Power and Sample Size for Unordered Multi-Sample Multinomial Response

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.

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.

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 $\alpha$.

Author

Kaifeng Lu, kaifenglu@gmail.com

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))
#>   alpha     power  n ngroups ncats effectsize
#> 1  0.05 0.9082873 40       3     4  0.4466015