`brunner_munzel.Rd`

This is a generic function that performs a generalized asymptotic Brunner-Munzel test in a fashion similar to t.test.

```
brunner_munzel(
x,
...,
paired = FALSE,
alternative = c("two.sided", "less", "greater"),
mu = 0.5,
alpha = 0.05,
perm = FALSE,
max_n_perm = 10000
)
# S3 method for default
brunner_munzel(
x,
y,
paired = FALSE,
alternative = c("two.sided", "less", "greater"),
mu = 0.5,
alpha = 0.05,
perm = FALSE,
max_n_perm = 10000,
...
)
# S3 method for formula
brunner_munzel(formula, data, subset, na.action, ...)
```

- x
a (non-empty) numeric vector of data values.

- ...
further arguments to be passed to or from methods.

- paired
a logical indicating whether you want a paired test.

- alternative
a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.

- mu
a number specifying an optional parameter used to form the null hypothesis (Default = 0.5). This can be thought of as the null in terms of the relative effect, p = P (X < Y ) + 0.5 * P (X = Y); See ‘Details’.

- alpha
alpha level (default = 0.05)

- perm
a logical indicating whether or not to perform a permutation test over approximate t-distribution based test (default is FALSE). Highly recommend to set perm = TRUE when sample size per condition is less than 15.

- max_n_perm
the maximum number of permutations (default is 10000).

- y
an optional (non-empty) numeric vector of data values.

- formula
a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.

- data
an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).

- subset
an optional vector specifying a subset of observations to be used.

- na.action
a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

A list with class `"htest"`

containing the following components:

"statistic": the value of the test statistic.

"parameter": the degrees of freedom for the test statistic.

"p.value": the p-value for the test.

"conf.int": a confidence interval for the relative effect appropriate to the specified alternative hypothesis.

"estimate": the estimated relative effect.

"null.value": the specified hypothesized value of the relative effect.

"stderr": the standard error of the relative effect.

"alternative": a character string describing the alternative hypothesis.

"method": a character string indicating what type of test was performed.

"data.name": a character string giving the name(s) of the data.

This function is made to provide a test of stochastic equality between two samples (paired or independent), and is referred to as the Brunner-Munzel test.

This tests the hypothesis that the relative effect, discussed below, is equal to the null value (default is mu = 0.5).

The estimate of the relative effect, which can be considered as value similar to the probability of superiority, refers to the following:

$$\hat p = p(X<Y) + \frac{1}{2} \cdot P(X=Y)$$

Note, for paired samples, this does *not* refer to the probability of an increase/decrease in paired sample but rather the probability that a randomly sampled value of X.
This is also referred to as the "relative" effect in the literature. Therefore, the results will differ from the concordance probability provided by the ses_calc function.

The brunner_munzel function is based on the npar.t.test and npar.t.test.paired functions within the nparcomp package (Konietschke et al. 2015).

Brunner, E., Munzel, U. (2000). The Nonparametric Behrens-Fisher Problem: Asymptotic Theory and a Small Sample Approximation. Biometrical Journal 42, 17 -25.

Neubert, K., Brunner, E., (2006). A Studentized Permutation Test for the Nonparametric Behrens-Fisher Problem. Computational Statistics and Data Analysis.

Munzel, U., Brunner, E. (2002). An Exact Paired Rank Test. Biometrical Journal 44, 584-593.

Konietschke, F., Placzek, M., Schaarschmidt, F., & Hothorn, L. A. (2015). nparcomp: an R software package for nonparametric multiple comparisons and simultaneous confidence intervals. Journal of Statistical Software 64 (2015), Nr. 9, 64(9), 1-17. http://www.jstatsoft.org/v64/i09/

Other Robust tests:
`boot_log_TOST()`

,
`boot_t_TOST()`

,
`boot_t_test()`

,
`log_TOST()`

,
`wilcox_TOST()`

```
data(mtcars)
brunner_munzel(mpg ~ am, data = mtcars)
#> Sample size in at least one group is small. Permutation test (perm = TRUE) is highly recommended.
#>
#> two-sample Brunner-Munzel test
#>
#> data: mpg by am
#> t = -4.2653, df = 20.893, p-value = 0.0003479
#> alternative hypothesis: true relative effect is not equal to 0.5
#> 95 percent confidence interval:
#> 0.009114802 0.330966169
#> sample estimates:
#> p(X>Y) + .5*P(X=Y)
#> 0.1700405
#>
```