Using Statistics To Make Inferences 6

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Using Statistics To Make Inferences 6

• Summary
•  
• Non-parametric tests.
• Wilcoxon Signed Ranks Test.
• Wilcoxon Matched Pairs Signed Ranks Test.
• Wilcoxon Rank Sum Test /
• Mann-Whitney Test.

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Goals

• Perform and interpret Wilcoxon Signed Ranks
  Test.
• Perform and interpret Wilcoxon Matched Pairs
  Signed Ranks Test.
• Perform and interpret Wilcoxon Rank Sum Test /
  Mann-Whitney Test.
• Know when each test is appropriate.

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Practical

• Perform a series of Mann-Whitney tests and
  compare the results to those obtained from
  t-tests.

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Recall

• Which tests might be appropriate when testing the
  mean of normal data?
• What criteria would you employ to select the
  appropriate test?

z or t
For z need s, for t calculate s
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Today

• What if we cannot assume normality?
• We shall develop a more powerful test than the
  sign test for the median (see lecture 2).

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Non-parametric Tests

• A single sample test
•  
• Wilcoxon Signed Ranks Test
•  

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Wilcoxon Signed Ranks Test Procedure

• Take the difference between each observation and
  the median (? - eta).

• Rank the absolute differences from 1 to n,
  allowing for ties. (1 smallest, n largest)

• Sum the rank values for those observations above
  ?, let this be W

• Sum the rank values for those observations below
  ?, let this be W-

• Use the smaller of W or W- to use as the test
  statistic. Call this Wcalc (or simply W).
  Critical values of the test statistic are given
  in tables for various significance levels.

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Wilcoxon Signed Ranks Test Notes

• If two or more differences are equal (tied) they
  are assigned the average of the ranks.

If a difference is zero, it is omitted.
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Example

• It has been established that an individuals
  median reaction time is 0.250 seconds. Twelve
  trials are conducted after the individual has
  consumed alcohol. The measured times are

0.235 0.252 0.312 0.264 0.323 0.241 0.284 0.306
0.248 0.284 0.298 0.320
Test whether the data are consistent with the
median value.
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Step 1

• The first step is to subtract the median (0.25)
  from every value.

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Step 1
Median 0.250 0.235 0.250 -0.015
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Step 2

• Now calculate and rank on the absolute
  differences

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Step 2
Now rank the data
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Step 2
Is the ranking unambiguous?
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Step 2
ltlt Tied Values, so average ranks
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Step 2
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Step 3/4

• Now separate the contributions for positive
  (negative) differences

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Step 3/4
Now sum the contributions for positive and
negative differences
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Step 3/4
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Step 5
W68.5 W-9.5
As a cross check Note that WW- ½ n (n1) Here
n 12 and ½ n (n1) ½ 12 (121) 78 As
expected
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Step 5
W68.5 W-9.5
Therefore the result is significant at the 5
level (14 gt 9.5), so the null hypothesis can be
rejected. The median is apparently not consistent
with 0.250 seconds.
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Normal approximation

• W 9.5 n 12
• Employing the normal approximation,

In this case the continuity correction is added
since a lower tail is being considered.
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Normal approximation

• Z -2.27

The p value is 2 x 0.0115 0.02 remarkably
close to the exact value
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Wilcoxon Matched Pairs Signed Ranks Test -
Example

• Certain mental tasks are performed before and
  after exercise. The scores were recorded.

Is there any evidence of a significant difference
in the levels of performance under the two
conditions?
Still effectively a single sample since we seek a
change.
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Find Differences

Find differences
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Find Absolute Differences

Find absolute differences
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Find Absolute Differences

Now rank the absolute differences
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Rank the Absolute Differences
Now deal with ties
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Rank the Absolute Differences
Now find the true ranks.
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Rank the Absolute Differences
Now separate positive and negative values
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Separate the Contributions for Positive
(Negative) Differences
Now form the totals
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Separate the Contributions for Positive
(Negative) Differences
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Conclusion
W50 W-5
As a cross check Note that WW- ½ n (n1) Here
n 10 and ½ n (n1) ½ 10 (101) 55 As
expected
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Conclusion
W50 W-5
Therefore the result is significant at the 5
level (8 gt 5), so the null hypothesis can be
rejected. The scores appear to differ.
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Aside

• It is claimed that it takes 15 minutes to mark
  an examination question.
• Test this claim using the following timings (x)
  for marking 10 questions.
• 12 15 14 16 12 15 14 16 11 15
• You might find the following sums useful
• Sx 140 and Sx2 1988

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Aside

• What are we testing?
• Which test is appropriate?

µ the population mean
a
t, we dont know s
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Aside
n 10 Sx 140 Sx2 1988
mean
variance
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Aside
n 10 Sx 140 Sx2 1988
C C C C C C C C C C c
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Aside
n 10 µ 15
? n 1 9
Highlighting the values for 95 and 99
confidence for a two tail test.
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Aside

• Since tcalc lt tcrit (1.79 lt 2.262) the
  experiment is consistent with a population mean
  of 15 at 95 confidence.
• This is supported by software which has a 95
  confidence interval that includes 15 and a
  p-value of 0.107 (gt0.05).

C C C C C C C C C C C c
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Aside
C C C C C C C C C C C C c
t value p-value Confidence interval
Confidence interval (15-2.26,15.26)
(12.74,15.26)
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Now for Two Samples

• For normal data we have the two sample t test of
  lecture 4.
• What if data is not normal?

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Wilcoxon Rank Sum Test / Mann-Whitney Test

• A two sample test

• Combine the observations from the two samples
  (sizes n1 and n2).

• Rank the sorted data from 1 to (n1n2).

• Calculate R1, as the sum of the ranks of the
  first sample and R2 for the second.

• Form the Mann-Whitney test statistic.

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Normal Approximation

• If n1 and n2 are greater than 8 a normal
  approximation may be employed where

In this case the continuity correction is added
since a lower tail is being considered. The
approximation is particularly relevant if the
tables are not extensive enough.
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Wilcoxon Rank Sum Test Example

• A study of 22 patients suffering from Parkinsons
  disease was conducted. An operation was performed
  on 8 of them, while it improved their general
  condition it might adversely affect their speech.
  In the data a higher value indicates a greater
  difficulty in speaking.

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Wilcoxon Rank Sum Test Data
Operated 2.6 2.0 1.7 2.7 2.5 2.6 2.5
3.0 Others 1.2 1.8 1.9 2.3 1.3 3.0 2.2 1.3 1.5
1.6 1.3 1.5 2.7 2.0
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Ranked Data
Speech Source Rank 1.2 Others
1 1.3 Others 2   1.3 Others
3   1.3 Others 4   1.5 Others
5   1.5 Others 6  
1.6 Others 7   1.7 Operated 8
1.8 Others 9   1.9 Others
10 2.0 Operated 11 2.0 Others
12   2.2 Others 13  
2.3 Others 14 2.5 Operated 15
2.5 Operated 16 2.6 Operated 17
2.6 Operated 18 2.7 Operated 19
2.7 Others 20  
3.0 Operated 21 3.0 Others 22
 
Note the ties
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Ranked Data
Speech Source Rank 1.2 Others
1   1.3 Others 2  
1.3 Others 3   1.3 Others 4
  1.5 Others 5  
1.5 Others 6   1.6 Others 7
  1.7 Operated 8 1.8 Others
9   1.9 Others 10
2.0 Operated 11 2.0 Others 12  
2.2 Others 13   2.3 Others 14
2.5 Operated 15 2.5 Operated 16
2.6 Operated 17 2.6 Operated 18
2.7 Operated 19 2.7 Others 20
  3.0 Operated 21 3.0 Others
22  
Now find the true ranks
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Ranked Data
Speech Source Rank True Rank 1.2 Others
1 1   1.3 Others 2 3  
1.3 Others 3 3   1.3 Others
4 3   1.5 Others 5 5.5  
1.5 Others 6 5.5   1.6 Others
7 7   1.7 Operated 8 8
1.8 Others 9 9   1.9 Others 10
10 2.0 Operated 11 11.5
2.0 Others 12 11.5   2.2 Others 13
13   2.3 Others 14 14
2.5 Operated 15 15.5 2.5 Operated 16
15.5 2.6 Operated 17 17.5
2.6 Operated 18 17.5 2.7 Operated 19
19.5 2.7 Others 20 19.5  
3.0 Operated 21 21.5 3.0 Others 22
21.5  
Now separate the contributions for each source.
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Ranked Data
Speech Source Rank True Rank Others Operated
1.2 Others 1 1 1  
1.3 Others 2 3 3  
1.3 Others 3 3 3  
1.3 Others 4 3 3  
1.5 Others 5 5.5 5.5  
1.5 Others 6 5.5 5.5  
1.6 Others 7 7 7  
1.7 Operated 8 8   8
1.8 Others 9 9 9  
1.9 Others 10 10 10  
2.0 Operated 11 11.5   11.5
2.0 Others 12 11.5 11.5  
2.2 Others 13 13 13  
2.3 Others 14 14 14  
2.5 Operated 15 15.5   15.5
2.5 Operated 16 15.5   15.5
2.6 Operated 17 17.5   17.5
2.6 Operated 18 17.5   17.5
2.7 Operated 19 19.5   19.5
2.7 Others 20 19.5 19.5  
3.0 Operated 21 21.5   21.5
3.0 Others 22 21.5 21.5  
Now sum the contributions for each source
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Ranked Data
Speech Source Rank True Rank Others Operated
1.2 Others 1 1 1  
1.3 Others 2 3 3  
1.3 Others 3 3 3  
1.3 Others 4 3 3  
1.5 Others 5 5.5 5.5  
1.5 Others 6 5.5 5.5  
1.6 Others 7 7 7  
1.7 Operated 8 8   8
1.8 Others 9 9 9  
1.9 Others 10 10 10  
2.0 Operated 11 11.5   11.5
2.0 Others 12 11.5 11.5  
2.2 Others 13 13 13  
2.3 Others 14 14 14  
2.5 Operated 15 15.5   15.5
2.5 Operated 16 15.5   15.5
2.6 Operated 17 17.5   17.5
2.6 Operated 18 17.5   17.5
2.7 Operated 19 19.5   19.5
2.7 Others 20 19.5 19.5  
3.0 Operated 21 21.5   21.5
3.0 Others 22 21.5 21.5  
Total 126.5 126.5
It is pure chance that the two sums are equal
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Calculations
R1126.5, R2126.5, n1 8, n2 14
Note, only need to evaluate one.
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Conclusion
U1 21.5 U2 90.5
(mid-point ½ n1 n2 56 so only need calculate
one)
For n18, n214, the critical value from the
tables for p0.05 is 26. The result is
significant at the 5 level (26 gt 21.5), the two
samples appear to differ.
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Normal Approximation

• Employing the normal approximation for U1 21.5

The p value is 2 x 0.0102 0.02 remarkably
close to the exact value
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SPSS Verification

• Note that the groups must have numerical
  identifiers
• 1 operated 2 others.
• Analyze gt Nonparametric tests gt Legacy Dialogs gt
• 2 independent samples

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SPSS Verification
Note the need to define group variables (1/2)
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SPSS Verification
The previous rank sums are reproduced.
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SPSS Verification
The previous U and Z values are reproduced.
The p value is less than .05
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Parametric vs Non-Parametric Tests

• Parametric Tests
•  
• They are robust with respect to violations of
  their assumptions.
• They are more powerful - more likely to detect
  an effect when one is present.
• They are more versatile there are tests for
  every experimental design.

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Parametric vs Non-Parametric Tests

• Non-Parametric Tests
•  
• They make fewer assumptions.
• They are ideal for ordinal data, which is common
  in Psychology, where as parametric tests require
  interval or ratio data.

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Light Back Ground Reading
Last year, the BBC ran a six-part primer by
Michael Blastland on understanding statistics in
the news. Blastland takes on the medias handling
of surveys/polls, counting, percentages,
averages, causation and doubt. Wouldn't it be
good, Blastland said, to have the mental
agility to separate the wheat from the chaff? He
then proceeds, in six weekly articles, to point
out the obvious vs. the correct ways to interpret
the data. Topics covered are Surveys Counting Perc
entages Averages Causation Doubt
A Statistical Primer from the BBC
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Read

• Read Howitt and Cramer pages 168-177
• Read Howitt and Cramer (e-text) pages 154-164,
  167-173
• Read Russo (e-text) pages 168-175
• Read Davis and Smith pages 448-459

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General Reading

• The Mann-Whitney U A Test for Assessing Whether
  Two Independent Samples Come
• from the Same Distribution
• N. Nachar
• Tutorials in Quantitative Methods for Psychology
  2008, vol. 4(1), p. 13-20.
• Paper

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Practical 6

• This material is available from the module web
  page.
• http//www.staff.ncl.ac.uk/mike.cox

Module Web Page
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Practical 6

• This material for the practical is available.

Instructions for the practical Practical 6
Material for the practical Practical 6
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