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Power and Sample Size for Difference in Two-Sample Multinomial Responses

Source: R/getDesignProportions.R
getDesignTwoMultinom.Rd

Obtains the power given sample size or obtains the sample size given power for difference in two-sample multinomial responses.

Usage

getDesignTwoMultinom(
  beta = NA_real_,
  n = NA_real_,
  ncats = NA_integer_,
  pi1 = NA_real_,
  pi2 = NA_real_,
  allocationRatioPlanned = 1,
  rounding = TRUE,
  alpha = 0.05
)

Arguments

beta

The type II error.

n

The total sample size.

ncats

The number of categories of the multinomial response.

pi1

The prevalence of each category for the treatment group. Only need to specify the valued for the first ncats-1 categories.

pi2

The prevalence of each category for the control group. Only need to specify the valued for the first ncats-1 categories.

allocationRatioPlanned

Allocation ratio for the active treatment versus control. Defaults to 1 for equal randomization.

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 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.

Details

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 \(\alpha\).

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


(design1 <- getDesignTwoMultinom(
  beta = 0.1, ncats = 3, pi1 = c(0.3, 0.35),
  pi2 = c(0.2, 0.3), alpha = 0.05))
#>   alpha     power   n ncats effectsize
#> 1  0.05 0.9000174 503     3 0.02515837