mpe (version 1.0)

power.unknown.var: Multiple Co-Primary Endpoints with Unknown Covariance

Description

The function calculates either sample size or power for continuous multiple co-primary endpoints with unknown covariance.

Usage

power.unknown.var(K, n = NULL, delta = NULL, Sigma, SD, rho, sig.level = 0.05, power = NULL, M = 10000, min.n = NULL, max.n = NULL, tol = .Machine$double.eps^0.25, use.uniroot = TRUE)

Arguments

K

number of co-primary endpoints

n

optional: sample size

delta

expected effect size (length K)

Sigma

unknown covariance matrix (dimension K x K)

SD

unknown standard deviations (length K)

rho

unknown correlations (length 0.5*K*(K-1))

sig.level

significance level (Type I error probability)

power

optional: power of test (1 minus Type II error probability)

M

Number of replications for the required simulations.

min.n

Starting point of search interval for sample size

max.n

End point of search interval for sample size, must be larger than min.n

tol

the desired accuracy for uniroot

use.uniroot

Finds one root of one equation

Value

Object of class power.mpe.test, a list of arguments (including the computed one) augmented with method and note elements.

Details

The function can be used to either compute sample size or power for continuous multiple co-primary endpoints with unknown covariance. The implementation is based on the formulas given in the references below.

The null hypothesis reads $mu_Tk-mu_Ck <= 0$="" for="" at="" least="" one="" $k="" in="" {1,...,k}$="" where="" tk="" is="" treatment="" k,="" ck="" control="" k="" and="" the="" number="" of="" co-primary="" endpoints.<="" p="">

One has to specify either n or power, the other parameter is determined. An approach to calculate sample size n, is to first call power.known.var and use the result as min.n. The input for max.n must be larger then min.n. Moreover, either covariance matrix Sigma or standard deviations SD and correlations rho must be given.

The sample size is calculated by simulating Wishart distributed random matrices, hence the results include a certain random variation.

References

Sugimoto, T. and Sozu, T. and Hamasaki, T. (2012). A convenient formula for sample size calculations in clinical trials with multiple co-primary continuous endpoints. Pharmaceut. Statist., 11: 118-128. doi:10.1002/pst.505

Sozu, T. and Sugimoto, T. and Hamasaki, T. and Evans, S.R. (2015). Sample Size Determination in Clinical Trials with Multiple Endpoints. Springer Briefs in Statistics, ISBN 978-3-319-22005-5.

See Also

power.known.var, mpe.t.test

Examples

## compute power
## Not run: 
# power.unknown.var(K = 2, n = 20, delta = c(1,1), Sigma = diag(c(1,1)))
# 
# ## To compute sample size, first assume covariance as known
# power.known.var(K = 2, delta = c(1,1), Sigma = diag(c(2,2)), power = 0.9,
#                   sig.level = 0.025)
# 
# ## The value of n, which is 51, is used as min.n and max.n must be larger
# ## then min.n so we try 60.
# power.unknown.var(K = 2, delta = c(1,1), Sigma = diag(c(2,2)), power = 0.9,
#                   sig.level = 0.025, min.n = 51, max.n = 60)
# 
# ## More complex example with unknown covariance matrix assumed to be
# Sigma <- matrix(c(1.440, 0.840, 1.296, 0.840,
#                   0.840, 1.960, 0.168, 1.568,
#                   1.296, 0.168, 1.440, 0.420,
#                   0.840, 1.568, 0.420, 1.960), ncol = 4)
# ## compute power
# power.unknown.var(K = 4, n = 90, delta = c(0.5, 0.75, 0.5, 0.75), Sigma = Sigma)
# ## equivalent: unknown SDs and correlation rho
# power.unknown.var(K = 4, n = 90, delta = c(0.5, 0.75, 0.5, 0.75),
#                   SD = c(1.2, 1.4, 1.2, 1.4),
#                   rho = c(0.5, 0.9, 0.5, 0.1, 0.8, 0.25))
# ## End(Not run)