getDesignUnorderedMultinom: Power and Sample Size for Unordered Multi-Sample Multinomial...

In lrstat: Power and Sample Size Calculation for Non-Proportional Hazards and Beyond

Power and Sample Size for Unordered Multi-Sample Multinomial Response

Description

Obtains the power given sample size or obtains the sample size given power for the chi-square test for unordered multi-sample multinomial response.

Usage

getDesignUnorderedMultinom(
  beta = NA_real_,
  n = NA_real_,
  ngroups = NA_integer_,
  ncats = NA_integer_,
  pi = NA_real_,
  allocationRatioPlanned = NA_integer_,
  rounding = TRUE,
  alpha = 0.05
)

Arguments

beta	The type II error.
n	The total sample size.
ngroups	The number of treatment groups.
ncats	The number of categories of the multinomial response.
pi	The matrix of response probabilities for the treatment groups. It should have ngroups rows and ncats-1 or ncats columns.
allocationRatioPlanned	Allocation ratio for the treatment groups.
rounding	Whether to round up sample size. Defaults to 1 for sample size rounding.
alpha	The two-sided significance level. Defaults to 0.05.

Details

A multi-sample multinomial response design is used to test whether the response probabilities differ among multiple treatment arms. Let \pi_{gi} denote the response probability for category i = 1,\ldots,C in group g = 1,\ldots,G, where G is the total number of treatment groups, and C is the total number of categories for the response variable.

The chi-square test statistic is given by

X^2 = \sum_{g=1}^{G} \sum_{i=1}^{C} \frac{(n_{gi} - n_{g+}n_{+i}/n)^2}{n_{g+} n_{+i}/n}

where n_{gi} is the number of subjects in category i for group g, n_{g+} is the total number of subjects in group g, and n_{+i} is the total number of subjects in category i across all groups, and n is the total sample size.

Let r_g denote the randomization probability for group g, and define the weighted average response probability for category i across all groups as

\bar{\pi_i} = \sum_{g=1}^{G} r_g \pi_{gi}

• Under the null hypothesis, X^2 follows a chi-square distribution with (G-1)(C-1) degrees of freedom.

• Under the alternative hypothesis, X^2 follows a non-central chi-square distribution with non-centrality parameter

  \lambda = n \sum_{g=1}^{G} \sum_{i=1}^{C} \frac{r_g (\pi_{gi} - \bar{\pi_i})^2} {\bar{\pi_i}}

The sample size is chosen such that the power to reject the null hypothesis is at least 1-\beta for a given significance level \alpha.

Value

An S3 class designUnorderedMultinom object with the following components:

• power: The power to reject the null hypothesis.

• alpha: The two-sided significance level.

• n: The maximum number of subjects.

• ngroups: The number of treatment groups.

• ncats: The number of categories of the multinomial response.

• pi: The response probabilities for the treatment groups.

• effectsize: The effect size for the chi-square test.

• allocationRatioPlanned: Allocation ratio for the treatment groups.

• rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, [e-mail]

Examples

(design1 <- getDesignUnorderedMultinom(
  beta = 0.1, ngroups = 3, ncats = 4,
  pi = matrix(c(0.230, 0.320, 0.272,
                0.358, 0.442, 0.154,
                0.142, 0.036, 0.039),
              3, 3, byrow = TRUE),
  allocationRatioPlanned = c(2, 2, 1),
  alpha = 0.05))

lrstat documentation built on Aug. 8, 2025, 7:36 p.m.