when your data don’t meet the assumptions

• If your data meet (or approximate) assumptions of parametrics, they are generally more powerful

• Monte-Carlo techniques are also often more powerful than non-parametrics

• However, non-parametrics simpler to use than MC

• Non-parametrics are known as rank-order tests, because they work by ranking observations and analyzing these ranks, rather than the data themselves.

• To use non-parametrics with continuous values, you have to discard a lot of information.

• We will talk about non-parametrics in relation to their parametric equivalents.

• Non-parametric regression techniques exist but are not commonly used.

• There are, however, several non-parametric correlation techniques that are widely used.

X and Y values are ranked separately, and the Pearson’s product-moment coefficient (\(r\)) is computed on these ranks

x <- c(0.9, 6.8, 3.2, 2.4, 1.2, 1.1)
y <- c(0.1, 4.5, 5.4, 1.5, 1.9, 4.1)

rank_x <- rank(x)
rank_y <- rank(y)
rank_x

## [1] 1 6 5 4 3 2

rank_y

## [1] 1 5 6 2 3 4

cor(x, y, method = "pearson")

## [1] 0.5590485

cor(x, y, method = "spearman")

## [1] 0.7142857

cor(rank_x, rank_y, method = "pearson")

## [1] 0.7142857

Alternative to Spearman….

• Rank observations
• Examine each pair of observations, determine whether they match or not
• Compute \(\tau\)

• \[\tau = \frac{(number\ of\ matched\ pairs) - (number\ of \ non\ matched\ pairs)}{\frac{1}{2}n(n-1)}\]

• Note: the denominator is the total number of pairwise comparisons.

cor(var1, var2, method="kendall")

## [1] 0.8222222

Mann-Whitney U, also known as the Wilcoxon Rank-Sum

• rank observations, ignoring group
• sum the ranks belonging to each group
• calculate the test statistic

• \[U = R - \frac{n(n+1)}{2}\]

• \(R\) is the summed ranks, and \(n\) is the group sample size
• do this for both groups, and take the smallest as the test statistic
• compare to known distribution under null hypothesis

x <- rnorm(10, mean=5)
y <- rnorm(10, mean=7)
wilcox.test(x,y)

## 
##  Wilcoxon rank sum exact test
## 
## data:  x and y
## W = 9, p-value = 0.00105
## alternative hypothesis: true location shift is not equal to 0

Kruskal-Wallis

• Rank all observations
• Calculate the average rank within each group
• Compare the average rank within group the the overall average of ranks, using a weighted sum-of-squares technique
• Compare p value of test statistic using chi-square approximation

kruskal.test(var~group)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  var by group
## Kruskal-Wallis chi-squared = 2.2946, df = 1, p-value = 0.1298

Kolmogorov-Smirnov Test

Non-parametric test to determine whether two distributions differ

• The single largest deviation of the empirical from the theoretical is the KS statistic. This is used to compute a p-value.

• Can be used for any distribution, not just the normal distribution.

ks.test(rnorm(100), "punif")

## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  rnorm(100)
## D = 0.42597, p-value = 3.469e-16
## alternative hypothesis: two-sided

ks.test(rnorm(100)^2, "pnorm")

## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  rnorm(100)^2
## D = 0.50003, p-value < 2.2e-16
## alternative hypothesis: two-sided

ks.test(runif(100), rnorm(100))

## 
##  Asymptotic two-sample Kolmogorov-Smirnov test
## 
## data:  runif(100) and rnorm(100)
## D = 0.5, p-value = 2.778e-11
## alternative hypothesis: two-sided