This function analyzes the frequency with which values get
repeated within a set of numbers. Unlike Benford's law, and its
generalizations, this approach examines the entire number at once, not only
the first or last digit(s).

```
repeated_test(
x,
check = c("last", "lasttwo", "all"),
method = c("af", "entropy"),
samples = 2000
)
```

## Arguments

- x
a numeric vector of values from which the digits should be
analyzed.

- check
which digits to shuffle during the procedure. Can be
`last`

or `lasttwo`

.

- method
which statistics is used. Defaults to `af`

for average
frequency, but can also be `entropy`

for entropy.

- samples
how many samples to use in the bootstraping procedure.

## Value

An object of class `jfaRv`

containing:

- x
input data.

- frequencies
frequencies of observations in `x`

.

- samples
vector of simulated samples.

- integers
counts for extracted integers.

- decimals
counts for extracted decimals.

- n
the number of observations in `x`

.

- statistic
the value the average frequency or entropy statistic.

- p.value
the p-value for the test.

- cor.test
correlation test for the integer portions of the number
versus the decimals portions of the number.

- method
method used.

- check
checked digits.

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

## Details

To determine whether the data show an excessive amount of bunching,
the null hypothesis that `x`

does not contain an unexpected amount of
repeated values is tested against the alternative hypothesis that `x`

has more repeated values than expected. The statistic can either be the
average frequency (\(AF = sum(f_i^2)/sum(f_i))\) of the data or the
entropy (\(E = - sum(p_i * log(p_i))\), with \(p_i=f_i/n\)) of the
data. Average frequency and entropy are highly correlated, but the average
frequency is often more interpretable. For example, an average frequency of
2.5 means that, on average, your observations contain a value that appears
2.5 times in the data set.To quantify what is expected, this test requires
the assumption that the integer portions of the numbers are not associated
with their decimal portions.

## References

Simohnsohn, U. (2019, May 25). Number-Bunching: A New Tool for
Forensic Data Analysis. Retrieved from https://datacolada.org/77.

## Examples

```
set.seed(1)
x <- rnorm(50)
# Repeated values analysis shuffling last digit
repeated_test(x, check = "last", method = "af", samples = 2000)
#>
#> Classical Repeated Values Test
#>
#> data: x
#> n = 50, AF = 1, p-value = 1
#> alternative hypothesis: average frequency in data is greater than for random data.
```