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Nonparametric tests

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Point estimate for ETA1-ETA2 is -4.500. 96.3 Percent CI for ETA1-ETA2 is (-10.998,1.999) ... Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.0662 ... – PowerPoint PPT presentation

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Title: Nonparametric tests


1
Non-parametric tests
  • Parametric tests
  • are based on the assumption that data are from a
    normal distribution
  • test for differences in mean values
  • Non-parametric tests
  • make no assumptions about the distribution of the
    data
  • test for differences in median values

2
Parametric tests
  • are generally more powerful than non-parametric
    tests.
  • are better at detecting differences from the null
    hypothesis
  • but only if the assumption of normality is valid.

3
If data is not normal
  • Transform the data so that the transformed data
    is normal e.g. use logs.
  • Parametric tests can then be used on the
    transformed data.
  • OR
  • Use non-parametric tests

4
xp
5
Non-parametric tests
  • Do not rely on assumptions of normality
  • Can be used on ranked data
  • Can be used on censored data
  • If two or more populations are being compared, it
    is necessary to assume the same distributions for
    each of the populations.

6
tests
7
Sign test
  • Counts of mayfly data were recorded in 12 small
    streams at low flow before and after industrial
    outfalls.
  • Does the presence of the outfall make a
    difference to the quality of the water and hence
    the number of mayfly recorded?

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11
Sign test
  • H0 median difference is 0
  • H1 median difference is greater than 0
  • 1 negative sign
  • 11 positive signs
  • 0 zeros

12
p-value
  • Probability of getting 11 or more positive signs
    out of 12 if the null hypothesis is true
  • If H0 is true then the probability of obtaining a
    positive sign is 0.5
  • P-value is calculated using the binomial
    distribution with n 12, p 0.5
  • From table 3
  • p-value 0.00020.00290.0031

13
StatgtNonparametricsgt1-sample sign
14
Minitab
  • Sign test of median 0.0 versus gt 0.0
  • N Below Equal Above P
    Median
  • diff 12 1 0 11 0.0032
    31.50

15
Note
  • The sign test takes no account of the size of the
    difference.
  • It is useful in certain situations where no
    quantitative measure is used e.g.
  • Better or worse
  • Preference for A or B
  • Test is based on non-zero differences only.

16
Tests based on ranks
  • Most other nonparametric tests are based on
    ranked data. e.g.
  • Data 10 14 15 15 25
  • Rank 1 2 3.5 3.5 5

17
Wilcoxon Test
  • The test is then based in the sum of the ranks.
  • Sum of ranks for negative signs is 6
  • Sum of ranks for positive signs is 72

18
Wilcoxon test
  • H0 median difference is 0
  • H1 median difference is greater than 0
  • If the null hypothesis is true then the sum of
    ranks for positive differences should be about
    the same as the sum of ranks for negative
    differences.

19
Minitab
  • Wilcoxon Signed Rank Test
  • Test of median 0.0 versus median gt 0.0
  • N for Wilcoxon
    Estimated
  • N Test Statistic P
    Median
  • difference 12 12 72.0 0.005
    32.00

20
Mann-Whitney Test
  • Also known as the Wilcoxon two-sample test
  • Nonparametric equivalent of the two-sample t-test
  • Sample sizes in the two groups do not need to be
    the same.

21
Example
  • Bacteria counts per unit volume for two types of
    cultures are measured as follows
  • Culture 1 27 31 26 25
  • Culture 2 32 29 35 28 36
  •  
  • Do these results present sufficient evidence that
    the cultures come from different populations?

22
Ranking
  • Obs 25 26 27 28 29 31
    32 35 36
  • Culture 1 1 1 2 2 1 2
    2 2
  • Rank 1 2 3 4 5 6 7
    8 9
  • The sum of ranks are
  • culture 1 1 2 3 6 12
  • culture 2 4 5 7 8 9 33

23
Hypotheses
  • H0 median for culture 1 is the same as for
    culture 2.
  • H1 the medians are not the same.
  •  If the null hypothesis is true then the sums of
    ranks for each sample should be proportional to
    the sample sizes.

24
Minitab output
  • Mann-Whitney Test and CI culture 1, culture 2
  •  
  • culture N 4 Median 25.500
  • culture N 5 Median 32.000
  • Point estimate for ETA1-ETA2 is -4.500
  • 96.3 Percent CI for ETA1-ETA2 is (-10.998,1.999)
  • W 12.0
  • Test of ETA1 ETA2 vs ETA1 not ETA2 is
    significant at 0.0662
  • The test is significant at 0.0651 (adjusted for
    ties)
  •  
  • Cannot reject at alpha 0.05
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