This function carries out an hypothesis test in which the null hypothesis is that the two samples are governed by the same underlying generative probability distribution against the alternative hypothesis that they are governed by two different generative probability distributions.

```
two_sample_test(
x,
y,
stats = list(stat_t),
B = 1000L,
M = NULL,
alternative = "two_tail",
combine_with = "tippett",
type = "exact",
seed = NULL,
...
)
```

- x
A numeric vector or a numeric matrix or a list representing the 1st sample. Alternatively, it can be a distance matrix stored as an object of class

`dist`

, in which case test statistics based on inter-point distances (marked with the`_ip`

suffix) should be used.- y
A numeric vector if

`x`

is a numeric vector, or a numeric matrix if`x`

is a numeric matrix, or a list if`x`

is a list, representing the second sample. Alternatively, if`x`

is an object of class`dist`

, it should be a numeric scalar specifying the size of the first sample.- stats
A list of functions produced by

`as_function`

specifying the chosen test statistic(s). A number of test statistic functions are implemented in the package and can be used as such. Alternatively, one can provide its own implementation of test statistics that (s)he deems relevant for the problem at hand. See the section*User-supplied statistic function*for more information on how these user-supplied functions should be structured for compatibility with the**flipr**framework. Default is`list(stat_t)`

.- B
The number of sampled permutations. Default is

`1000L`

.- M
The total number of possible permutations. Defaults to

`NULL`

, which means that it is automatically computed from the given sample size(s).- alternative
A single string or a character vector specifying whether the p-value is right-tailed, left-tailed or two-tailed. Choices are

`"right_tail"`

,`"left_tail"`

and`"two_tail"`

. Default is`"two_tail"`

. If a single string is provided, it is assumed that it should be applied to all test statistics provided by the user. Alternative, the length of`alternative`

should match the length of the`stats`

parameter and it is assumed that there is a one-to-one correspondence.- combine_with
A string specifying the combining function to be used to compute the single test statistic value from the set of p-value estimates obtained during the non-parametric combination testing procedure. For now, choices are either

`"tippett"`

or`"fisher"`

. Default is`"tippett"`

, which picks Tippett's function.- type
A string specifying which formula should be used to compute the p-value. Choices are

`exact`

(default),`upper_bound`

and`estimate`

. See Phipson & Smith (2010) for details.- seed
An integer specifying the seed of the random generator useful for result reproducibility or method comparisons. Default is

`NULL`

.- ...
Extra parameters specific to some statistics.

A `list`

with three components: the value of the
statistic for the original two samples, the p-value of the resulting
permutation test and a numeric vector storing the values of the permuted
statistics.

A user-specified function should have at least two arguments:

the first argument is

`data`

which should be a list of the`n1 + n2`

concatenated observations with the original`n1`

observations from the first sample on top and the original`n2`

observations from the second sample below;the second argument is

`perm_data`

which should be an integer vector giving the indices in`data`

that are considered to belong to the first sample.

It is possible to use the `use_stat`

function with `nsamples = 2`

to have **flipr** automatically generate a template file for writing down
your own test statistics in a way that makes it compatible with the **flipr**
framework.

See the `stat_t`

function for an example.

```
n <- 10L
mx <- 0
sigma <- 1
# Two different models for the two populations
x <- rnorm(n = n, mean = mx, sd = sigma)
delta <- 10
my <- mx + delta
y <- rnorm(n = n, mean = my, sd = sigma)
t1 <- two_sample_test(x, y)
t1$pvalue
#> [1] 0
# Same model for the two populations
x <- rnorm(n = n, mean = mx, sd = sigma)
delta <- 0
my <- mx + delta
y <- rnorm(n = n, mean = my, sd = sigma)
t2 <- two_sample_test(x, y)
t2$pvalue
#> [1] 0.4335637
```