Obtains the power given sample size or obtains the sample size given power for the chi-square test for unordered multi-sample binomial response.
getDesignUnorderedBinom(
beta = NA_real_,
n = NA_real_,
ngroups = NA_integer_,
pi = NA_real_,
allocationRatioPlanned = NA_integer_,
rounding = TRUE,
alpha = 0.05
)An S3 class designUnorderedBinom 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.
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 response probabilities for the treatment groups.
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 binomial response design is used to test whether the response probabilities differ among multiple treatment arms. Let \(\pi_{g}\) denote the response probability in group \(g = 1,\ldots,G\), where \(G\) is the total number of treatment groups.
The chi-square test statistic is given by $$X^2 = \sum_{g=1}^{G} \sum_{i=1}^{2} \frac{(n_{gi} - n_{g+}n_{+i}/n)^2}{n_{g+} n_{+i}/n}$$ 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 across all groups as $$\bar{\pi} = \sum_{g=1}^{G} r_g \pi_g$$
Under the null hypothesis, \(X^2\) follows a chi-square distribution with \(G-1\) degrees of freedom.
Under the alternative hypothesis, \(X^2\) follows a non-central chi-square distribution with non-centrality parameter $$\lambda = n \sum_{g=1}^{G} \frac{r_g (\pi_{g} - \bar{\pi})^2} {\bar{\pi} (1-\bar{\pi})}$$
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 <- getDesignUnorderedBinom(
beta = 0.1, ngroups = 3, pi = c(0.1, 0.25, 0.5), alpha = 0.05))
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