Nonparametric tests II
as randomisation tests
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
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
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
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
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 >
H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign
WilcoxonSignedRanks
Mann-Whitney
H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
WilcoxonSignedRanks
Mann-Whitney
H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
WilcoxonSignedRanks
Mann-Whitney
H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
WilcoxonSignedRanks
Mann-Whitney
H0 FixedInformation
RandomisedInformation
Test Statistic Notes
Sign Prob of +equalsProb of -
Number ofnon-zerodifferences
Sign of eachdifference
Number of +’s
WilcoxonSignedRanks
Mann-Whitney
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
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 >
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)
So if we take ten items that might be plus or minus,
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.
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.
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 >
109876543210
200
100
0
C3
109876543210
200
100
0
C3
The truth about confidence intervals
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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 >
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 >
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
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
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
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
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
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
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
. .. . . .: ..---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
. .. . . .: ..---+---------+---------+---------+---------+---------+---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.
. .. . . .: ..---+---------+---------+---------+---------+---------+---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.
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”.)
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 >
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
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%
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
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
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
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
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
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
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
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
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
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
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
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
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
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 >
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
Sign vs Wilcoxon Signed Ranks
Sign vs Wilcoxon Signed Ranks
• Less powerful • More powerful
Sign vs Wilcoxon Signed Ranks
• Less powerful– Less sensitive– Wider confidence
intervals
• More powerful– More sensitive– Narrower confidence
intervals
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
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
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
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
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
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
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
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
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
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
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
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
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.