), Placebo group within subgroup ~N(0,\sigma^2_S), Treatment group outside of subgroup ~N(\Delta_F\S,\sigma^2_F\S), Treatment group within subgroup ~N(\Delta_S,\sigma^2_S).

If fix.tau=YES the subgroup size is generated according to the prevalence tau, i.e. n_S=\tau*n. If fix.tau=YES, then each new generated observations probability to belong to the subgroup is Ber(\code{tau}) distributed and therefore only E(n_s)=\tau*n holds.

The argument k is the sample size allocation factor, i.e. let n_C and n_T denote the sample sizes of of the control and treatment group, respectively, then k = n_T/n_C.

Value

r.1subgroup returns a data matrix of dimension n x 3. The first column TrPl defines whether the observation belongs to the treatment group (TrPl=0) or to the placebo group (TrPl=1). Second column contains the grouping variable FS. For FS=1 the observation stems from the subgroup, for FS=0 from the full population without the subgroup. In the last column value the observation can be found. between time points.

Source

r.1subgroup uses code contributed by Marius Placzek.

Examples

set.seed(142)
random<-r.1subgroup(n=50, delta=c(0,1), sigma=c(1,1), tau=0.4, fix.tau="YES", k=2)
random 

Generate dataset of normal distributed repeated observations in a one subgroup design

Description

r.gee.1subgroup generates data for a design with one subgroup within a full population. Each baseline-observation is normal distributed with mean

\beta_0

in placebo group and

\beta_0+\beta_1

in treatment group. Measurements after baseline have mean

\beta_0+\beta_2*t

in placebo group and

\beta_0+\beta_1+\beta_2*t+\beta_3*t

in treatment group where

t

is the measurement time. Whether the effect can be found solely in the subgroup or additionally a certain amount outside of the subgroup can be specified as well as a potential different covariance-structure within subgroup and in the complementary subgroup.

Usage

r.gee.1subgroup(n, reg, sigma, rho, theta, tau, k, Time, OD)

Arguments

Details

For reglist(c(\beta_0^F\S,\beta_1^F\S,\beta_2^F\S,\beta_3^F\S), c(\beta_0^S,\beta_1^S,\beta_2^S,\beta_3^S)) and variances sigma=(\sigma_F\S, \sigma_S) function r.gee.1subgroup generates data given correlation-variables \rho and \theta as follows (and let t=0 be the baseline measurement):

Placebo group - complementary population y_{it}=N(\beta_0+\beta_2*t,\sigma_F\S), Placebo group - within subgroup y_{it}=N(\beta_0+\beta_2*t,\sigma_S), Treatment group - complementary population y_{it}=N(\beta_0+\beta_1+\beta_2*t+\beta_3*t,\sigma_F\S), Treatment group - within subgroup y_{it}=N(\beta_0+\beta_1+\beta_2*t+\beta_3*t,\sigma_S). Correlation between measurements - corr(\epsilon_it,\epsilon_io)=\rho^{(t-o)^\theta}

Argument k is the sample size allocation factor, i.e. the ratio between control and treatment. Let n_C and n_T denote sample sizes of control and treatment groups respectively, then k = n_T/n_C.

Argument Time is the vector denoting all measuring-times, i. e. every value for t.

Argument OD sets the overall dropout rate observed at the last timepoint. For OD=0.5, 50 percent of all observation had a dropout event at some point. If a subject experienced a dropout the starting time of the dropout is equally distributed over all timepoints.

Value

r.gee.1subgroup returns a list with 7 different entries. Every Matrix rows are the simulated subjects and the columns are the observed time points.

The first list element is a vector containing subject ids. The second element contains a matrix with the outcomes of a subject with row being the subjects and columns being the measuring-timepoints Elements 3 to 5 return matrices with the information of which patients have baseline-measurements, which patients belong to treatment and which to control and what are the observed timepoints for each patient respectively. The sixth entry returns a matrix which contains the residuals of each measurement. The seventh entry returns the sub-population identification.

Source

r.gee.1subgroup uses code contributed by Roland Gerard Gera

Examples

set.seed(2015)
dataset<-r.gee.1subgroup(n=200, reg=list(c(0,0,0,0.1),c(0,0,0,0.1)), sigma=c(3,2.5),
tau=0.5, rho=0.25, theta=1, k=1.5, Time=c(0:5), OD=0)
dataset

Generate Time Series with Negative Binomial Distribution and Multivariate Gamma Frailty with Autoregressive Correlation Structure of Order One

Description

rnbinom.gf generates one or more independent time series following the Gamma frailty model. The generated data has negative binomial marginal distribution and the underlying multivariate Gamma frailty an autoregressive covariance structure.

Usage

rnbinom.gf(n, size, mu, rho, tp)

Arguments

Details

The generated marginal negative binomial distribution with mean mu = \mu and size = \eta has density

(\mu/(\mu+\eta))^x \Gamma(x + \eta)/(\Gamma(x+1)\Gamma(\eta)) (\eta/(\mu+\eta))^\eta

for 0 < \mu, 0 < \eta and x=0, 1, 2, .... Hereby, each entry of vector mu corresponds to one time point. Therefore, each timepoint can have its distinct mean.

Within the Gamma frailty model, the correlation between two frailties of time points t and s for rho = \rho is given by

\rho^|t-s|

for 0 \le \rho \le 1. Note: this does not correspond to the correlation of observations.

Value

rnbinom.gf returns a matrix of dimension n x tp with marginal negative binomial distribution with means mu, common dispersion parameter size and a correlation induce by the autoregressive multivariate Gamma frailty.

Source

rnbinom.gf computes observations from a Gamma frailty model by Fiocco et. al. 2009 using code contributed by Thomas Asendorf.

References

Fiocco M, Putter H, Van Houwelingen JC, (2009), A new serially correlated gamma-frailty process for longitudinal count data Biostatistics Vol. 10, No. 2, pp. 245-257.

Examples

set.seed(8)
random<-rnbinom.gf(n=1000, size=0.6, mu=1:6, rho=0.8, tp=6)
cor(random)

#Check the marginal distribution of time point 3
plot(table(random[,3])/1000, xlab="Probability", ylab="Observation")
lines(0:26, dnbinom(0:26, mu=3, size=0.6), col="red")
legend("topright",legend=c("Theoretical Marginal Distribution", "Observed Distribution"),
  col=c("red", "black"), lty=1, lwd=c(1,2))

Generate Time Series with Negative Binomial Distribution and Autoregressive Correlation Structure of Order One: NB-INAR(1)

Description

rnbinom.inar1 generates one or more independent time series following the NB-INAR(1) model. The generated data has negative binomial marginal distribution and an autoregressive covariance structure.

Usage

rnbinom.inar1(n, size, mu, rho, tp)

Arguments

Details

The generated marginal negative binomial distribution with mean mu = \mu and size = \eta has density

(\mu/(\mu+\eta))^x \Gamma(x + \eta)/(\Gamma(x+1)\Gamma(\eta)) (\eta/(\mu+\eta))^\eta

for 0 < \mu, 0 < \eta and x=0, 1, 2, ....

Within the NB-INAR(1) model, the correlation between two time points t and s for rho = \rho is given through

\rho^|t-s|

for 0 \le \rho \le 1.

Value

rnbinom.inar1 returns a matrix of dimension n x tp with marginal negative binomial distribution with mean mu and dispersion parameter size, and an autoregressive correlation structure between time points.

Source

rnbinom.inar1 computes a reparametrization of the NB-INAR(1) model by McKenzie 1986 using code contributed by Thomas Asendorf.

References

McKenzie Ed (1986), Autoregressive Moving-Average Processes with Negative-Binomial and Geometric Marginal Distributions. Advances in Applied Probability Vol. 18, No. 3, pp. 679-705.

Examples

set.seed(8)
random<-rnbinom.inar1(n=1000, size=0.6, mu=2, rho=0.8, tp=6)
cor(random)

#Check the marginal distribution of time point 3
plot(table(random[,3])/1000, xlab="Probability", ylab="Observation")
lines(0:26, dnbinom(0:26, mu=2, size=0.6), col="red")
legend("topright",legend=c("Theoretical Marginal Distribution", "Observed Distribution"), 
col=c("red", "black"), lty=1, lwd=c(1,2))

Calculate the robust covariance estimator for GEE given an

Description

sandwich calculates the covariance structure between timepoints given matrices yCov, D,V and correctionmatrix. This is done to be able to account for missingness in the Data.

Usage

sandwich(
  yCov,
  D,
  V,
  correctionmatrix,
  missing = rep(0, dim(yCov)[[2]]),
  missingtype = c("none", "monotone", "intermittened")
)

Arguments

Details

yCov is either empirical or the estimated covariance-matrix between timepoints which is needed to calculate the sandwich estimator. This matrix can either be generated by estimating the empirical covariance matrix using existing data or by using function gen_cov_cor to calculate a estimation for the covariance.

D denotes the estimation of n^-1* \sum_i^N \Delta\mu_i/\delta\beta, which means that D=E(D_i). As of yet this has the unfortunate side effect that E(D_i

Value

sandwich returns the robust covariance estimator of regression coefficients which are implicitly defined by D.

Source

sandwich computes the asymptotic sandwich covariance estimator and uses code contributed by Roland Gerard Gera.

References

Liang Kung-Yee, Zeger Scott L. (1986); Jung Sin-Ho, Ahn Chul (2003); Wachtlin Daniel Kieser Meinhard (2013)

Examples

#Let's assume we wish to calculate the robust variance estimator for  equation
#\eqn{y_{it}=\beta_0+\beta_1*I_{treat}+\beta_2*t+\beta_3*I _{treat}*t+\epsilon_{it}}.
#Furthermore we use the identitiy matrix as the working covariance matrix.
#The chance to get treatment is 60 percent and the observed timerange ranges from 0:5.

  ycov = gen_cov_cor(var = 3,rho = 0.25,theta = 1,Time = 0:5,cov = TRUE)
  D = matrix(c(1,0.6,0,0,
               1,0.6,1,0.6,
               1,0.6,2,1.2,
               1,0.6,3,1.8,
               1,0.6,4,2.4,
               1,0.6,5,3.0),nrow=4)

 D=t(D)
 V=diag(1,length(0:5))
 #We correct entries where E(D_i %*% D_i) is unequal to E(D_i)%*%E(D_i) (D %*% D).
 correctionmatrix=matrix(c(1,1,1,1,1,1/0.6,1,1/0.6,1,1,1,1,1,1/0.6,1,1/0.6),nrow=4)
 missingtype = "none"

 robust=sandwich(yCov=ycov,D=D,V=V,missingtype=missingtype,correctionmatrix=correctionmatrix)
 robust

Generate Time Series with Negative Binomial Distribution and Autoregressive Correlation Structure of Order One: NB-INAR(1)

Description

rnbinom.inar1 generates one or more independent time series following the NB-INAR(1) model. The generated data has negative binomial marginal distribution and an autoregressive covariance structure.

Usage

sandwich2(sigma, rho, theta, k, Time, dropout, Model)

Arguments

Details

The generated marginal negative binomial distribution with mean mu = \mu and size = \eta has density

(\mu/(\mu+\eta))^x \Gamma(x + \eta)/(\Gamma(x+1)\Gamma(\eta)) (\eta/(\mu+\eta))^\eta

for 0 < \mu, 0 < \eta and x=0, 1, 2, ....

Within the NB-INAR(1) model, the correlation between two time points t and s for rho = \rho is given through

\rho^|t-s|

for 0 \le \rho \le 1.

Value

rnbinom.inar1 returns a matrix of dimension n x tp with marginal negative binomial distribution with mean mu and dispersion parameter size, and an autoregressive correlation structure between time points.

Source

rnbinom.inar1 computes a reparametrization of the NB-INAR(1) model by McKenzie 1986 using code contributed by Thomas Asendorf.

References

McKenzie Ed (1986), Autoregressive Moving-Average Processes with Negative-Binomial and Geometric Marginal Distributions. Advances in Applied Probability Vol. 18, No. 3, pp. 679-705.

Examples

set.seed(8)
random<-rnbinom.inar1(n=1000, size=0.6, mu=2, rho=0.8, tp=6)
cor(random)

#Check the marginal distribution of time point 3
plot(table(random[,3])/1000, xlab="Probability", ylab="Observation")
lines(0:26, dnbinom(0:26, mu=2, size=0.6), col="red")
legend("topright",legend=c("Theoretical Marginal Distribution", "Observed Distribution"), 
col=c("red", "black"), lty=1, lwd=c(1,2))

Simulation of a One Subgroup Design with Internal Pilot Study

Description

Given estimates of the treatment effects to be proven, the variances, and the prevalence, sim.bssr.1subgroup calculates a initial sample size and performes a blinded sample size recalculation after a prespecified number of subjects have been enrolled. Each oberservation is simulated and a final analysis executed. Several variations are included, such as different approximations or sample size allocation.

Usage

sim.bssr.1subgroup(
  nsim = 1000,
  alpha,
  beta,
  delta,
  sigma,
  tau,
  vdelta,
  vsigma,
  vtau,
  rec.at = 1/2,
  eps = 0.001,
  approx = c("conservative.t", "liberal.t", "normal"),
  df = c("n", "n1"),
  fix.tau = c("YES", "NO"),
  k = 1,
  adjust = c("YES", "NO")
)

Arguments

Details

This function combines sample size estimation, blinded sample size reestimation and analysis in a design with a subgroup within a full population where we want to test for treatment effects between a control and a treatment group. The required sample size for the control and treatment group to prove an existing alternative delta with a specified power 1-beta when testing the global null hypothesis H_0: \Delta_F=\Delta_S=0 to level alpha is calculated prior to the study and then recalculated in an internal pilot study.

For sample sizes n_C and n_T of the control and treatment group, respectively, the argument k is the sample size allocation factor, i.e. k = n_T/n_C.

The parameter df provides a difference to the standard sample size calculation procedure implemented in n.1subgroup. When applying a multivariate t distribution approximation to approximate the joint distribution of the standardized test statistics it gives the opportunity to use degrees of freedom depending on the number of subjects in the IPS instead of degrees of freedom depending on the projected final sample size. Note that this leads to better performance when dealing with extremely small subgroup sample sizes but significantly increases the calculated final sample size.

Value

sim.bssr.1subgroup returns a data.frame containing the mean recalculated sample size within the control group and treatment group and the achieved simulated power along with all relevant parameters.

Source

sim.bssr.1subgroup uses code contributed by Marius Placzek.

See Also

sim.bssr.1subgroup makes use of n.1subgroup, bssr.1subgroup, and r.1subgroup.

Examples

sim.bssr.1subgroup(nsim=10,alpha=0.025,beta=0.1,delta=c(0,1),sigma=c(1,1.3),tau=0.2,
vdelta=c(0,1),vsigma=c(1,1),vtau=0.3,eps=0.002, approx="conservative.t",df="n",
fix.tau="YES",k=1,adjust="NO")

Simulation of a longitudinal one subgroup design with internal pilot Study

Description

Given estimates of the treatment effects to be proven, the variances, and the prevalence, sim.bssr.gee.1subgroup calculates an initial sample size and performs a blinded sample size recalculation after a pre-specified number of subjects have been enrolled. Each observation is simulated and a final analysis executed. Several variations are included, such as different approximations or sample size allocation.

Usage

sim.bssr.gee.1subgroup(
  nsim = 1000,
  alpha = 0.05,
  tail = "both",
  beta = 0.2,
  delta = c(0.1, 0.1),
  vdelta = c(0.1, 0.1),
  sigma_pop = c(3, 3),
  vsigma_pop = c(3, 3),
  tau = 0.5,
  rho = 0.25,
  vrho = 0.25,
  theta = 1,
  vtheta = 1,
  Time = 0:5,
  rec.at = 0.5,
  k = 1,
  model = 1,
  V = diag(rep(1, length(Time))),
  OD = 0,
  vdropout = rep(0, length(Time)),
  missingtype = "none",
  vmissingtype = "none",
  seed = 2015
)

Arguments

Details

This function combines sample size estimation, blinded sample size re-estimation and analysis in a design with a subgroup within a full population where we want to test for treatment effects between a control and a treatment group. The required sample size for the control and treatment group to prove an existing alternative delta with a specified power 1-beta when testing the global null hypothesis H_0: \Delta_F=\Delta_S=0 to level alpha is calculated prior to the study and then recalculated in an internal pilot study.

For sample sizes n_C and n_T of the control and treatment group, respectively, the argument k is the sample size allocation factor, i.e. k = n_T/n_C.

Value

sim.bssr.1subgroup returns a data.frame containing the mean and variance of recalculated sample sizes within the control group and treatment group respectively and the achieved simulated power along with all relevant parameters.

Source

sim.bssr.gee.1subgroup uses code contributed by Roland Gerard Gera.

See Also

sim.bssr.gee.1subgroup makes use of n.gee.1subgroup, bssr.gee.1subgroup, and r.gee.1subgroup.

Examples

sim.bssr.gee.1subgroup(nsim = 5,missingtype = "intermittened")

Summarizing Blinded Sample Size Reestimation

Description

summary method for class "bssrest".

Usage

## S3 method for class 'bssrest'
summary(object, ...)

Arguments

Details

summary.bssrest gives back blinded sample size estimates. Furthermore, inputs are displayed for double checking.

See Also

n.nb.inar1 for initial sample size estimates within the NB-INAR(1) model.

Examples

#Calculate required sample size to find significant difference with
#80% probability when testing the Nullhypothesis H_0: mu_T/mu_C >= 1
#assuming the true effect delta is 0.8 and rate, size and correlation
#parameter in the control group are 2, 1 and 0.5, respectively.

estimate<-n.nb.inar1(alpha=0.025, power=0.8, delta=0.8, muC=2, size=1, rho=0.5, tp=7, k=1)

#Simulate data
set.seed(8)
placebo<-rnbinom.inar1(n=50, size=1, mu=2, rho=0.5, tp=7)
treatment<-rnbinom.inar1(n=50, size=1, mu=1.6, rho=0.5, tp=7)

#Blinded sample size reestimation
estimate<-bssr.nb.inar1(alpha=0.025, power=0.8, delta=0.8, x=rbind(placebo, treatment),
  n=c(50,50), k=1)
summary(estimate)

Summarizing Initial Sample Size Estimates

Description

summary method for class "ssest".

Usage

## S3 method for class 'ssest'
summary(object, ...)

Arguments

Details

summary.ssest gives back initial sample size estimates required. Furthermore, inputs are displayed for double checking.

See Also

n.nb.inar1 for initial sample size estimates within the NB-INAR(1) model.

Examples

#Calculate required sample size to find significant difference with
#80% probability when testing the Nullhypothesis H_0: mu_T/mu_C >= 1
#assuming the true effect delta is 0.8 and rate, size and correlation
#parameter in the control group are 2, 1 and 0.5, respectively.

estimate<-n.nb.inar1(alpha=0.025, power=0.8, delta=0.8, muC=2, size=1, rho=0.5, tp=7, k=1)
summary(estimate)

Testing Hypotheses in Gamma Frailty models

Description

test.nb.gf tests hypotheses for certain trends in Gamma frailty models

Usage

test.nb.gf(
  dataC,
  dataE,
  h,
  hgrad,
  h0 = 0,
  trend = c("constant", "exponential", "custom"),
  H0 = FALSE,
  one.sided = TRUE,
  ...
)

Arguments

Details

the function test.nb.gf tests for the null hypothesis h(\eta, \lambda) = h_0 against the alternative h(\eta, \lambda) \neq h_0. The fitting function allows for incomplete follow up, but not for intermittent missingness.

If parameter H0 is set to TRUE, the hessian and outer gradient are calculated under the assumption that lambda[2] \geq h0 if trend = "constant" or lambda[3] \geq h0 if trend = "exponential".

Value

test.nb.gf returns effect size, standard error, Z-statistic and p-value attained through standard normal approximation.

Source

test.nb.gf uses code contributed by Thomas Asendorf.

References

Fiocco M, Putter H, Van Houwelingen JC, (2009), A new serially correlated gamma-frailty process for longitudinal count data Biostatistics Vol. 10, No. 2, pp. 245-257.

See Also

rnbinom.gf for information on the Gamma Frailty model, n.nb.gf for calculating initial sample size required when performing inference, fit.nb.gf for calculating initial parameters required when performing sample size estimation.

Examples

#Create data from two groups
random<-get.groups(n=c(100,100), size=c(0.7, 0.7), lambda=c(0.8, 0), rho=c(0.6, 0.6),
  tp=7, trend="constant")

#Define hypothesis
h<-function(lambda.eta){
  lambda.eta[2]
}
hgrad<-function(lambda.eta){
  c(0, 1, 0)
}
test.nb.gf(dataC=random[101:200,], dataE=random[1:100,], h=h, hgrad=hgrad, h0=0,
  trend="constant", H0=FALSE)

Testing Hypotheses in NB-INAR(1) model

Description

test.nb.inar1 tests hypotheses for rate ratios of two groups in an NB-INAR(1) model

Usage

test.nb.inar1(dataC, dataE, h0 = 1)

Arguments

Details

the function test.nb.inar1 tests for the null hypothesis \lambda_T/\lambda_C = h0 against the alternative \lambda_T/\lambda_C \neq h_0. For attaining estimates, method of moments estimators are used.

Value

test.nb.inar1 returns effect size, standard error, Z-statistic and p-value attained through standard normal approximation.

Source

test.nb.inar1 uses code contributed by Thomas Asendorf.

See Also

rnbinom.inar1 for information on the NB-INAR(1) model, n.nb.inar1 for calculating initial sample size required when performing inference, fit.nb.inar1 for calculating initial parameters required when performing sample size estimation

Examples

set.seed(8)
groupE<-rnbinom.inar1(n=1000, size=0.6, mu=2, rho=0.8, tp=6)
groupC<-rnbinom.inar1(n=1000, size=0.6, mu=2, rho=0.8, tp=6)

test.nb.inar1(dataC=groupC, dataE=groupE, h0=1)