PMCMRplus (version 1.9.12)

power.williams.test: Power calculations for minimum detectable difference of the Williams' test

Description

Compute the power of a Williams' test, or determine parameters to obtain a target power.

Usage

power.williams.test(n = NULL, k, delta, sd = 1, power = NULL, ...)

Value

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

Arguments

n

number of observations (per group).

k

number of treatment groups.

delta

clinically meaningful minimal difference (between a treatment group and control).

sd

common standard deviation.

power

power of test (1 minus Type II error probability).

...

further arguments, currently ignored.

Details

Exactly one of the parameters n or power must be passed as NULL, and that parameter is determined from the others.

The function has implemented the following Eq. in order to estimate power (Chow et al. 2008):

$$ 1 - \beta = 1 - \Phi \left(T_{K \alpha v} - |\Delta| / \sigma \sqrt{2/n}\right) $$

with \(|\Delta|\) the clinically meaningful minimal difference, \(T_{K \alpha v}\) the critical Williams' t-statistic for \(\alpha = 0.05\), \(v = \infty\) degree of freedom and \(\Phi\) the probability function of the standard normal function.

The required sample size (balanced design) is estimated based on the expression as given by the PASS manual, p. 595-2:

$$ n = 2 \sigma^2 ~ \left(T_{K \alpha v} + z_{\beta} \right)^2 ~ / ~ \Delta^2 $$

References

Chow, S.-C., Shao, J., Wan, H., 2008, Sample Size Calculations in Clinical Research, 2nd ed, Chapman & Hall/CRC: Boca Raton, FL.

See Also

optimise williamsTest

Examples

## Chow et al. 2008, p. 288 depicts 53 (rounded),
## better use ceiling for rounding
power.williams.test(power = 0.8, k = 3, delta = 11, sd = 22)
power.williams.test(n = 54, k = 3, delta = 11, sd = 22)

## PASS manual example:
## up-rounded n values are:
## 116, 52, 29, 14, 8 and 5
## according to PASS manual, p. 595-5
D <- c(10, 15, 20, 30, 40, 50)
y <- sapply(D, function(delta) {
 power.williams.test(power = 0.9, k = 4, delta = delta, sd = 25)$n
 })
ceiling(y)

if (FALSE) {
 ## compare with power.t.test
 ## and bonferroni correction
 power.t.test(power = 0.9, delta = 50, sd = 25,
 sig.level = 0.05 / 4, alternative = "one.sided")
}