Nonparametric statistics

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Nonparametric statistics
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Nonparametric?

• What are they? Tests that dont assume data is
  normal, or t-distributed, or ?2, or ...
• so they dont 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)

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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 1s.

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(Side issue)

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

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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(W7.5) is not lt 5, so
  even with 1-sided test, do not reject H0.
• Apparently Excel doesnt know Wilcoxon test

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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