Power Calculations for 2x2 Tables with Independent Samples

Calculates the power of the design for known sample sizes and true probabilities.

power.exact.test(p1, p2, n1, n2, alternative = c("two.sided", "less", "greater"),
    alpha = 0.05, npNumbers = 100, np.interval = FALSE, beta = 0.001,
    method = c("z-pooled", "z-unpooled", "boschloo", "santner and snell", "csm",
               "fisher", "pearson chisq", "yates chisq"),
    tsmethod = c("square", "central"), simulation = FALSE, nsim = 100,
    delta = 0, convexity = TRUE, useStoredCSM = TRUE)

Arguments

p1: The probability of success given in first group
p2: The probability of success given in second group
n1: The sample size in first group
n2: The sample size in second group
alternative: Indicates the alternative hypothesis: must be either "two.sided", "less", or "greater"
alpha: Significance level
npNumbers: Number: The number of nuisance parameters considered
np.interval: Logical: Indicates if a confidence interval on the nuisance parameter should be computed
beta: Number: Confidence level for constructing the interval of nuisance parameters considered. Only used if np.interval=TRUE
method: Indicates the method for finding more extreme tables: must be either "Z-pooled", "Z-unpooled", "Santner and Snell", "Boschloo", "CSM", "Fisher", "Pearson Chisq", or "Yates Chisq"
tsmethod: Indicates two-sided method: must be either "square" or "central"
simulation: Logical: Indicates if the power calculation is exact or estimated by simulation
nsim: Number of simulations run. Only used if simulation=TRUE
delta: Number: null hypothesis of the difference in proportion
convexity: Logical: assumes convexity for interval approach. Only used if np.interval=TRUE
useStoredCSM: Logical: uses stored CSM ordering matrix. Only used if method="csm"

Details

The power calculations are for binomial models with independent samples. The design must know the fixed sample sizes in advance. There are (n1+1) x (n2+1) possible tables that could be produced. There are two ways to calculate the power: simulate the tables under two independent binomial distributions or determine the rejection region for all possible tables and calculate the exact power. The calculations can be done using any exact.test computation, Fisher's exact test, or chi-square tests (Yates' or Pearson's; note: these are not exact tests). The power calculations utilize the convexity property, which greatly speeds up computation time (see exact.reject.region documentation).

Value

A list with class "power.htest" containing the following components:

n1, n2: The respective sample sizes
p1, p2: The respective success probabilities
alpha: Significance level
power: Power of the test
alternative: A character string describing the alternative hypothesis
delta: Null hypothesis of the difference in proportion
method: A character string describing the method to determine more extreme tables

References

Berger, R. (1994) Power comparison of exact unconditional tests for comparing two binomial proportions. Institute of Statistics Mimeo Series No. 2266

Berger, R. (1996) More powerful tests from confidence interval p values. American Statistician, 50, 314-318

Boschloo, R. D. (1970), Raised Conditional Level of Significance for the 2x2-table when Testing the Equality of Two Probabilities. Statistica Neerlandica, 24, 1-35

Author

Peter Calhoun

Note

Pearson's and Yates' chi-square tests are not exact tests, so the function name may be a misnomer. These tests may have inflated type 1 error rates. These options were added to compute the power efficiently when using asymptotic tests.

Examples

# Superiority power #
power.exact.test(p1=0.15, p2=0.60, n1=15, n2=30, method="Z-pooled")
#> 
#>      Z-pooled Exact Test 
#> 
#>          n1, n2 = 15, 30
#>          p1, p2 = 0.15, 0.60
#>           alpha = 0.05
#>           power = 0.8512212
#>     alternative = two.sided
#>           delta = 0
#> 
power.exact.test(p1=0.15, p2=0.60, n1=15, n2=30, method="Fisher")
#> 
#>      Fisher's Exact Test 
#> 
#>          n1, n2 = 15, 30
#>          p1, p2 = 0.15, 0.60
#>           alpha = 0.05
#>           power = 0.8049468
#>     alternative = two.sided
#>           delta = 0
#> 
power.exact.test(p1=0.15, p2=0.60, n1=15, n2=30, method="Boschloo",
                 np.interval=TRUE, beta=0.001)
#> 
#>      Boschloo's Exact Test with Interval Approach 
#> 
#>          n1, n2 = 15, 30
#>          p1, p2 = 0.15, 0.60
#>           alpha = 0.05
#>           power = 0.8588912
#>     alternative = two.sided
#>           delta = 0
#> 
if (FALSE) { # \dontrun{
# Ensure that the ExactData R package is available before running the CSM test.
if (requireNamespace("ExactData", quietly = TRUE)) {
power.exact.test(p1=0.15, p2=0.60, n1=15, n2=30, method="CSM")
}
} # }

# Non-inferiority power #
power.exact.test(p1=0.30, p2=0.30, n1=65, n2=65, method="Z-pooled",
                 delta=0.2, alternative="less")
#> 
#>      Z-pooled Exact Test 
#> 
#>          n1, n2 = 65, 65
#>          p1, p2 = 0.3, 0.3
#>           alpha = 0.05
#>           power = 0.8005406
#>     alternative = less
#>           delta = 0.2
#>