09 Nonparametric Statistics by yaofenji

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```									NON-PARAMETRIC
STATISTICS

Hairul Hafiz Mahsol
Institute for Tropical Biology &
Conservation
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
assumptions.
 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
instances.
Parametric or Nonparametric
Tests?
 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
techniques.
   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
sizes).
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
prefer.
The Wilcoxon Signed Rank
Test
 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.
lawyers”).
   The Mann-Whitney U Test is a commonly
used alternative non-parametric test, which is
extremely similar to the Wilcoxon Rank Sum
Test.
   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
above).
   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
same.
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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