A two-sample test based on the Cramer-Von Mises test statistic (`cvm_stat`

).

## Usage

```
cvm_test(a, b, nboots = 2000, p = default.p, keep.boots = T, keep.samples = F)
cvm_stat(a, b, power = def_power)
```

## Arguments

- a
a vector of numbers (or factors -- see details)

- b
a vector of numbers

- nboots
Number of bootstrap iterations

- p
power to raise test stat to

- keep.boots
Should the bootstrap values be saved in the output?

- keep.samples
Should the samples be saved in the output?

- power
power to raise test stat to

## Value

Output is a length 2 Vector with test stat and p-value in that order. That vector has 3 attributes -- the sample sizes of each sample, and the number of bootstraps performed for the pvalue.

## Details

The CVM test compares two ECDFs by looking at the sum of the squared differences between them -- evaluated at each point in the joint sample. Formally -- if E is the ECDF of sample 1 and F is the ECDF of sample 2, then $$CVM = \sum_{x\in k}|E(x)-F(x)|^p$$ where k is the joint sample. The test p-value is calculated by randomly resampling two samples of the same size using the combined sample. Intuitively the CVM test improves on KS by using the full joint sample, rather than just the maximum distance -- this gives it greater power against shifts in higher moments, like variance changes.

In the example plot below, the CVM statistic is the sum of the heights of the vertical black lines.

Inputs `a`

and `b`

can also be vectors of ordered (or unordered) factors, so long as both have the same levels and orderings. When possible, ordering factors will substantially increase power.

## Functions

`cvm_test`

: Permutation based two sample Cramer-Von Mises test`cvm_stat`

: Permutation based two sample Cramer-Von Mises test

## See also

`dts_test()`

for a more powerful test statistic. See `ks_test()`

or `kuiper_test()`

for the predecessors to this test statistic. See `wass_test()`

and `ad_test()`

for the successors to this test statistic.

## Examples

```
set.seed(314159)
vec1 = rnorm(20)
vec2 = rnorm(20,0.5)
out = cvm_test(vec1,vec2)
out
#> Test Stat P-Value
#> 3.180 0.009
summary(out)
#> CVM Test
#> =========================
#> Test Statistic: 3.18
#> P-Value: 0.009 *
#> - - - - - - - - - - - - -
#> n1 n2 n.boots
#> 20 20 2000
#> =========================
#> Test stat rejection threshold for alpha = 0.05 is: 2.0105
#> Null rejected: samples are from different distributions
plot(out)
# Example using ordered factors
vec1 = factor(LETTERS[1:5],levels = LETTERS,ordered = TRUE)
vec2 = factor(LETTERS[c(1,2,2,2,4)],levels = LETTERS, ordered=TRUE)
cvm_test(vec1,vec2)
#> Test Stat P-Value
#> 0.760 0.524
```