09 Nonparametric Statistics by yaofenji



Hairul Hafiz Mahsol
Institute for Tropical Biology &
School of Science & Technology
   In addition to the standard sampling
    assumptions that
    – 1) a population has been drawn by
      random sampling and that
    – 2) the observations are independent,
   standard parametric statistical
    techniques have several additional
 Alternative parameteric methods are
  available for most statistical tests when
  the assumption of equal variances of
  comparison groups is not met.
 In addition, when the dependent
  variable is not normally or near-
  normally distributed, the Central
  Limit Theorem supports waiving this
  requirement if the sample size is
  sufficiently large.
 Another assumption is that the interval
  between each unit of the dependent
  variable has approximately equal value
  (ie., the distance between 1 and 2 is
  comparable to the difference between
  100 and 101).
 Violations of this assumption can
  sometimes be dealt with by
  transforming the dependent variable
  and can be tested (in part) during
  diagnostic testing.
   Although deciding when using
    parametric statistical approaches (e.g.,
    t-testing, ANOVA) can be complex or
    controversial, the following rules of
    thumb will serve you well in most
Parametric or Nonparametric
 If all assumptions are met, use
  Parametric techniques.
 If the dependent variable is a rank
  order, the distribution is bimodal or
  otherwise clearly does not represent
  near-equal intervals between it’s values
  (e.g., the distance between 1 and 2 is
  unlikely to be similar to the distance
  between 4 and 5), do not use
  parametric methods.
   If the dependent variable is reasonably
    symmetric (ie, Pearson kurtosis
    between two and four [or Fisher
    kurtosis between –1 and +1], and skew
    is between –1 and +1) and n >25, you
    are generally safe using parametric
   If the dependent variable is heavily
    skewed, use parametric statistics only
    when your sample is large (n > 50-100)
    and you may wish to consider
    transforming the dependent variable or
    use non parametric techniques anyway.
What Are Non-parametric Tests?
  Non-Parametric statistical tests have
   much less restrictive assumptions
   concerning the distributions of the
   variables and the variances of
   comparison groups.
  Indeed these techniques tend to rely
   upon the rank of the individual
   observations rather than their absolute
   numeric values.
   The computations are also generally
    easier to understand.

   However, they have the following

    – 1. You lose the metric and numerical
      values of the results (since only the ranks,
      not the numerical values are used)
– 2. when the sample size is small, you can
  lose considerable statistical power
  compared to using Parametric statistics
  (although the statistical power of
  parametric and non-parametric testing
  tends to be very similar at larger sample
          The Sign Test
   The sign test is used for paired data (such
    as in a pre-post study).
   The sign test is a simple test in which you
    count up the number of cases for which the
    second value is greater than the first value
    (positive signs) and you see whether the
    number of positives are greater than you
    would have expected by chance alone.
   You could count up negative signs if you
The Wilcoxon Signed Rank
 This test is also used for paired data
  and is analogous to the Parametric test,
  the paired t-Test.
 The Wilcoxon Signed Test is similar to
  the Sign Test but also ranks results by
  the magnitude of the difference of the
  paired values.
 It tests for the median difference
  between pairs being zero.
The Wilcoxon Rank Sum Test
   This test is analogous to the Parametric test,
    the Student t-test.
   It is used in instances where you have
    independent data for which you want to
    compare data for two different groups. (eg.
    “comparing the income of doctors vs.
   The Mann-Whitney U Test is a commonly
    used alternative non-parametric test, which is
    extremely similar to the Wilcoxon Rank Sum
   It tests for the median of the two groups
    being the same.
    The Kruskal-Wallis Test
   This test is analogous to the Parametric test;
    one-way ANOVA, and is just a slight variation
    on the Wilcoxon Rank Sum Test (shown
   This is used in instances where you want to
    compare results for more than two groups
    (e.g. “compare the income of doctors vs.
    lawyers vs. MBA’s”).
   It tests for the median of all groups being the
Parametric Test   Analogous Non-Parametric test

Student t-test    Wilcoxon Rank Sum Test or Mann-Whitney U Test

ANOVA             Kruskal-Wallis Test

Paired t-test     Wilcoxon Signed Rank Test or the Sign Test

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