Nonparametric statistics

1. Nonparametric statistics

2. Nonparametric? • What are they? Tests that don’t assume data is normal, or t-distributed, or χ2, or ... • so they don’t involve the “parameters” like mean and std dev of population distribution • Why bother? Can handle small samples without assuming something about the original data distribution. • (Recall normal distribution comes from averages of large samples)

3. Wilcoxon rank sum test • Two categories; want to know whether one ranks significantly differently from (higher or lower than) the other. • so it can handle ordinal data • Suppose data from n samples, each in one of the two categories, is ranked 1-st, 2-nd, 3-rd, ..., n-th. • H0: Categories are equally distributed among ranks • so if k of data are in cat 1, then expect rank sum k(avg(1,2,...n)) of cat 1’s.

4. (Side issue) • Text starts with, instead of rank sum for category 1’s, difference between avg ranks of cat 1’s and cat 2’s. • This is equivalent, because sum of all ranks is constant, so from sum of cat 1’s ranks, we can figure out avg ranks of cat 1’s and cat 2’s, and hence difference.

5. Wilcoxon example • Are foreign cars cheaper to run than domestic? Annual costs for seven cars are listed: (foreign, domestic) • 2000, 2200, 2200, 2400, 2700, 3200, 7000 • ranks:1,2.5,4 (2.5,5,6,7) • rank sum W (of foreign): 7.5 • From software: Assuming all choices of 3 ranks out of 7 are foreign, P(W≤7.5) is not < 5%, so even with 1-sided test, do not reject H0. • Apparently Excel doesn’t know Wilcoxon test

7. Remarks on example • Includes how to handle rank ties • Shows how outlier (7000), which would throw off means and std devs, becomes just one datum with high rank