# exact2x2: Exact Conditional Tests for 2 by 2 Tables of Count Data In exact2x2: Exact Tests and Confidence Intervals for 2x2 Tables

## Description

Performs exact conditional tests for two by two tables. For independent binary responses, performs either Fisher's exact test or Blaker's exact test for testing hypotheses about the odds ratio. The commands follow the style of `fisher.test`, the difference is that for two-sided tests there are three methods for calculating the exact test, and for each of the three methods its matching confidence interval is returned (see details). For paired binary data resulting in a two by two table, performs an exact McNemar's test.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```exact2x2(x, y = NULL, or = 1, alternative = "two.sided", tsmethod = NULL, conf.int = TRUE, conf.level = 0.95, tol = 0.00001, conditional = TRUE, paired=FALSE, plot=FALSE, midp=FALSE) fisher.exact(x, y = NULL, or = 1, alternative = "two.sided", tsmethod = "minlike", conf.int = TRUE, conf.level = 0.95, tol = 0.00001, midp=FALSE) blaker.exact(x, y = NULL, or = 1, alternative = "two.sided", conf.int = TRUE, conf.level = 0.95, tol = 0.00001) mcnemar.exact(x,y=NULL, conf.level=.95) ```

## Arguments

 `x` either a two-dimensional contingency table in matrix form, or a factor object. `y` a factor object; ignored if `x` is a matrix. `or` the hypothesized odds ratio. Must be a single numeric. `alternative` indicates the alternative hypothesis and must be one of `"two.sided"`, `"greater"` or `"less"`. if "two.sided" uses method defined by tsmethod. `tsmethod` one of "minlike","central", or "blaker". NULL defaults to "minlike" when paired=FALSE and "central" when paired=TRUE or midp=TRUE. Defines type of two-sided method (see details). Ignored if alternative="less" or "greater". `conf.int` logical indicating if a confidence interval should be computed. `conf.level` confidence level for the returned confidence interval. Only used if `conf.int = TRUE`. `tol` tolerance for confidence interval estimation. `conditional` TRUE. Unconditional exact tests should use `uncondExact2x2`. `paired` logical. TRUE gives exact McNemar's test, FALSE are all other tests `midp` logical. TRUE gives mid p-values and mid-p CIs. Not supported for tsmethod='minlike' or 'blaker' `plot` logical. TRUE gives basic plot of point null odds ratios by p-values, for greater plot control use `exact2x2Plot`. Not supported for midp=TRUE.

## Details

The motivation for this package is to match the different two-sided conditional exact tests for 2x2 tables with the appropriate confidence intervals.

There are three ways to calculate the two-sided conditional exact tests, motivated by three different ways to define the p-value. The usual two-sided Fisher's exact test defines the p-value as the sum of probability of tables with smaller likelihood than the observed table (`tsmethod`="minlike"). The central Fisher's exact test defines the p-value as twice the one-sided p-values (but with a maximum p-value of 1). Blaker's (2000) exact test defines the p-value as the sum of the tail probibility in the observed tail plus the largest tail probability in the opposite tail that is not greater than the observed tail probability.

In `fisher.test` the p-value uses the two-sample method associated with `tsmethod`="minlike", but the confidence interval method associated with `tsmethod`="central". The probability that the lower central confidence limit is less than the true odds ratio is bounded by `1-(1-conf.level)/2` for the central intervals, but not for the other two two-sided methods. The confidence intervals in for `exact2x2` match the test associated with alternative. In other words, the confidence interval is the smallest interval that contains the confidence set that is the inversion of the associated test (see Fay, 2010). The functions `fisher.exact` and `blaker.exact` are just wrappers for certain options in `exact2x2`.

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.

P-values are obtained directly using the (central or non-central) hypergeometric distribution.

The null of conditional independence is equivalent to the hypothesis that the odds ratio equals one. ‘Exact’ inference can be based on observing that in general, given all marginal totals fixed, the first element of the contingency table has a non-central hypergeometric distribution with non-centrality parameter given by the odds ratio (Fisher, 1935). The alternative for a one-sided test is based on the odds ratio, so `alternative = "greater"` is a test of the odds ratio being bigger than `or`.

When paired=TRUE, this denotes there is some pairing of the data. For example, instead of Group A and Group B, we may have pretest and posttest binary responses. The proper two-sided test for such a setup is McNemar's Test, which only uses the off-diagonal elements of the 2x2 table, and tests that both are equal or not. The exact version is based on the binomial distribution on one of the off-diagonal values conditioned on the total of both off-diagonal values. We use `binom.exact` from the `exactci` package, and convert the p estimates and confidence intervals (see note) to odds ratios (see Breslow and Day, 1980, p. 165). The function `mcnemar.exact` is just a wrapper to call `exact2x2` with `paired=TRUE, alternative="two.sided",tsmethod="central"`. One-sided exact McNemar-type tests may be calculated using the `exact2x2` function with `paired=TRUE`. For details of McNemar-type tests see Fay (2010, R Journal).

The mid p-value is an adjusted p-value to account for discreteness. The mid-p adjustment is not guaranteed to give type I error rates that are less than or equal to nominal levels, but gives p-values that lead to the probability of rejection that is sometimes less than the nominal level and sometimes greater than the nominal level. This adjustment is sometimes used because exact p-values for discrete data cannot give actual type I error rates equal to the nominal value unless randomization is done (and that is not typically done because two researchers doing the same method could get different answers). Essentially, exact p-values lead to the probability of rejecting being less than the nominal level for most parameter values in the null hypothesis in order to make sure that it is not greater than the nominal level for ANY parameter values in the null hypothesis. The mid p-value was studied by Lancaster (1961), and for the 2x2 case by Hirji et al (1991).

## Value

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

 `p.value` the p-value of the test `conf.int` a confidence interval for the odds ratio `estimate` an estimate of the odds ratio. Note that the conditional Maximum Likelihood Estimate (MLE) rather than the unconditional MLE (the sample odds ratio) is used. `null.value` the odds ratio under the null, `or`. `alternative` a character string describing the alternative hypothesis `method` a character string, changes depending on alternative and tsmethod `data.name` a character string giving the names of the data

## Note

The default exact confidence intervals for the odds ratio when paired=TRUE (those matching the exact McNemar's test) are transformations of the Clopper-Pearson exact confidence intervals for a single binomial parameter which are central intervals. See note for `binom.exact` for discussion of exact binomial confidence intervals.

Michael Fay

## References

Blaker, H. (2000) Confidence curves and improved exact confidence intervals for discrete distributions. Canadian Journal of Statistics 28: 783-798.

Breslow, NE and Day NE (1980). Staistical Methods in Cancer Research: Vol 1-The analysis of Case-Control Studies. IARC Scientific Publications. IARC, Lyon.

Fay, M. P. (2010). Confidence intervals that Match Fisher's exact and Blaker's exact tests. Biostatistics, 11: 373-374 (go to doc directory for earlier version or https://www.niaid.nih.gov/about/brb-staff-fay for link to official version).

Fay M.P. (2010). Two-sided Exact Tests and Matching Confidence Intervals for Discrete Data. R Journal 2(1):53-58.

Fisher, R.A. (1935) The logic of inductive inference. Journal of the Royal Statistical Society Series A 98:39-54.

Hirji, K.F., Tan, S-J, and Elashoff, R.M. (1991). A quasi-exact test for comparing two binomial proportions. Statistics in Medicine 10: 1137-1153.

Lancaster, H.O. (1961). Significance tests in discrete distributions. JASA 56: 223-234.

`fisher.test` or `mcnemar.test`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20``` ```## In example 1, notice how fisher.test rejects the null at the 5 percent level, ## but the 95 percent confidence interval on the odds ratio contains 1 ## The intervals do not match the p-value. ## In fisher.exact you get p-values and the matching confidence intervals example1<-matrix(c(6,12,12,5),2,2,dimnames=list(c("Group A","Group B"),c("Event","No Event"))) example1 fisher.test(example1) fisher.exact(example1,tsmethod="minlike") fisher.exact(example1,tsmethod="central") blaker.exact(example1) ## In example 2, this same thing happens, for ## tsmethod="minlike"... this cannot be avoided because ## of the holes in the confidence set. ## example2<-matrix(c(7,255,30,464),2,2,dimnames=list(c("Group A","Group B"),c("Event","No Event"))) example2 fisher.test(example2) exact2x2(example2,tsmethod="minlike") ## you can never get a test-CI inconsistency when tsmethod="central" exact2x2(example2,tsmethod="central") ```

### Example output

```Loading required package: exactci
Event No Event
Group A     6       12
Group B    12        5

Fisher's Exact Test for Count Data

data:  example1
p-value = 0.04371
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
0.03888003 1.05649145
sample estimates:
odds ratio
0.2189021

Two-sided Fisher's Exact Test (usual method using minimum likelihood)

data:  example1
p-value = 0.04371
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
0.0435 0.9170
sample estimates:
odds ratio
0.2189021

Central Fisher's Exact Test

data:  example1
p-value = 0.06059
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
0.03888003 1.05649145
sample estimates:
odds ratio
0.2189021

Blaker's Exact Test

data:  example1
p-value = 0.04371
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
0.0423 0.9170
sample estimates:
odds ratio
0.2189021

Event No Event
Group A     7       30
Group B   255      464

Fisher's Exact Test for Count Data

data:  example2
p-value = 0.04996
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
0.1553144 1.0055843
sample estimates:
odds ratio
0.4249891

Two-sided Fisher's Exact Test (usual method using minimum likelihood)

data:  example2
p-value = 0.04996
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
0.1773 1.0138
sample estimates:
odds ratio
0.4249891

Central Fisher's Exact Test

data:  example2
p-value = 0.0518
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
0.1553144 1.0055843
sample estimates:
odds ratio
0.4249891
```

exact2x2 documentation built on Dec. 11, 2021, 9:43 a.m.