| Type: | Package |
| Title: | Study Planning and Adaptation of Sample Size |
| Version: | 1.3 |
| Date: | 2020-12-01 |
| Author: | T. Asendorf, R. Gera, S. Islam, M. Harden and M. Placzek |
| Maintainer: | Marius Placzek <marius.placzek@med.uni-goettingen.de> |
| Description: | Sample size estimation and blinded sample size reestimation in Adaptive Study Design. |
| License: | GPL-2 |GPL-3 [expanded from: GPL] |
| Encoding: | UTF-8 |
| LazyData: | true |
| Imports: | Rcpp,mvtnorm,multcomp,MASS,geepack,stats,utils |
| LinkingTo: | Rcpp |
| RoxygenNote: | 6.0.1 |
| NeedsCompilation: | yes |
| Packaged: | 2020-12-02 16:01:42 UTC; tasendorf |
| Repository: | CRAN |
| Date/Publication: | 2020-12-07 22:40:02 UTC |
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
data | data matrix with data from ongoing trial: see 'Details'. |
alpha | level (type I error) to which the hypothesis is tested. |
beta | type II error (power=1-beta) to which an alternative should be proven. |
delta | vector of treatment effects to be proven, c(outside subgroup, inside subgroup). |
eps | precision parameter concerning the power calculation in the iterative sample size search algorithm. |
approx | approximation method: Use a conservative multivariate t distribution ("conservative.t"), a liberal multivariate t distribution ("liberal.t") or a multivariate normal distribution ("normal") to approximate the joint distribution of the standardized test statistics. |
df | in case of a multivariate t distribution approximation, recalculate sample size with degrees of freedom depending on the size of the IPS (df=n1) or depending on the final sample size (df=n). |
adjust | adjust blinded estimators for assumed treatment effect ("YES","No"). |
k | sample size allocation factor between groups: see 'Details'. |
nmax | maximum total sample size. |
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 existingalternativedelta with a specified power 1-beta when testing the global null hypothesisH_0: \Delta_F=\Delta_S=0 to levelalpha is calculated.
The data matrixdata 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 sizesn_C andn_T of the control and treatment group, respectively, the argumentk is thesample size allocation factor, i.e.k = n_T/n_C.
The parameterdf provides a difference to the standard sample size calculation procedure implemented inn.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. Usesummary.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 forn.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
alpha | level (type I error) to which the hypothesis is tested. |
tail | which type of test is used, e.g. which quartile und H0 is calculated. |
beta | type II error (power=1-beta) to which an alternative should be proven. |
delta | vector of estimated treatment effect in overall and sub population, c(overall population, only subpopulation). |
estsigma | vector of re-estimated standard deviations, c(full population, subpopulation). See 'Details'. |
tau | ratio between complementary F/S and sub-population S. |
k | treatment allocation factor between groups: see 'Details'. |
Details
This function provides a simple warped forn.gee.1subgroup where instead of initial assumptions, reestimated nuisance parameter are used.For more information seen.gee.1subgroup.Required samplesize to test alternativedelta with specified power 1-beta when testing the global null hypothesisH_0: \beta_3^F=\beta_3^S=0 to levelalpha is estimated. When testing outcomes have varianceestsigma.
For sample sizesn_C andn_T of the control and treatment group respectively, the argumentk is thesample size allocation factor, i.e.k = n_T/n_C andtau represents the ratio of the sub-population.
Value
bssr.gee.1subgroup returns a list containing the recalculated sample sizes along with all relevant parameters. Usesummary.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 andestimcov how the re-estimate nuisance parameters. Seesim.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 fortrend 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
data | a matrix or data frame containing count data which is to be fitted. Columns correspond to time points, rows to observations. |
alpha | level (type I error) to which the hypothesis is tested. |
power | power (1 - type II error) to which an alternative should be proven. |
delta | the relevant effect size, which is assumed to be true, see 'Details'. |
h0 | the value against which h is tested, see 'Details'. |
tp | number of observed time points. (see |
k | sample size allocation factor between groups: see 'Details'. |
trend | the trend which assumed to underlying in the data. |
approx | numer of iterations in numerical calculation of the sandwich estimator, see 'Details'. |
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 effectdelta is therefore equal to\lambda_2.
Under an exponential trend, the means in control and experiment group are equal toexp(\lambda_1+t \cdot \lambda_2) and\lambda_1 + t\cdot \lambda_2 + t\cdot \lambda_3, respectively.The treatment effectdelta is therefore equal to\lambda_3.
bssr.nb.gf returns the required sample size for the control and treatment group required to prove an existingalternativedelta with a specified powerpower when testing the null hypothesisH_0: \delta \ge h_0 at levelalpha.Nuisance parameters are estimated through the blinded observationsdata, thus not further required.For sample sizesn_C andn_T of the control and treatment group, respectively, the argumentk is the desiredsample 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 calculatinginitial sample size required when performing inference,fit.nb.gf for calculatinginitial 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 fora rate ratio between two groups unequal to one. See 'Details' for more information.
Usage
bssr.nb.inar1(alpha, power, delta, x, n, k)Arguments
alpha | level (type I error) to which the hypothesis is tested. |
power | power (1 - type II error) to which an alternative should be proven. |
delta | the rate ratio which is to be proven. |
x | a matrix or data frame containing count data which is to be fitted. Columns correspond to time points, rows to observations. |
n | a vector giving the sample size within the control group and the treatment group, respecitvely. |
k | planned sample size allocation factor between groups: see 'Details'. |
Details
When testing for differences between rates\mu_C and\mu_T of two groups, a control and a treatment group respectively, we usuallytest 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 existingalternativetheta with a specified powerpower when testing the null hypothesisH_0: \mu_T/\mu_C \ge 1 to levelalpha.Nuisance parameters are estimated through the blinded observationsx, thus not further required.
for sample sizesn_C andn_T of the control and treatment group, respectively, the argumentk is the desiredsample 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 calculatinginitial sample size required when performing inference,fit.nb.inar1 for calculatinginitial 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 dataplacebo<-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 reestimationblinded.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
data | matrix with the dataset which is used to estimate the covariance and dropout structure. |
Time | vector with observation-times. |
Startvalues | vector with starting values for variance, |
stepwidth | vector describing the step length of previously mentioned values. |
maxiter | maximum amount of iterations |
lower | vector with minimum for the parameters described in Startvalues |
upper | vector with maximum for the parameters described in Startvalues |
Details
This function is designed to estimate the variance,rho andtheta 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 andtheta 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
dataC | a matrix containing count data from the control group, which is to be fitted. Columns correspond to time points, rows to observations. |
dataE | a matrix containing count data from the experiment group, which is to be fitted. Columns correspond to time points, rows to observations. |
trend | the trend which assumed to underlying in the data. |
lower | vector of lower bounds for estimated parameters |
upper | vector of upper bounds for estimated parameters |
method | algorithm used for minimization of the likelihood, see |
start | vector of starting values for estimated parameters |
approx | numer of iterations in numerical calculation of the sandwich estimator, see 'Details'. |
rho | indicates whether or not to calculate the correlation coefficient of Gamma frailties. Must be TRUE or FALSE. |
H0 | indicates whether or not to calculate the hessian and outer gradient matrix under the null hypothesis, see 'Details'. |
h0 | the value against which is tested under the null |
Details
the functionfit.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 haveto be approximated numerically. The valueapprox defines how close the approximation is to the true expected sandwich estimator.High values ofapprox 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 thatlambda[2]\geqh0 iftrend = "constant" orlambda[3]\geqh0 iftrend = "exponential".
Value
fit.nb.gf returns estimates of the trend parameterslambda, dispersion parametersize,Hessian matrixhessian, outer gradient product matrixogradient and, if inquired, correlation coefficientrho.
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 dataBiostatistics Vol. 10, No. 2, pp. 245-257.
See Also
rnbinom.gf for information on the Gamma frailty model,n.nb.gf for calculatinginitial sample size required when performing inference,bssr.nb.gf for blindedsample size reestimation within a running trial,optim for more information on the used minimization algorithms.
Examples
#Generate data from the Gamma frailty modelrandom<-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
x | a matrix or data frame containing count data which is to be fitted. Columns correspond to time points, rows to observations. |
lower | vector of lower bounds for estimated parameters |
upper | vector of upper bounds for estimated parameters |
method | algorithm used for minimization of the likelihood, see |
start | vector of starting values for estimated parameters |
Details
the functionfit.nb.inar1 fits a reparametrization of the NB-INAR(1) model as found in McKenzie (1986). The reparametrized modelassumes equal means and dispersion parameter between time points with an autoregressive correlation structure. The function is especially usefulfor estimating parameters for an initial sample size calculation usingn.nb.inar1. The fitting function allows for incomplete follow up,but not for intermittent missingness.
Value
fit.nb.inar1 return estimates of the meanmu, dispersion parametersize and correlation coefficientrho.
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 calculatinginitial sample size required when performing inference,bssr.nb.inar1 for blindedsample size reestimation within a running trial,optim for more information on the used minimization algorithms.
Examples
#Generate data from the NB-INAR(1) modelset.seed(8)random<-rnbinom.inar1(n=1000, size=1.5, mu=2, rho=0.6, tp=7)estimate<-fit.nb.inar1(random)estimateGeneration of a covariance or a correlation matrix
Description
Generate a covariance or correlation matrix given parametersvar,rho,theta for the covariance structure,Time for the observed timepoints andcov=TRUE if a covariance orcov=FALSE if a correlation-matrix is generated.
Usage
gen_cov_cor(var = 1, rho, theta, Time, cov = TRUE)Arguments
var | variance at each timepoint |
rho | correlation between two adjacent timepoints 1 timeunit appart |
theta | variable specifying the type of the correlation structure: see 'Details' |
Time | list with time measures which are used to generate the covariance- or correlation-structure: see 'Details' |
cov | TRUE/FALSE statement which determines if a covariance- or a correlation-matrix is generated. |
Details
gen_cov_cor is used to generate either a covariance or a correlation matrix. Given vectorTime and parametersvar,rho andtheta 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
@seealsor.gee.1subgroup for information on the generated longitudinal data andn.gee.1subgroup for the calculation ofinitial sample sizes for longitudinal GEE-models andbssr.gee.1subgroup for blindedsample size re-estimation within a trial. Seeestimcov 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 valuescorr<-gen_cov_cor(rho=0.25,theta=1,Time=c(0,1,1.5,2,5),cov=FALSE)corrGenerate 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
n | number of observations. |
size | dispersion parameter (the shape parameter of the gamma mixing distribution). Must be strictly positive, need not be integer. |
lambda | vector of means of trend parameters. |
rho | correlation coefficient of the underlying autoregressive Gamma frailty. Must be between 0 and 1. |
tp | number of observed time points. |
trend | a string giving the trend which is to be simulated. |
Details
The function relies onrnbinom.gf for creating data with underlying constant or exponential trends.
Value
get.groups returns a matrix of dimensionn xtp with marginal negative binomialdistribution with means corresponding to trend parameterslambda, common dispersion parametersize and a correlation induce byrho,the correlation coefficient of the autoregressive multivariate Gamma frailty.
Source
rnbinom.gf computes observations from a Gamma frailty model byFiocco 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 dataBiostatistics 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 foran 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
alpha | level (type I error) to which the hypothesis is tested. |
beta | type II error (power=1-beta) to which an alternative should be proven. |
delta | vector of treatment effects to be proven, c(outside subgroup, inside subgroup). |
sigma | vector of standard deviations, c(outside subgroup, inside subgroup). |
tau | subgroup prevalence. |
eps | precision parameter concerning the power calculation in the iterative sample size search algorithm. |
approx | approximation method: Use a conservative multivariate t distribution ("conservative.t"), a liberal multivariate t distribution ("liberal.t") or a multivariate normal distribution ("normal") to approximate the joint distribution of the standardized teststatistics. |
k | sample size allocation factor between groups: see 'Details'. |
nmax | maximum total sample size. |
nmin | minimum total sample size. |
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 therequired sample size for the control and treatment group to prove an existingalternativedelta with a specified power 1-beta when testing the global null hypothesisH_0: \Delta_F=\Delta_S=0 to levelalpha.
For sample sizesn_C andn_T of the control and treatment group, respectively, the argumentk is thesample 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
alpha | level (type I error) to which the hypothesis is tested. |
tail | which type of test is used, e.g. which quartile und H0 is calculated. |
beta | type II error (power=1-beta) to which an alternative should be proven. |
delta | vector of estimated treatment effect in overall and sub population, c(overall population, only subpopulation). |
sigma | vector of estimated standard deviations, c(full population, subpopulation). See 'Details'. |
tau | subgroup prevalence. |
k | sample size allocation factor between control and treatment: see 'Details'. |
npow | calculates power of a test if |
nmax | maximum total sample size. |
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 asdelta.sigma is the variance of that regressor.The power for the global null hypothesis is given by 1-beta andalpha specifies the false positve level for rejectingH_0: \Delta_F=\Delta_S=0 to levelalpha.
Here argumentk denotes thesample 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 andsandwich 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
alpha | level (type I error) to which the hypothesis is tested. |
power | power (1 - type II error) to which an alternative should be proven. |
lambda | the set of trend parameters assumed to be true at the beginning prior to trial onset |
size | dispersion parameter (the shape parameter of the gamma mixing distribution). Must be strictly positive, need not be integer (see |
rho | correlation coefficient of the autoregressive correlation structure of the underlying Gamma frailty. Must be between 0 and 1 (see |
tp | number of observed time points. (see |
k | sample size allocation factor between groups: see 'Details'. |
h | hypothesis to be tested. The function must return a single value when evaluated on lambda. |
hgrad | gradient of function h |
h0 | the value against which h is tested, see 'Details'. |
trend | the trend which assumed to underlying in the data. |
approx | numer of iterations in numerical calculation of the sandwich estimator, see 'Details'. |
Details
The function calculates required samples sizes for testing trend parameters of trends in longitudinal negative binomial data. The underlyingone-sided null-hypothesis is defined byH_0: h(\eta, \lambda) \geq h_0 vs. the alternativeH_A: h(\eta, \lambda) < h_0. For testingthese hypothesis, the program therefore requires a functionh and a valueh0.
n.nb.gf gives back the required sample size for the control and treatment group, to prove an existing alternativeh(\eta, \lambda) - h_0with a power ofpower when testing at levelalpha. For sample sizesn_C andn_T of the control and treatment group, respectively, the argumentk is thesample 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 haveto be approximated numerically. The valueapprox defines how close the approximation is to the true expected sandwich estimator.High values ofapprox 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 calculatinginitial parameters required when performing sample size estimation,bssr.nb.gf for blindedsample 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 fora 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
alpha | level (type I error) to which the hypothesis is tested. |
power | power (1 - type II error) to which an alternative should be proven. |
delta | the rate ratio which is to be proven. |
muC | the rate observed within the control group. |
size | dispersion parameter (the shape parameter of the gamma mixing distribution). Must be strictly positive, need not be integer (see |
rho | correlation coefficient of the underlying autoregressive correlation structure. Must be between 0 and 1 (see |
tp | number of observed time points. (see |
k | sample size allocation factor between groups: see 'Details'. |
npow | sample size for which a power is to be calculated. Can not be specified if power is also specified. |
nmax | maximum total sample size of both groups. If maximum is reached a warning message is broadcasted. |
Details
When testing for differences between rates\mu_C and\mu_T of two groups, a control and a treatment group respectively, we usuallytest 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 existingalternativetheta with a specified powerpower when testing the null hypothesisH_0: \mu_T/\mu_C \ge 1 to levelalpha.Ifpower is not specified but insteadnpow, the power achieved with a total sample size ofnpow is calculated.
For sample sizesn_C andn_T of the control and treatment group, respectively, the argumentk is thesample 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 calculatinginitial parameters required when performing sample size estimation,bssr.nb.inar1 for blindedsample 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
n | number of observations. If length(n) > 1, the length is taken to be the number required. |
delta | vector of treatment effects in the treatment group, c(outside subgroup, within subgroup). |
sigma | vector of standard deviations, c(outside subgroup, inside subgroup). |
tau | subgroup prevalence. |
fix.tau | subgroup prevalence fix or simulated according to tau, see 'Details'. |
k | sample size allocation factor between groups: see 'Details'. |
Details
Fordelta=(\Delta_F\S, \Delta_S)' andsigma=(\sigma_F\S, \sigma_S)'this functionr.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).
Iffix.tau=YES the subgroup size is generated according to the prevalencetau, i.e.n_S=\tau*n.Iffix.tau=YES, then each new generated observations probability to belong to the subgroup isBer(\code{tau})distributed and therefore onlyE(n_s)=\tau*n holds.
The argumentk is thesample size allocation factor, i.e. letn_C andn_T denote the sample sizes of of the control andtreatment group, respectively, thenk = n_T/n_C.
Value
r.1subgroup returns a data matrix of dimensionn x3. The first columnTrPl defines whetherthe observation belongs to the treatment group (TrPl=0) or to the placebo group (TrPl=1). Second columncontains the grouping variableFS. ForFS=1 the observation stems from the subgroup, forFS=0 fromthe full population without the subgroup. In the last columnvalue 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)randomGenerate 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
n | overall sample size for the overall population |
reg | list containing coefficients
to
for complementary population, |
sigma | vector with standard deviations for generated observations c(complementary population, subpopulation). |
rho | variable used together with |
theta | variable used together with |
tau | subgroup prevalence. |
k | sample size allocation factor between treatment groups: see 'Details'. |
Time | list of timepoints |
OD | percentage of observed overall dropout at last timepoint: see 'Details'. |
Details
Forreglist(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 variancessigma=(\sigma_F\S, \sigma_S) functionr.gee.1subgroup generates data given correlation-variables\rho and\theta as follows (and let t=0 be the baseline measurement):
Placebo group - complementary populationy_{it}=N(\beta_0+\beta_2*t,\sigma_F\S),Placebo group - within subgroupy_{it}=N(\beta_0+\beta_2*t,\sigma_S),Treatment group - complementary populationy_{it}=N(\beta_0+\beta_1+\beta_2*t+\beta_3*t,\sigma_F\S),Treatment group - within subgroupy_{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}
Argumentk is the sample size allocation factor, i.e. the ratio between control and treatment. Letn_C andn_T denote sample sizes of control and treatment groups respectively, thenk = n_T/n_C.
ArgumentTime is the vector denoting all measuring-times, i. e. every value fort.
ArgumentOD sets the overall dropout rate observed at the last timepoint. ForOD=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-timepointsElements 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)datasetGenerate 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
n | number of observations. If length(n) > 1, the length is taken to be the number required. |
size | dispersion parameter (the shape parameter of the gamma mixing distribution). Must be strictly positive, need not be integer. |
mu | vector of means of time points: see 'Details'. |
rho | correlation coefficient of the underlying autoregressive Gamma frailty. Must be between 0 and 1. |
tp | number of observed time points. |
Details
The generated marginal negative binomial distribution with meanmu =\mu andsize =\eta has density
(\mu/(\mu+\eta))^x \Gamma(x + \eta)/(\Gamma(x+1)\Gamma(\eta)) (\eta/(\mu+\eta))^\eta
for0 < \mu,0 < \eta andx=0, 1, 2, .... Hereby, each entry of vectormu corresponds toone time point. Therefore, each timepoint can have its distinct mean.
Within the Gamma frailty model, the correlation between two frailties of time pointst ands forrho =\rho is given by
\rho^|t-s|
for0 \le \rho \le 1. Note: this does not correspond to the correlation of observations.
Value
rnbinom.gf returns a matrix of dimensionn xtp with marginal negative binomialdistribution with meansmu, common dispersion parametersize and a correlation induce by the autoregressivemultivariate Gamma frailty.
Source
rnbinom.gf computes observations from a Gamma frailty model byFiocco 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 dataBiostatistics 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 3plot(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
n | number of observations. If length(n) > 1, the length is taken to be the number required. |
size | dispersion parameter (the shape parameter of the gamma mixing distribution). Must be strictly positive, need not be integer. |
mu | parametrization via mean: see 'Details'. |
rho | correlation coefficient of the underlying autoregressive correlation structure. Must be between 0 and 1. |
tp | number of observed time points. |
Details
The generated marginal negative binomial distribution with meanmu =\mu andsize =\eta has density
(\mu/(\mu+\eta))^x \Gamma(x + \eta)/(\Gamma(x+1)\Gamma(\eta)) (\eta/(\mu+\eta))^\eta
for0 < \mu,0 < \eta andx=0, 1, 2, ....
Within the NB-INAR(1) model, the correlation between two time pointst ands forrho =\rho is given through
\rho^|t-s|
for0 \le \rho \le 1.
Value
rnbinom.inar1 returns a matrix of dimensionn xtp with marginal negative binomialdistribution with meanmu and dispersion parametersize, and an autoregressive correlation structurebetween time points.
Source
rnbinom.inar1 computes a reparametrization of the NB-INAR(1) model byMcKenzie 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 3plot(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 matricesyCov,D,V andcorrectionmatrix. 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
yCov |
|
D |
|
V |
|
correctionmatrix | As of this version this matrix is needed to correct some calculations. see 'Details' to see for more details and how to correctly select matrices. |
missing | vector which denotes the probability to experience a dropout at each timepoint. If |
missingtype | String which describes the type of missingness occuring in the data. |
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 functiongen_cov_cor to calculate a estimation for the covariance.
D denotes the estimation ofn^-1* \sum_i^N \Delta\mu_i/\delta\beta, which means thatD=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 byD.
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) robustGenerate 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
sigma | assymptotic standard deviation for Full and subpupulation |
rho | correlation coefficient of the underlying autoregressive correlation structure. Must be between 0 and 1. |
theta | correlation absorption coefficient if tinepoints are farther appart |
k | sample size allocation factor between groups: see 'Details'. |
Time | vector of measured timepoints |
dropout | vector describing the percentage of dropout in every timepoint |
Model | either 1 or 2, describing if 4-regressor or 3-regressor model was used. |
Details
The generated marginal negative binomial distribution with meanmu =\mu andsize =\eta has density
(\mu/(\mu+\eta))^x \Gamma(x + \eta)/(\Gamma(x+1)\Gamma(\eta)) (\eta/(\mu+\eta))^\eta
for0 < \mu,0 < \eta andx=0, 1, 2, ....
Within the NB-INAR(1) model, the correlation between two time pointst ands forrho =\rho is given through
\rho^|t-s|
for0 \le \rho \le 1.
Value
rnbinom.inar1 returns a matrix of dimensionn xtp with marginal negative binomialdistribution with meanmu and dispersion parametersize, and an autoregressive correlation structurebetween time points.
Source
rnbinom.inar1 computes a reparametrization of the NB-INAR(1) model byMcKenzie 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 3plot(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 recalculationafter 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
nsim | number of simulation runs. |
alpha | level (type I error) to which the hypothesis is tested. |
beta | type II error (power=1-beta) to which an alternative should be proven. |
delta | vector of true treatment effects, c(outside subgroup, inside subgroup). |
sigma | vector of true standard deviations, c(outside subgroup, inside subgroup). |
tau | subgroup prevalence. |
vdelta | vector of treatment effects to be proven, c(outside subgroup, inside subgroup). |
vsigma | vector of assumed standard deviations, c(outside subgroup, inside subgroup). |
vtau | expected subgroup prevalence. |
rec.at | blinded sample size review is performed after |
eps | precision parameter concerning the power calculation in the iterative sample size search algorithm. |
approx | approximation method: Use a conservative multivariate t distribution ("conservative.t"), a liberal multivariate t distribution ("liberal.t") or a multivariate normal distribution ("normal") to approximate the joint distribution of the standardized teststatistics. |
df | in case of a multivariate t distribution approximation, recalculate sample size with degrees of freedom depending on the size of the IPS (df=n1) or depending on the final sample size (df=n). |
fix.tau | subgroup prevalence is fixed by design (e.g. determined by recruitment) or is simulated and has to be reestimated during the blinded review. |
k | sample size allocation factor between groups: see 'Details'. |
adjust | adjust blinded estimators for assumed treatment effect ("YES","No"). |
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 existingalternativedelta with a specified power 1-beta when testing the global null hypothesisH_0: \Delta_F=\Delta_S=0 to levelalpha is calculated prior to the study and then recalculated in an internal pilot study.
For sample sizesn_C andn_T of the control and treatment group, respectively, the argumentk is thesample size allocation factor, i.e.k = n_T/n_C.
The parameterdf provides a difference to the standard sample size calculation procedure implemented inn.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 ofn.1subgroup,bssr.1subgroup, andr.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 recalculationafter 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
nsim | number of simulation runs. |
alpha | level (type I error) to which the hypothesis is tested. |
tail | which type of test is used, e.g. which quartile und H0 is calculated |
beta | type II error (power=1-beta) to which an alternative should be proven. |
delta | vector of true treatment effects, c(overall population, inside subgroup). |
vdelta | vector of treatment effects to be proven, c(overall population, inside subgroup). |
sigma_pop | vector of true standard deviations of the treatment effects, c(overall population, subgroup). |
vsigma_pop | vector of assumed standard deviations, c(overall population, inside subgroup). |
tau | subgroup prevalence. |
rho | true correlation coefficient between two adjacent timepoints |
vrho | initial expectation of the correlation coefficient between two adjacent timepoints |
theta | true correlation absorption coefficient if timepoints are farther apart |
vtheta | expected correlation absorption coefficient if timepoints are farther apart |
Time | vector of measured timepoints |
rec.at | blinded sample size review is performed after |
k | sample size allocation factor between groups: see 'Details'. |
model | which of the two often revered statistical models should be used?: see 'Details'. |
V | working covariance matrix. |
OD | overall dropout measured at last timepoint |
vdropout | vector of expected dropouts per timepoint if missingness is to be expected |
missingtype | true missingtype underlying the missingness |
vmissingtype | initial assumptions about the missingtype underlying the missingness |
seed | set seed value for the simulations to compare results. |
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 existingalternativedelta with a specified power 1-beta when testing the global null hypothesisH_0: \Delta_F=\Delta_S=0 to levelalpha is calculated prior to the study and then recalculated in an internal pilot study.
For sample sizesn_C andn_T of the control and treatment group, respectively, the argumentk is thesample 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 ofn.gee.1subgroup,bssr.gee.1subgroup, andr.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
object | an object of class " |
... | Arguments to be passed to or from other methods. |
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 dataset.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 reestimationestimate<-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
object | an object of class " |
... | Arguments to be passed to or from other methods. |
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
dataC | a matrix or data frame containing count data from the control group. Columns correspond to time points, rows to observations. |
dataE | a matrix or data frame containing count data from the experiment group. Columns correspond to time points, rows to observations. |
h | hypothesis to be tested. The function must return a single value when evaluated on lambda. |
hgrad | gradient of function h |
h0 | the value against which h is tested, see 'Details'. |
trend | the trend which assumed to be underlying in the data. |
H0 | indicates if the sandwich estimator is calculated under the null hypothesis or alternative. |
one.sided | indicates if the hypothesis should be tested one- or two-sided |
... | Arguments to be passed to function |
Details
the functiontest.nb.gf tests for the null hypothesish(\eta, \lambda) = h_0 against the alternativeh(\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 thatlambda[2]\geqh0 iftrend = "constant" orlambda[3]\geqh0 iftrend = "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 dataBiostatistics Vol. 10, No. 2, pp. 245-257.
See Also
rnbinom.gf for information on the Gamma Frailty model,n.nb.gf for calculatinginitial sample size required when performing inference,fit.nb.gf for calculatinginitial parameters required when performing sample size estimation.
Examples
#Create data from two groupsrandom<-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 hypothesish<-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
dataC | a matrix or data frame containing count data from the control group. Columns correspond to time points, rows to observations. |
dataE | a matrix or data frame containing count data from the experiment group. Columns correspond to time points, rows to observations. |
h0 | the value against which h is tested, see 'Details'. |
Details
the functiontest.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 calculatinginitial sample size required when performing inference,fit.nb.inar1 for calculatinginitial 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)