Obtains the power given sample size or obtains the sample size given power for the chi-square test for unordered multi-sample multinomial response.
getDesignUnorderedMultinom(
beta = NA_real_,
n = NA_real_,
ngroups = NA_integer_,
ncats = NA_integer_,
pi = NA_real_,
allocationRatioPlanned = NA_integer_,
rounding = TRUE,
alpha = 0.05
)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.
The type II error.
The total sample size.
The number of treatment groups.
The number of categories of the multinomial response.
The matrix of response probabilities for the treatment groups.
It should have ngroups rows and ncats-1 or ncats
columns.
Allocation ratio for the treatment groups.
Whether to round up sample size. Defaults to 1 for sample size rounding.
The two-sided significance level. Defaults to 0.05.
Kaifeng Lu, kaifenglu@gmail.com
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 \(\alpha\).
(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))
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