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] 0.9555556
```

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 test
data: x and y
W = 3, p-value = 7.578e-05
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 = 1.0566, df = 1, p-value = 0.304
```

Kolmogorov-Smirnov Test

Non-parametric test to determine whether two distributions differ