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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 before 640.0 1050.0 70.0 84.0 83.0 77.0 64.0 110.0 420.0 440.0 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 change 640.0 1050.0 -410.0 70.0 84.0 -14.0 83.0 77.0 6.0 64.0 110.0 -46.0 420.0 440.0 -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 change 640.0 1050.0 -410.0 70.0 84.0 -14.0 83.0 77.0 6.0 64.0 110.0 -46.0 420.0 440.0 -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 -13.8 1.6 6.0
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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 MEDIAN change 10 8 0 2 0.1094 -21.00 MTB >
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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 MEDIAN change 10 8 0 2 0.1094 -21.00 MTB >
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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 MEDIAN change 10 8 0 2 0.1094 -21.00 MTB > 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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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 MEDIAN change 10 8 0 2 0.1094 -21.00 MTB >
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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 MEDIAN change 10 8 0 2 0.1094 -21.00 MTB >
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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 MEDIAN change 10 8 0 2 0.1094 -21.00 MTB >
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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 Median C3 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 Median C3 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 -21.00 0 10 8 0 2 0.1094 -21.00 5 10 9 0 1 0.0215 -21.00 10 10 10 0 0 0.0020 -21.00 15 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 -21.00 0 10 8 0 2 0.1094 -21.00 5 10 9 0 1 0.0215 -21.00 10 10 10 0 0 0.0020 -21.00 15 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 -21.00 0 10 8 0 2 0.1094 -21.00 5 10 9 0 1 0.0215 -21.00 10 10 10 0 0 0.0020 -21.00 15 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 -21.00 0 10 8 0 2 0.1094 -21.00 5 10 9 0 1 0.0215 -21.00 10 10 10 0 0 0.0020 -21.00 15 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 -21.00 0 10 8 0 2 0.1094 -21.00 5 10 9 0 1 0.0215 -21.00 10 10 10 0 0 0.0020 -21.00 15 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 -21.00 0 10 8 0 2 0.1094 -21.00 5 10 9 0 1 0.0215 -21.00 10 10 10 0 0 0.0020 -21.00 15 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 TION change 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 -21.00 0 10 8 0 2 0.1094 -21.00 5 10 9 0 1 0.0215 -21.00 10 10 10 0 0 0.0020 -21.00 15 10 10 0 0 0.0020 -21.00......:.. ---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0
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after before change 640.0 1050.0 -410.0 70.0 84.0 -14.0 83.0 77.0 6.0 64.0 110.0 -46.0 420.0 440.0 -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 -13.8 1.6 6.0 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 TION change 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 change 640.0 1050.0 -410.0 70.0 84.0 -14.0 83.0 77.0 6.0 64.0 110.0 -46.0 420.0 440.0 -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 -13.8 1.6 6.0
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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 MEDIAN change 10 10 3.0 0.014 -46.00 MTB > MTB > stest 'change' Sign Test for Median Sign test of median=0.000 versus N.E. 0.000 N BELOW EQUAL ABOVE P-VALUE MEDIAN change 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 MEDIAN change 10 10 3.0 0.014 -46.00 MTB > MTB > stest 'change' Sign Test for Median Sign test of median=0.000 versus N.E. 0.000 N BELOW EQUAL ABOVE P-VALUE MEDIAN change 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 MEDIAN change 10 10 3.0 0.014 -46.00 MTB > MTB > stest 'change' Sign Test for Median Sign test of median=0.000 versus N.E. 0.000 N BELOW EQUAL ABOVE P-VALUE MEDIAN change 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 TION change 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 INTERVAL change 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 TION change 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 INTERVAL change 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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Less powerfulMore powerful
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Sign vs Wilcoxon Signed Ranks Less powerful –Less sensitive –Wider confidence intervals More powerful –More sensitive –Narrower confidence intervals
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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
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after before change 640.0 1050.0 -410.0 70.0 84.0 -14.0 83.0 77.0 6.0 64.0 110.0 -46.0 420.0 440.0 -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 -13.8 1.6 6.0
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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
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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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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
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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.
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