spass (version 1.2)

n.nb.gf: 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.inar1 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 rnbinom.gf).

rho

correlation coefficient of the autoregressive correlation structure of the underlying Gamma frailty. Must be between 0 and 1 (see rnbinom.gf).

tp

number of observed time points. (see rnbinom.gf)

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'.

Value

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

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.

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

# NOT RUN {
##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)

# }