Obtains the power given sample size or obtains the sample size given power for difference in two-sample multinomial responses.
getDesignTwoMultinom(
beta = NA_real_,
n = NA_real_,
ncats = NA_integer_,
pi1 = NA_real_,
pi2 = NA_real_,
allocationRatioPlanned = 1,
rounding = TRUE,
alpha = 0.05
)An S3 class designTwoMultinom 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.
ncats: The number of categories of the multinomial response.
pi1: The prevalence of each category for the treatment group.
pi2: The prevalence of each category for the control group.
effectsize: The effect size for the chi-square test.
allocationRatioPlanned: Allocation ratio for the active treatment
versus control.
rounding: Whether to round up sample size.
The type II error.
The total sample size.
The number of categories of the multinomial response.
The prevalence of each category for the treatment group.
Only need to specify the valued for the first ncats-1 categories.
The prevalence of each category for the control group.
Only need to specify the valued for the first ncats-1 categories.
Allocation ratio for the active treatment versus control. Defaults to 1 for equal randomization.
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 two-arm multinomial response design is used to test whether the prevalence of each category differs between two treatment arms. Let \(\pi_{gi}\) denote the prevalence of category \(i\) in group \(g\), where \(g=1\) for the treatment group and \(g=2\) for the control group. The chi-square test statistic is given by $$X^2 = \sum_{g=1}^{2} \sum_{i=1}^{C} \frac{(n_{gi} - n_{g+} n_{+i}/n)^2}{n_{g+} n_{+i}/n}$$ 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 both groups, and \(n\) is the total sample size.
Under the null hypothesis, \(X^2\) follows a chi-square distribution with \(C-1\) degrees of freedom.
Under the alternative hypothesis, \(X^2\) follows a non-central chi-square distribution with non-centrality parameter $$\lambda = n r (1-r) \sum_{i=1}^{C} \frac{(\pi_{1i} - \pi_{2i})^2} {r \pi_{1i} + (1-r)\pi_{2i}}$$ where \(r\) is the randomization probability for the active treatment.
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 <- getDesignTwoMultinom(
beta = 0.1, ncats = 3, pi1 = c(0.3, 0.35),
pi2 = c(0.2, 0.3), alpha = 0.05))
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