BarnardTest: Barnard's Unconditional Test

Description

Barnard's unconditional test for superiority applied to $2 \times 2$ contingency tables using Score or Wald statistics for the difference between two binomial proportions.

Usage

BarnardTest(x, y = NULL, dp = 0.001, pooled = TRUE, alternative = "two.sided")

Arguments

either a two-dimensional contingency table in matrix form, or a factor object.

a factor object; ignored if x is a matrix.

The resolution of the search space for the nuisance parameter

pooled

Z statistic with pooled (Score) or unpooled (Wald) variance

alternative

indicates the alternative hypothesis and must be one of "two.sided", "greater" or "less". You can specify just the initial letter.

Value

A list with class "htest" containing the following components:
p.valuethe p-value of the test.
estimatean estimate of the nuisance parameter where the p-value is maximized.
alternativea character string describing the alternative hypothesis.
methodthe character string "Barnards Unconditional 2x2-test".
data.namea character string giving the names of the data.
statistic.tableThe contingency tables considered in the analysis represented by 'n1' and 'n2', their scores, and whether they are included in the one-sided (1), two-sided (2) tests, or not included at all (0)
nuisance.matrixNuisance parameters, $p$, and the corresponding p-values for both one- and two-sided tests

Details

If x is a matrix, it is taken as a two-dimensional contingency table, and hence its entries should be nonnegative integers. Otherwise, both x and y must be vectors of the same length. Incomplete cases are removed, the vectors are coerced into factor objects, and the contingency table is computed from these. For a 2x2 contingency table, such as $X=[n_1,n_2;n_3,n_4]$, the normalized difference in proportions between the two categories, given in each column, can be written with pooled variance (Score statistic) as $$T(X)=\frac{\hat{p}_2-\hat{p}_1}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{c_1}+\frac{1}{c_2})}},$$ where $\hat{p}=(n_1+n_3)/(n_1+n_2+n_3+n_4)$, $\hat{p}_2=n_2/(n_2+n_4)$, $\hat{p}_1=n_1/(n_1+n_3)$, $c_1=n_1+n_3$ and $c_2=n_2+n_4$. Alternatively, with unpooled variance (Wald statistic), the difference in proportions can we written as $$T(X)=\frac{\hat{p}_2-\hat{p}_1}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{c_1}+\frac{\hat{p}_2(1-\hat{p}_2)}{c_2}}}.$$ The probability of observing $X$ is $$P(X)=\frac{c_1!c_2!}{n_1!n_2!n_3!n_4!}p^{n_1+n_2}(1-p)^{n_3+n_4},$$ where $p$ is the unknown nuisance parameter. Barnard's test considers all tables with category sizes $c_1$ and $c_2$ for a given $p$. The p-value is the sum of probabilities of the tables having a score in the rejection region, e.g. having significantly large difference in proportions for a two-sided test. The p-value of the test is the maximum p-value calculated over all $p$ between 0 and 1.

References

Barnard, G.A. (1945) A new test for 2x2 tables. Nature, 156:177. Barnard, G.A. (1947) Significance tests for 2x2 tables. Biometrika, 34:123-138. Suissa, S. and Shuster, J. J. (1985), Exact Unconditional Sample Sizes for the 2x2 Binomial Trial, Journal of the Royal Statistical Society, Ser. A, 148, 317-327. Cardillo G. (2009) MyBarnard: a very compact routine for Barnard's exact test on 2x2 matrix. http://www.mathworks.com/matlabcentral/fileexchange/25760 Galili T. (2010) http://www.r-statistics.com/2010/02/barnards-exact-test-a-powerful-alternative-for-fishers-exact-test-implemented-in-r/ Lin C.Y., Yang M.C. (2009) Improved p-value tests for comparing two independent binomial proportions. Communications in Statistics-Simulation and Computation, 38(1):78-91. Trujillo-Ortiz, A., R. Hernandez-Walls, A. Castro-Perez, L. Rodriguez-Cardozo N.A. Ramos-Delgado and R. Garcia-Sanchez. (2004). Barnardextest:Barnard's Exact Probability Test. A MATLAB file. [WWW document]. http://www.mathworks.com/

Examples

Run this code

tab <- as.table(matrix(c(8, 14, 1, 3), nrow=2,
                dimnames=list(treat=c("I","II"), out=c("I","II"))))
BarnardTest(tab)

# Plotting the search for the nuisance parameter for a one-sided test
bt <- BarnardTest(tab)
plot(bt$nuisance.matrix[, 1:2],
     t="l", xlab="nuisance parameter", ylab="p-value")

# Plotting the tables included in the p-value
ttab <- as.table(matrix(c(40, 14, 10, 30), nrow=2,
                 dimnames=list(treat=c("I","II"), out=c("I","II"))))

bt <- BarnardTest(ttab)
bts <- bt$statistic.table
plot(bts[, 1], bts[, 2],
     col=hsv(bts[, 4] / 4, 1, 1),
     t="p", xlab="n1", ylab="n2")

# Plotting the difference between pooled and unpooled tests
bts <- BarnardTest(ttab, pooled=TRUE)$statistic.table
btw <- BarnardTest(ttab, pooled=FALSE)$statistic.table
plot(bts[, 1], bts[, 2],
     col=c("black", "white")[1 + as.numeric(bts[, 4]==btw[, 4])],
     t="p", xlab="n1", ylab="n2")

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