paired.exact.test: Unconditional Exact Tests for 2x2 Tables with Paired Samples

In Exact: Unconditional Exact Test

Unconditional Exact Tests for 2x2 Tables with Paired Samples

Description

Calculates unconditional exact test for paired samples

Usage

paired.exact.test(data, alternative = c("two.sided", "less", "greater"), npNumbers = 100,
    np.interval = FALSE, beta = 0.001,
    method = c("uam", "ucm", "uamcc", "csm"),
    tsmethod = c("square", "central"),
    conf.int = FALSE, conf.level = 0.95, to.plot = TRUE,
    ref.pvalue = TRUE, delta = 0,
    reject.alpha = NULL, useStoredCSM = TRUE)

Arguments

data	A two dimensional contingency table in matrix form
alternative	Indicates the alternative hypothesis: must be either "two.sided", "less", or "greater"
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 the more extreme tables: must be either "UAM", "UCM", "UAMCC", or "CSM"
tsmethod	A character string describing the method to implement two-sided tests.
conf.int	Logical: Indicates if a confidence interval on the difference in proportion should be computed
conf.level	Number: Confidence level of interval on difference in proportion. Only used if conf.int=TRUE
to.plot	Logical: Indicates if plot of p-value vs. nuisance parameter should be generated
ref.pvalue	Logical: Indicates if p-value should be refined by maximizing the p-value function after the nuisance parameter is selected
delta	Number: null hypothesis of the difference in proportion
reject.alpha	Number: significance level for exact test. Optional and primarily used to speed up paired.reject.region function
useStoredCSM	Logical: uses stored CSM ordering matrix. Only used if method="csm"

Details

This function performs unconditional exact tests to compare two paired proportions. The null hypothesis is the difference of two paired proportions is equal to 0. Under the null hypothesis, the probability of a 2x2 table is a trinomial distribution. The p-value is calculated by maximizing a nuisance parameter and summing the as or more extreme tables. The method parameter specifies the method to determine the more extreme tables (see references for more details):

• UAM (Unconditional Asymptotic McNemar) - Uses McNemar's Z-statistic

• UCM (Unconditional Conditional McNemar) - Uses McNemar's conditional p-value

• UAMCC (Unconditional Asyptotic McNemar with Continuity Correction) - Uses McNemar's Z-statistic with Continuity Correction

• CSM - Starts with the most extreme table and sequentially adds more extreme tables based on the smallest p-value (calculated by maximizing the probability of a 2x2 table). This is extending Barnard's original method to test two paired proportions

There is little research comparing two paired proportions. The author of this R package recommends using the CSM (Convexity, Symmetry, and Minimization) test as this test is often the most powerful, but is much more computationally intensive.

Once the more extreme tables are determined, the p-value is calculated by maximizing over the common discordant probability – a nuisance parameter. The p-value computation has many local maxima and can be computationally intensive. The code performs an exhaustive search by considering many values of the nuisance parameter from 0 to 0.5, represented by npNumbers. If ref.pvalue = TRUE, then the code will also use the optimise function near the nuisance parameter to refine the p-value. Increasing npNumbers and using ref.pvalue ensures the p-value is correctly calculated at the expense of slightly more computation time.

Another approach, proposed by Berger and Sidik, is to calculate the Clopper-Pearson confidence interval of the nuisance parameter (represented by np.interval) and only maximize the p-value function for nuisance parameters within the confidence interval; this approach adds a small penalty to the p-value to control for the type 1 error rate (cannot be used with CSM). If the CSM test is too computationally intensive, the author of this R package generally recommends using the UAM test with confidence interval approach.

The tests can also be implemented for non-inferiority hypotheses by changing the delta. A confidence interval for the difference in proportion can be constructed by determining the smallest delta such that all greater deltas are significant (essentially the delta that crosses from non-significant to significant, but since the p-value is non-monotonic as a function of delta, this code takes the supremum). We note the "UAM" method uses a delta-projected Z-statistic, while "UCM" method ignores the delta and uses the same ordering procedure.

For two-sided tests, the code will either sum the probabilities for both sides of the table if tsmethod = "square" (Default) or construct a one-sided test and double the p-value if tsmethod = "central". The two methods give the same results when delta is zero. For a non-zero delta, the "square" procedure is generally more powerful and conventional, although there are some advantages with using the "central" procedure. Mainly, UCM test cannot order tables (i.e., use McNemar's conditional p-value) for a two-sided alternative with non-zero delta. Thus, to calculate a two-sided confidence interval with UCM test, one has to resort to using the "central" approach. For other tests, there is an equivalent statistic based on delta, and the two-sided p-value is determined by either the Agresti-Min interval (tsmethod = "square") or Chan-Zhang interval (tsmethod = "central").

The CSM test is computationally intensive due to iteratively maximizing the p-value calculation to order the tables. The CSM ordering matrix has been stored for total sample sizes less than or equal to 200. Thus, using the useStoredCSM = TRUE can greatly improve computation time. However, the stored ordering matrix was computed with npNumbers=100 and it is possible that the ordering matrix was not optimal for larger npNumbers. Increasing npNumbers and setting useStoredCSM = FALSE ensures the p-value is correctly calculated at the expense of significantly greater computation time. The stored ordering matrix is not used in the calculation of confidence intervals or non-inferiority tests, so CSM can still be very computationally intensive.

Value

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

p.value	The p-value of the test
statistic	The observed test statistic to determine more extreme tables
estimate	An estimate of the difference in proportion
null.value	The difference in proportion under the null
conf.int	A confidence interval for the difference in proportion. Only present if conf.int = TRUE
alternative	A character string describing the alternative hypothesis
method	A character string describing the method to determine more extreme tables
tsmethod	A character string describing the method to implement two-sided tests
np	The nuisance parameter that maximizes the p-value
np.range	The range of nuisance parameters considered
data.name	A character string giving the names of the data
parameter	The sample sizes

Warning

CSM tests with confidence intervals may take a very long time, even for small sample sizes.

Note

CSM test is much more computationally intensive. Increasing the number of nuisance parameters considered and refining the p-value will increase the computation time, but more likely to ensure accurate calculations. Performing confidence intervals also greatly increases computation time.

Author(s)

Peter Calhoun

References

Berger, R.L. and Sidik, K. (2003) Exact unconditional tests for 2 x 2 matched-pairs design. Statistical Methods in Medical Research, 12, 91–108

Hsueh, H., Liu, J., and Chen, J.J. (2001) Unconditional exact tests for equivalence or noninferiority for paired binary endpoints. Biometrics, 57, 478–483

See Also

mcnemar.test and exact2x2

Examples

data <- matrix(c(3,6,1,0), 2, 2,
               dimnames=list(c("Population 1 Success", "Population 1 Failure"),
                             c("Population 2 Success", "Population 2 Failure")))
paired.exact.test(data, method="UAM", alternative="less")

## Not run: 
# Ensure that the ExactData R package is available before running the CSM test.
if (requireNamespace("ExactData", quietly = TRUE)) {
paired.exact.test(data, method="CSM", alternative="less")
}

## End(Not run)

Exact documentation built on Sept. 11, 2024, 6:17 p.m.