Obtains the power given sample size or obtains the sample size given power for the Cochran-Armitage trend test for ordered multi-sample binomial response.
getDesignOrderedBinom(
beta = NA_real_,
n = NA_real_,
ngroups = NA_integer_,
pi = NA_real_,
w = NA_real_,
allocationRatioPlanned = NA_integer_,
rounding = TRUE,
alpha = 0.05
)An S3 class designOrderedBinom 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.
w: The scores assigned to the treatment groups.
trendstat: The Cochran-Armitage trend test statistic.
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.
The scores assigned to the treatment groups. This should reflect the ordinal nature of the treatment groups, e.g. dose levels. Defaults to equally spaced scores.
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
An ordered multi-sample binomial response design is used to test whether the response probabilities differ across multiple treatment groups. The null hypothesis is that the response probabilities are equal across all treatment groups, while the alternative hypothesis is that the response probabilities are ordered, i.e. the response probability increases with the treatment group index. The Cochran-Armitage trend test is used to test this hypothesis. This test effectively regresses the response probabilities against treatment group scores, and test whether the slope of the regression line is significantly different from zero.
The trend parameter is defined as $$\theta = \sum_{g=1}^{G} r_g (w_g - \bar{w}) \pi_g$$ where \(G\) is the number of treatment groups, \(r_g\) is the randomization probability for treatment group \(g\), \(w_g\) is the score assigned to treatment group \(g\), \(\pi_g\) is the response probability for treatment group \(g\), and \(\bar{w} = \sum_{g=1}^{G} r_g w_g\) is the weighted average score across all treatment groups.
Since \(\hat{\theta}\) is a linear combination of the estimated response probabilities, its variance is given by $$Var(\hat{\theta}) = \frac{1}{n}\sum_{g=1}^{G} r_g (w_g - \bar{w})^2 \pi_g(1-\pi_g)$$ where \(n\) is the total sample size.
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 <- getDesignOrderedBinom(
beta = 0.1, ngroups = 3, pi = c(0.1, 0.25, 0.5), alpha = 0.05))
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