This week, we'll be covering non-parametric (or assumption-free) statistical tests (brief overview). Parametric techniques, which include the correlation r and the t-test, refer to the use of sample statistics to estimate population parameters (e.g., rho, mu).* Thus far, we've come across a number of assumptions that technically are required to be met for doing parametric analyses, although in practice there's some leeway in meeting the assumptions.
Assumptions for parametric analyses are as follows (for further information, see here):
o Data for a given variable are normally distributed in the population.
o Equal-interval measurement.
o Random sampling is used.
o Homogeneity of variance between groups (for t-test).
One would generally opt for a non-parametric test when there's violation of one or more of the above assumptions and sample size is small. According to King, Rosopa, and Minium (2011), "...the problem of violation of assumptions is of great concern when sample size is small (< 25)" (p. 382). In other words, if assumptions are violated but sample size is large, you still may be able to use parametric techniques (example). The reason is something called the Central Limit Theory. I've annotated the following screenshot from Sabina's Stats Corner to show what the CLT does.
It is important first to distinguish between a frequency plot of raw data (which appears in the top row of Sabina's diagram) and something else known as a sampling distribution. A sampling distribution is what you get when you draw repeated random samples from a full population of individual persons (sometimes known as a "parent" population) and plot the means of all the samples you have drawn. Under the CLT, a frequency plot of these means (the sampling distribution) tends toward normality regardless of the parent shape. Further, the sampling distribution increasingly resembles a bell-curve shape as the size of the multiple samples increases. In essence, what the CLT does for us is get us back to normal distributions when the original parent population distribution is non-normal.
We'll be doing a neat demonstration with dice that conveys the role of large samples in salvaging data from the normal-distribution assumption, under the CLT.
Now that we've established that non-parametric statistics typically are used when one or more assumptions of parametrics statistics are violated and sample size is small, we can ask: What are some actual non-parametric statistical techniques?
To a large extent, the different non-parametric techniques represent analogues to parametric techniques. For example, the non-parametric Mann-Whitney U test (song below) is analogous to the parametric t-test, when comparing data from two independent groups, and the non-parametric Wilcoxon signed-ranks test is analogous to a repeated-measures t-test. This PowerPoint slideshow demonstrates these two non-parametric techniques corresponding to t-tests. As you'll see, non-parametric statistics operate on ranks (e.g., who has the highest score, the second highest, etc.) rather than original scores, which may have outliers or other problems.
The parametric Pearson correlation has the non-parametric analogue of a Spearman rank-order correlation. Let's work out an example involving the 2024-25 Miami Heat, a National Basketball Association team suggested by one of the students. The roster of one team gives us a small sample size of players, on whom we will correlate their annual salary with their career performance on a metric called Win Shares (i.e., how many wins are attributed to each player based on his points scored, rebounds, assists, etc.). Here are our data (as of November 1, 2024):
These data clearly have some outliers. Jimmy Butler is the highest-paid Heat player at $48.8 million per year and he is credited statistically with personally contributing 115.5 wins to his teams in his 14-year career. Other, younger players have smaller salaries (although still huge in layperson terms) and have had many fewer wins attributed to them. The ordinary Pearson correlation and Spearman rank-order correlations are shown at the bottom of this posting,** if you'd like to calculate them in suspense. Which do you think would be larger and why?
Finally, we'll close with our song...
Mann-Whitney U
Lyrics by Alan Reifman
(May be sung to the tune of “Suzie Q.,” Hawkins/Lewis, covered by John Fogerty)
Mann-Whitney U,
When your groups are two,
If your scaling’s suspect, and your cases are few,
Mann-Whitney U,
The cases are laid out,
Converted to rank scores,
You then add these up, done within each group,
Mann-Whitney U,
(Instrumental)
There is a formula,
That uses the summed ranks,
A distribution’s what you, compare the answer to,
Mann-Whitney U
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*According to King, Rosopa, and Minium (2011), "Many people call chi-square a nonparametric test, but it does in fact assume the central limit theorem..." (p. 382).
**The Pearson correlation is r = .57 (p = .03), whereas the Spearman correlation is rs = .40 (nonsignificant due to the small sample size).
