PHIL ROWE Statistics Lecture 1 Data Presentation

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• Lecture 24
• Non-parametric methods

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Parametric v Non-parametric tests
Another solution to non-normality is to use an
alternative test that does not require a normal
distribution. Non-parametric tests. Do the same
job as the equivalent parametric test, but much
less fussy about the data.
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Non-parametric alternatives
Parametric
Non-parametric Two sample t-test
Mann-Whitney Paired t-test
Wilcoxon Paired Samples One way
analysis of variance Kruskal-Wallis Pearson
correlation Spearman rank
correlation
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Use of a non-parametric test with non-normal data
Re-analyse the data from the cimetidine/
immuno-suppressant trial using the Mann-Whitney
test in place of a two sample t-test. Direct use
of data. No log transformation.
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Outline of Mann-Whitney test
Cimet Rank Control Rank 5.13 8
4.96 7 11.22 20 5.99
10 13.34 22 6.29 12 37.13 29
3.19 5 6.79 14 9.77
18 7.22 17 7.20 16 5.65 9
13.88 23 13.05 21 3.13
4 7.14 15 2.43 3 3.65 6
6.51 13 10.10 19 31.83
28 39.05 30 16.25 25 15.78 24
1.82 2 20.32 26 1.74
1 22.85 27 6.11 11 287 178
Compare rank totals Fairly similar
Non-significant Very different Significant
Highest
Lowest
Rank totals
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Ranking
All the non-parametric methods use this process
of converting the data to rankings. Then all
further calculations are based on the ranks,
instead of the initial data
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Performing the Mann-Whitney test
Follow the menus Stat Nonparametrics Mann-Whitn
ey ...
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Columns containing the data
Mann-Whitney test has no requirement for normal
distribution or equal SDs, so use the
untransformed data.
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Effect of cimetidine Mann-Whitney test Minitab
output
Mann-Whitney Confidence Interval and
Test Cimetidi N 15 Median
11.22 Placebo N 15 Median
6.11 Point estimate for ETA1-ETA2 is
4.79 95.4 Percent CI for ETA1-ETA2 is
(0.52,10.21) W 287.0 Test of ETA1 ETA2 vs
ETA1 not ETA2 is significant at 0.0251
Effectively the P value
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Conclusion
Two sample t-test (applied to log transformed
data) and this non-parametric test both agree
that there is a significant effect of cimetidine.
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Ordinal scale data
Data that ranks without providing a precise
measure. e.g. Result of therapy 1 Feel much
better 2 Feel a bit better 3 Feel about the
same 4 Feel a bit worse 5 Feel much worse
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Ordinal data
Ordinal data commonly severely non-normal and
cant be transformed to normality, so probably
use a non-parametric method.
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Ordinal data -an example
Comparison of medicine flavours Four different
formulations of a liquid medicine are to be
assessed for acceptability of flavour. Ten
subjects rank the medicines as 1 (Worst) to 4
(Best). A comparison of 4 averages - normally use
a one way analysis of variance. This data is
ordinal, so use non-parametric equivalent -
Kruskal-Wallis test.
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Rankings of the flavours of different formulations
Formulation no. 1 2 3
4 2 1 4 3 4 1 2 3 2 1 3 4 2 3 4 1 4 2 3 1 4 2
3 1 2 1 4 3 2 1 3 4 1 2 4 3 3 1 4 2
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Entering data for a Kruskal-Wallis test
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Performing the Kruskal-Wallis test
Follow the menus Stat Nonparametrics Kruskal-Wa
llis ...
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Results
Codes for formulations
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Kruskal-Wallis test Minitab output
Kruskal-Wallis Test Kruskal-Wallis Test on Rank
Formuln N Median Ave Rank
Z 1 10 2.000 21.5 0.31 2
10 1.000 10.5 -3.12 3
10 3.500 29.5 2.81 4
10 3.000 20.5 0.00 Overall 40
20.5 H 13.32 DF 3 P
0.004 H 14.20 DF 3 P 0.003 (adjusted for
ties)
Significant
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P values Adjusted for ties
When ranking the initial data there may be some
ties (equal values). P value tends to be
increased if ties are present. Should we correct
for that bias??? (Controversial) For all
non-parametric methods, my recommendation is to
use the Corrected P value, when one is produced.
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Conclusion
There is significant evidence that some of the
formulations are preferred to others. (P
0.003) Based on the median rankings Formulation
2 is very unpopular and Formulation 3 is the
favourite.
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Use parametric or non-parametric test?
Parametric methods are slightly more powerful
than the non-parametric equivalents, if the data
is appropriate. Aim is generally to try to use a
parametric method if possible.
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Use parametric or non-parametric test?
Parametric test
Severe deviation from requirements for parametric
test?
No
Interval
Yes
Can transform data to meet requirements ?
Yes
No
Non-parametric test
Ordinal
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Terms with which you should be familiar

• Mann-Whitney test
• Wilcoxon paired samples test
• Kruskal-Wallis test
• Spearman rank correlation
• Ranking

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What you should be able to do

• Recognise that non-parametric tests are very
  robust and can be used for almost any data set.
• Select a non-parametric equivalent for various
  parametric tests
• Use Minitab (etc) to carry out non-parametric
  tests
• Recognise ordinal scale data
• Select whether to use a parametric or a
  non-parametric method.