Test if Same Population

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] 7.061153
## 
## $`Upper Bound Effect`
## [1] 7.445108
## 
## $`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.

wilcox.test(mpg ~ am, data=mtcars) 

## 
##  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

Test if means are Equal

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] 0.007338401
## 
## $`Upper Bound Effect`
## [1] 0.1260941
## 
## $`Robust Certainty Estimate`
## [1] 0.9184417
## 
## $`Lower 95% CI`
## [1] 0.7799747
## 
## $`Upper 95% CI`
## [1] 0.9712966

t.test(x,y)

## 
##  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

Test if means are Unequal

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] 0.9648731
## 
## $`Upper Bound Effect`
## [1] 1.083629
## 
## $`Robust Certainty Estimate`
## [1] 0.1769465
## 
## $`Lower 95% CI`
## [1] 0.1569815
## 
## $`Upper 95% CI`
## [1] 0.205275

t.test(x,y)

## 
##  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.

Medians

In order to test medians instead of means, simply set both means.only = TRUE and medians = TRUE in NNS.ANOVA().

NNS.ANOVA(control = x, treatment = y,
          means.only = TRUE, medians = TRUE, robust = TRUE, plot = TRUE)

## $`Control Median`
## [1] 0.009209639
## 
## $`Treatment Median`
## [1] 1.054852
## 
## $`Grand Median`
## [1] 0.532031
## 
## $`Control CDF`
## [1] 0.704
## 
## $`Treatment CDF`
## [1] 0.305
## 
## $Certainty
## [1] 0.3497634
## 
## $`Lower Bound Effect`
## [1] 0.9592677
## 
## $`Upper Bound Effect`
## [1] 1.126243
## 
## $`Robust Certainty Estimate`
## [1] 0.3346173
## 
## $`Lower 95% CI`
## [1] 0.2991483
## 
## $`Upper 95% CI`
## [1] 0.3765887

Stochastic Dominant Efficient Sets

NNS also offers the ability to isolate a set of variables that do not have any dominated constituents with the NNS.SD.efficient.set() function.

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.

set.seed(123)
x1 = rnorm(1000)
x2 = x1 + 1
x3 = rnorm(1000)
x4 = x3 + 1
x5 = rnorm(1000)
x6 = x5 + 1
x7 = rnorm(1000)
x8 = x7 + 1

NNS.SD.efficient.set(cbind(x1, x2, x3, x4, x5, x6, x7, x8), degree = 1, status = FALSE)
[1] "x4" "x2" "x8" "x6"