MKpower (version 0.9)

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

Value

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

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.

n.min

Starting point of search interval for sample size

n.max

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

tol

the desired accuracy for uniroot

use.uniroot

Finds one root of one equation

Author

Srinath Kolampally, Matthias Kohl

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 \(\mu_{Tk}-\mu_{Ck}\le 0\) for at least one \(k\in\{1,\ldots,K\}\) where Tk is treatment k, Ck is control k and K is the number of co-primary endpoints.

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.mpe.known.var and use the result as n.min. The input for n.max must be larger then n.min. 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.mpe.known.var

Examples

## compute power
if (FALSE) {
power.mpe.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.mpe.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 n.min and n.max must be larger
## then n.min so we try 60.
power.mpe.unknown.var(K = 2, delta = c(1,1), Sigma = diag(c(2,2)), power = 0.9,
                  sig.level = 0.025, n.min = 51, n.max = 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.mpe.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.mpe.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))
}