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Nonparametric tests II
as randomisation tests
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Lecture Outline
• Background: Nonparametric tests as randomisation tests– The sign test– The Wilcoxon signed ranks test– The Mann-Whitney test
• General remarks on randomisation tests
• Brief Review of the course so far
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after before640.0 1050.0 70.0 84.0 83.0 77.0 64.0 110.0420.0 440.0 6.4 4.8 26.0 48.0 2.2 16.0 75.0 340.0 16.0 430.0
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after before change640.0 1050.0 -410.0 70.0 84.0 -14.0 83.0 77.0 6.0 64.0 110.0 -46.0420.0 440.0 -20.0 6.4 4.8 1.6 26.0 48.0 -22.0 2.2 16.0 -13.8 75.0 340.0 -265.0 16.0 430.0 -414.0
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after before change640.0 1050.0 -410.0 70.0 84.0 -14.0 83.0 77.0 6.0 64.0 110.0 -46.0420.0 440.0 -20.0 6.4 4.8 1.6 26.0 48.0 -22.0 2.2 16.0 -13.8 75.0 340.0 -265.0 16.0 430.0 -414.0
schange-414.0-410.0-265.0-46.0-22.0-20.0-14.0-13.81.66.0
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MTB > stest 'change'
Sign Test for Median
Sign test of median=0.000 versus N.E. 0.000
N BELOW EQUAL ABOVE P-VALUE MEDIANchange 10 8 0 2 0.1094 -21.00MTB >
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign
WilcoxonSignedRanks
Mann-Whitney
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
WilcoxonSignedRanks
Mann-Whitney
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
WilcoxonSignedRanks
Mann-Whitney
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
WilcoxonSignedRanks
Mann-Whitney
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s
WilcoxonSignedRanks
Mann-Whitney
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Mann-Whitney
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MTB > stest 'change'
Sign Test for Median
Sign test of median=0.000 versus N.E. 0.000
N BELOW EQUAL ABOVE P-VALUE MEDIANchange 10 8 0 2 0.1094 -21.00MTB >
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MTB > stest 'change'
Sign Test for Median
Sign test of median=0.000 versus N.E. 0.000
N BELOW EQUAL ABOVE P-VALUE MEDIANchange 10 8 0 2 0.1094 -21.00MTB >
which added = number of non-zero datapoints (in this case there are no zeroes)
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So if we take ten items that might be plus or minus,
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So if we take ten items that might be plus or minus,and randomly choose them, we get the set of relevant comparisons for our dataset of 8 minus and 2 plus. This is the randomisation part of the test.
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So if we take ten items that might be plus or minus,and randomly choose them, we get the set of relevant comparisons for our dataset of 8 minus and 2 plus. This is the randomisation part of the test.
To decide whether our actual dataset is extreme in the distribution, we calculate the test statistic in each case - just the number of plusses. We count in what fraction of cases, the relevant comparison has a more extreme number of plusses, that is, either 2 or fewer, or 8 or more.
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MTB > stest 'change'
Sign Test for Median
Sign test of median=0.000 versus N.E. 0.000
N BELOW EQUAL ABOVE P-VALUE MEDIANchange 10 8 0 2 0.1094 -21.00MTB >
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109876543210
200
100
0
C3
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109876543210
200
100
0
C3
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The truth about confidence intervals
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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MTB > stest 'change'
Sign Test for Median
Sign test of median=0.000 versus N.E. 0.000
N BELOW EQUAL ABOVE P-VALUE MEDIANchange 10 8 0 2 0.1094 -21.00MTB >
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MTB > stest 'change'
Sign Test for Median
Sign test of median=0.000 versus N.E. 0.000
N BELOW EQUAL ABOVE P-VALUE MEDIANchange 10 8 0 2 0.1094 -21.00MTB >
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MTB > stest 0 c3
Sign Test for Median: C3
Sign test of median = 0.00000 versus not = 0.00000
N Below Equal Above P MedianC3 10 8 0 2 0.1094 -21.00
MTB > stest 10 c3
Sign Test for Median: C3
Sign test of median = 10.00 versus not = 10.00
N Below Equal Above P MedianC3 10 10 0 0 0.0020 -21.00
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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H0median N Below Equal Above P Median
-50 10 3 0 7 0.3438 -21.00-40 10 4 0 6 0.7539 -21.00-35 10 4 0 6 0.7539 -21.00-30 10 4 0 6 0.7539 -21.00-25 10 4 0 6 0.7539 -21.00-20 10 5 1 4 1.0000 -21.00-15 10 6 0 4 0.7539 -21.00-10 10 8 0 2 0.1094 -21.00-5 10 8 0 2 0.1094 -21.000 10 8 0 2 0.1094 -21.005 10 9 0 1 0.0215 -21.0010 10 10 0 0 0.0020 -21.0015 10 10 0 0 0.0020 -21.00
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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H0median N Below Equal Above P Median
-50 10 3 0 7 0.3438 -21.00-40 10 4 0 6 0.7539 -21.00-35 10 4 0 6 0.7539 -21.00-30 10 4 0 6 0.7539 -21.00-25 10 4 0 6 0.7539 -21.00-20 10 5 1 4 1.0000 -21.00-15 10 6 0 4 0.7539 -21.00-10 10 8 0 2 0.1094 -21.00-5 10 8 0 2 0.1094 -21.000 10 8 0 2 0.1094 -21.005 10 9 0 1 0.0215 -21.0010 10 10 0 0 0.0020 -21.0015 10 10 0 0 0.0020 -21.00
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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H0median N Below Equal Above P Median
-50 10 3 0 7 0.3438 -21.00-40 10 4 0 6 0.7539 -21.00-35 10 4 0 6 0.7539 -21.00-30 10 4 0 6 0.7539 -21.00-25 10 4 0 6 0.7539 -21.00-20 10 5 1 4 1.0000 -21.00-15 10 6 0 4 0.7539 -21.00-10 10 8 0 2 0.1094 -21.00-5 10 8 0 2 0.1094 -21.000 10 8 0 2 0.1094 -21.005 10 9 0 1 0.0215 -21.0010 10 10 0 0 0.0020 -21.0015 10 10 0 0 0.0020 -21.00
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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H0median N Below Equal Above P Median
-50 10 3 0 7 0.3438 -21.00-40 10 4 0 6 0.7539 -21.00-35 10 4 0 6 0.7539 -21.00-30 10 4 0 6 0.7539 -21.00-25 10 4 0 6 0.7539 -21.00-20 10 5 1 4 1.0000 -21.00-15 10 6 0 4 0.7539 -21.00-10 10 8 0 2 0.1094 -21.00-5 10 8 0 2 0.1094 -21.000 10 8 0 2 0.1094 -21.005 10 9 0 1 0.0215 -21.0010 10 10 0 0 0.0020 -21.0015 10 10 0 0 0.0020 -21.00
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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H0median N Below Equal Above P Median
-50 10 3 0 7 0.3438 -21.00-40 10 4 0 6 0.7539 -21.00-35 10 4 0 6 0.7539 -21.00-30 10 4 0 6 0.7539 -21.00-25 10 4 0 6 0.7539 -21.00-20 10 5 1 4 1.0000 -21.00-15 10 6 0 4 0.7539 -21.00-10 10 8 0 2 0.1094 -21.00-5 10 8 0 2 0.1094 -21.000 10 8 0 2 0.1094 -21.005 10 9 0 1 0.0215 -21.0010 10 10 0 0 0.0020 -21.0015 10 10 0 0 0.0020 -21.00
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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H0median N Below Equal Above P Median
-50 10 3 0 7 0.3438 -21.00-40 10 4 0 6 0.7539 -21.00-35 10 4 0 6 0.7539 -21.00-30 10 4 0 6 0.7539 -21.00-25 10 4 0 6 0.7539 -21.00-20 10 5 1 4 1.0000 -21.00-15 10 6 0 4 0.7539 -21.00-10 10 8 0 2 0.1094 -21.00-5 10 8 0 2 0.1094 -21.000 10 8 0 2 0.1094 -21.005 10 9 0 1 0.0215 -21.0010 10 10 0 0 0.0020 -21.0015 10 10 0 0 0.0020 -21.00
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
The green values cannot be rejected at the 5% level, while the red values can.
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. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
The green values cannot be rejected at the 5% level, while the red values can.
The range of green values is therefore the 95% confidence interval for the median based on the sign test.
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The real definition of 95% confidence interval
• is “the set of values of a parameter that cannot be rejected at the 5% level”
• is therefore not “the set of values that the parameter has a 95% chance of belonging to”, as many textbooks claim. (This is called a “fiducial interval”.)
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MTB > sinterval 'change'
Sign Confidence Interval
Sign confidence interval for median
ACHIEVED POSI N MEDIAN CONFIDENCE CONFIDENCE INTERVAL TIONchange 10 -21.000 0.8906 (-265.000, -13.800) 3 0.9500 (-314.640, -8.528) NLI 0.9785 (-410.000, 1.600) 2
MTB >
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H0median N Below Equal Above P Median
-50 10 3 0 7 0.3438 -21.00-40 10 4 0 6 0.7539 -21.00-35 10 4 0 6 0.7539 -21.00-30 10 4 0 6 0.7539 -21.00-25 10 4 0 6 0.7539 -21.00-20 10 5 1 4 1.0000 -21.00-15 10 6 0 4 0.7539 -21.00-10 10 8 0 2 0.1094 -21.00-5 10 8 0 2 0.1094 -21.000 10 8 0 2 0.1094 -21.005 10 9 0 1 0.0215 -21.0010 10 10 0 0 0.0020 -21.0015 10 10 0 0 0.0020 -21.00
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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after before change640.0 1050.0 -410.0 70.0 84.0 -14.0 83.0 77.0 6.0 64.0 110.0 -46.0420.0 440.0 -20.0 6.4 4.8 1.6 26.0 48.0 -22.0 2.2 16.0 -13.8 75.0 340.0 -265.0 16.0 430.0 -414.0
schange-414.0-410.0-265.0-46.0-22.0-20.0-14.0-13.81.66.0
97.85%89.06%
97.85%89.06%
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MTB > sinterval 'change'
Sign Confidence Interval
Sign confidence interval for median
ACHIEVED POSI N MEDIAN CONFIDENCE CONFIDENCE INTERVAL TIONchange 10 -21.000 0.8906 (-265.000, -13.800) 3 0.9500 (-314.640, -8.528) NLI 0.9785 (-410.000, 1.600) 2
MTB > . .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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Why does Minitab give three confidence intervals for the sign test?
• the p-value for rejecting a value changes in a step function at observed values
• so exact confidence intervals are given between observed values, at whatever level of confidence is attained
• the NLI (Non-Linear Interpolation) confidence interval is a confidence trick
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Lecture Outline
• Background: Nonparametric tests as randomisation tests– The sign test– The Wilcoxon signed ranks test– The Mann-Whitney test
• General remarks on randomisation tests
• Brief Review of the course so far
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after before change640.0 1050.0 -410.0 70.0 84.0 -14.0 83.0 77.0 6.0 64.0 110.0 -46.0420.0 440.0 -20.0 6.4 4.8 1.6 26.0 48.0 -22.0 2.2 16.0 -13.8 75.0 340.0 -265.0 16.0 430.0 -414.0
schange-414.0-410.0-265.0-46.0-22.0-20.0-14.0-13.81.66.0
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Mann-Whitney
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Symmetryabout zero
Mann-Whitney
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Symmetryabout zero
Absolutevalues ofdifferences
Mann-Whitney
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Symmetryabout zero
Absolutevalues ofdifferences
Sign of eachdifference
Mann-Whitney
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Symmetryabout zero
Absolutevalues ofdifferences
Sign of eachdifference
Sum of ranks ofnegativedifferences
Mann-Whitney
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Symmetryabout zero
Absolutevalues ofdifferences
Sign of eachdifference
Sum of ranks ofnegativedifferences
Numberedcoin tossing
Mann-Whitney
![Page 48: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/48.jpg)
MTB > wtest 'change'
Wilcoxon Signed Rank Test
TEST OF MEDIAN = 0.000 VERSUS MEDIAN N.E. 0.000
N FOR WILCOXON ESTIMATED N TEST STATISTIC P-VALUE MEDIANchange 10 10 3.0 0.014 -46.00MTB >
MTB > stest 'change'
Sign Test for Median
Sign test of median=0.000 versus N.E. 0.000
N BELOW EQUAL ABOVE P-VALUE MEDIANchange 10 8 0 2 0.1094 -21.00
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MTB > wtest 'change'
Wilcoxon Signed Rank Test
TEST OF MEDIAN = 0.000 VERSUS MEDIAN N.E. 0.000
N FOR WILCOXON ESTIMATED N TEST STATISTIC P-VALUE MEDIANchange 10 10 3.0 0.014 -46.00MTB >
MTB > stest 'change'
Sign Test for Median
Sign test of median=0.000 versus N.E. 0.000
N BELOW EQUAL ABOVE P-VALUE MEDIANchange 10 8 0 2 0.1094 -21.00
![Page 50: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/50.jpg)
MTB > wtest 'change'
Wilcoxon Signed Rank Test
TEST OF MEDIAN = 0.000 VERSUS MEDIAN N.E. 0.000
N FOR WILCOXON ESTIMATED N TEST STATISTIC P-VALUE MEDIANchange 10 10 3.0 0.014 -46.00MTB >
MTB > stest 'change'
Sign Test for Median
Sign test of median=0.000 versus N.E. 0.000
N BELOW EQUAL ABOVE P-VALUE MEDIANchange 10 8 0 2 0.1094 -21.00
The Wilcoxon test is more powerful than the Sign Test
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MTB > sinterval 'change'
Sign Confidence Interval
Sign confidence interval for median ACHIEVED POSI N MEDIAN CONFIDENCE CONFIDENCE INTERVAL TIONchange 10 -21.000 0.8906 (-265.000, -13.800) 3 0.9500 (-314.640, -8.528) NLI 0.9785 (-410.000, 1.600) 2
MTB > winterval 'change'
Wilcoxon Signed Rank Confidence Interval
ESTIMATED ACHIEVED N MEDIAN CONFIDENCE CONFIDENCE INTERVALchange 10 -46 94.7 ( -218, -8)MTB >
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MTB > sinterval 'change'
Sign Confidence Interval
Sign confidence interval for median ACHIEVED POSI N MEDIAN CONFIDENCE CONFIDENCE INTERVAL TIONchange 10 -21.000 0.8906 (-265.000, -13.800) 3 0.9500 (-314.640, -8.528) NLI 0.9785 (-410.000, 1.600) 2
MTB > winterval 'change'
Wilcoxon Signed Rank Confidence Interval
ESTIMATED ACHIEVED N MEDIAN CONFIDENCE CONFIDENCE INTERVALchange 10 -46 94.7 ( -218, -8)MTB >
The Wilcoxon confidence interval is narrower
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Sign vs Wilcoxon Signed Ranks
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Sign vs Wilcoxon Signed Ranks
• Less powerful • More powerful
![Page 55: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/55.jpg)
Sign vs Wilcoxon Signed Ranks
• Less powerful– Less sensitive– Wider confidence
intervals
• More powerful– More sensitive– Narrower confidence
intervals
![Page 56: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/56.jpg)
Sign vs Wilcoxon Signed Ranks
• Less powerful– Less sensitive– Wider confidence
intervals
• Uses less information– only sign of
difference
• More powerful– More sensitive– Narrower confidence
intervals
• Uses more information– also size of
difference
![Page 57: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/57.jpg)
after before change640.0 1050.0 -410.0 70.0 84.0 -14.0 83.0 77.0 6.0 64.0 110.0 -46.0420.0 440.0 -20.0 6.4 4.8 1.6 26.0 48.0 -22.0 2.2 16.0 -13.8 75.0 340.0 -265.0 16.0 430.0 -414.0
schange-414.0-410.0-265.0-46.0-22.0-20.0-14.0-13.81.66.0
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Lecture Outline
• Background: Nonparametric tests as randomisation tests– The sign test– The Wilcoxon signed ranks test– The Mann-Whitney test
• General remarks on randomisation tests
• Brief Review of the course so far
![Page 59: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/59.jpg)
H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Symmetryabout zero
Absolutevalues ofdifferences
Sign of eachdifference
Sum of ranks ofnegativedifferences
Numberedcoin tossing
Mann-Whitney
![Page 60: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/60.jpg)
H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Symmetryabout zero
Absolutevalues ofdifferences
Sign of eachdifference
Sum of ranks ofnegativedifferences
Numberedcoin tossing
Mann-Whitney
Two groupsfrom samedistribution
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H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Symmetryabout zero
Absolutevalues ofdifferences
Sign of eachdifference
Sum of ranks ofnegativedifferences
Numberedcoin tossing
Mann-Whitney
Two groupsfrom samedistribution
Set of ranks;numbers ineach group
![Page 62: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/62.jpg)
H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Symmetryabout zero
Absolutevalues ofdifferences
Sign of eachdifference
Sum of ranks ofnegativedifferences
Numberedcoin tossing
Mann-Whitney
Two groupsfrom samedistribution
Set of ranks;numbers ineach group
Groupmembershipof datapoints
![Page 63: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/63.jpg)
H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Symmetryabout zero
Absolutevalues ofdifferences
Sign of eachdifference
Sum of ranks ofnegativedifferences
Numberedcoin tossing
Mann-Whitney
Two groupsfrom samedistribution
Set of ranks;numbers ineach group
Groupmembershipof datapoints
Sum of ranks infirst group
![Page 64: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/64.jpg)
H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s BinomialDistributionin Theory
WilcoxonSignedRanks
Symmetryabout zero
Absolutevalues ofdifferences
Sign of eachdifference
Sum of ranks ofnegativedifferences
Numberedcoin tossing
Mann-Whitney
Two groupsfrom samedistribution
Set of ranks;numbers ineach group
Groupmembershipof datapoints
Sum of ranks infirst group
Randomassignmentto groups
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Lecture Outline
• Background: Nonparametric tests as randomisation tests– The sign test– The Wilcoxon signed ranks test– The Mann-Whitney test
• General remarks on randomisation tests
• Brief Review of the course so far
![Page 66: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/66.jpg)
In these randomisation tests,
• there is a simple direct connection between the null hypothesis and the randomisation procedure
• there is freedom of choice of test statistic
• estimation relies on scales of measurement and so is not as ‘principled’ as hypothesis tests
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Lecture Outline
• Background: Nonparametric tests as randomisation tests– The sign test– The Wilcoxon signed ranks test– The Mann-Whitney test
• General remarks on randomisation tests
• Brief Review of the course so far
![Page 68: Nonparametric tests II as randomisation tests. Lecture Outline Background: Nonparametric tests as randomisation tests – The sign test – The Wilcoxon signed](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649d805503460f94a64480/html5/thumbnails/68.jpg)
Last remarks• Randomisation tests are powerful tools
• All parametric and nonparametric tests can be understood as randomisation tests
• Nowadays they are used when no others can be used.
• NEXT WEEK: Conclusion to course and some exam questions. READ Chapter 14 of textbook.