** NNS** offers a multitude of ways to test
if distributions came from the same population, or if they share the
same mean. The underlying function for these tests is

`NNS.ANOVA()`

The output from ** NNS.ANOVA()** is a

`Certainty`

statistic, which compares CDFs of distributions
from several shared quantiles and normalizes the similarity of these
points to be within the interval \([0,1]\), with 1 representing identical
distributions. For a complete analysis of `Certainty`

to
common p-values and the role of power, please see the References.Below we run the analysis to whether automatic transmissions and
manual transmissions have significantly different `mpg`

distributions per the `mtcars`

dataset.

The plot on the left shows the robust `Certainty`

estimate, reflecting the distribution of `Certainty`

estimates over 100 random permutations of both variables. The plot on
the right illustrates the control and treatment variables, along with
the grand mean among variables, and the confidence interval associated
with the control mean.

```
mpg_auto_trans = mtcars[mtcars$am==1, "mpg"]
mpg_man_trans = mtcars[mtcars$am==0, "mpg"]
NNS.ANOVA(control = mpg_man_trans, treatment = mpg_auto_trans, robust = TRUE)
```

```
## $`Control Mean`
## [1] 17.14737
##
## $`Treatment Mean`
## [1] 24.39231
##
## $`Grand Mean`
## [1] 20.76984
##
## $`Control CDF`
## [1] 0.9152794
##
## $`Treatment CDF`
## [1] 0.1670107
##
## $Certainty
## [1] 0.01009348
##
## $`Lower Bound Effect`
## [1] 5.502308
##
## $`Upper Bound Effect`
## [1] 9.035466
##
## $`Robust Certainty Estimate`
## [1] 0.0108829
##
## $`Lower 95% CI`
## [1] 0.002574409
##
## $`Upper 95% CI`
## [1] 0.104652
```

The `Certainty`

shows that these two distributions clearly
do not come from the same population. This is verified with the
Mann-Whitney-Wilcoxon test, which also does not assume a normality to
the underlying data as a nonparametric test of identical
distributions.

```
##
## Wilcoxon rank sum test with continuity correction
##
## data: mpg by am
## W = 42, p-value = 0.001871
## alternative hypothesis: true location shift is not equal to 0
```

Here we provide the output from
** NNS.ANOVA()** and

`t.test()`

functions on two Normal distribution samples, where we are pretty
certain these two means are equal.```
set.seed(123)
x = rnorm(1000, mean = 0, sd = 1)
y = rnorm(1000, mean = 0, sd = 2)
NNS.ANOVA(control = x, treatment = y,
means.only = TRUE, robust = TRUE, plot = TRUE)
```

```
## $`Control Mean`
## [1] 0.01612787
##
## $`Treatment Mean`
## [1] 0.08493051
##
## $`Grand Mean`
## [1] 0.05052919
##
## $`Control CDF`
## [1] 0.5218858
##
## $`Treatment CDF`
## [1] 0.4893919
##
## $Certainty
## [1] 0.912545
##
## $`Lower Bound Effect`
## [1] 0.004184327
##
## $`Upper Bound Effect`
## [1] 0.1277884
##
## $`Robust Certainty Estimate`
## [1] 0.9184417
##
## $`Lower 95% CI`
## [1] 0.7799747
##
## $`Upper 95% CI`
## [1] 0.9712966
```

```
##
## Welch Two Sample t-test
##
## data: x and y
## t = -0.96711, df = 1454.4, p-value = 0.3336
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.20835512 0.07074984
## sample estimates:
## mean of x mean of y
## 0.01612787 0.08493051
```

By altering the mean of the `y`

variable, we can start to
see the sensitivity of the results from the two methods, where both
firmly reject the null hypothesis of identical means.

```
set.seed(123)
x = rnorm(1000, mean = 0, sd = 1)
y = rnorm(1000, mean = 1, sd = 1)
NNS.ANOVA(control = x, treatment = y,
means.only = TRUE, robust = TRUE, plot = TRUE)
```

```
## $`Control Mean`
## [1] 0.01612787
##
## $`Treatment Mean`
## [1] 1.042465
##
## $`Grand Mean`
## [1] 0.5292966
##
## $`Control CDF`
## [1] 0.7862176
##
## $`Treatment CDF`
## [1] 0.2197938
##
## $Certainty
## [1] 0.1824463
##
## $`Lower Bound Effect`
## [1] 0.9617191
##
## $`Upper Bound Effect`
## [1] 1.085323
##
## $`Robust Certainty Estimate`
## [1] 0.1769465
##
## $`Lower 95% CI`
## [1] 0.1569815
##
## $`Upper 95% CI`
## [1] 0.205275
```

```
##
## Welch Two Sample t-test
##
## data: x and y
## t = -22.933, df = 1997.4, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -1.1141064 -0.9385684
## sample estimates:
## mean of x mean of y
## 0.01612787 1.04246525
```

The effect size from ** NNS.ANOVA()** is
calculated from the confidence interval of the control mean and the
specified

`y`

shift of 1 is within the provided lower and
upper effect boundaries.Another method of comparing distributions involves a test for
stochastic dominance. The first, second, and third degree stochastic
dominance tests are available in ** NNS**
via:

`NNS.FSD()`

`NNS.SSD()`

`NNS.TSD()`

`## [1] "Y FSD X"`

** NNS.FSD()** correctly identifies the
shift in the

`y`

variable we specified when testing for
unequal means.** NNS** also offers the ability to isolate
a set of variables that do not have any dominated constituents with the

`NNS.SD.efficient.set()`

`x2, x4, x6, x8`

all dominate their preceding
distributions yet do not dominate one another, and are thus included in
the first degree stochastic dominance efficient set.

If the user is so motivated, detailed arguments and proofs are provided within the following: