Nonparametric tests II as randomisation tests Lecture Outline

Lecture Outline • Background: Nonparametric tests as randomisation tests – The sign test –

after before 640. 0 1050. 0 70. 0 84. 0 83. 0 77. 0

after before change 640. 0 1050. 0 -410. 0 70. 0 84. 0 -14.

MTB > stest 'change' Sign Test for Median Sign test of median=0. 000 versus

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

The truth about confidence intervals . . . : . . ---+---------+---------+-----+---change -400 -320

MTB > stest 0 c 3 Sign Test for Median: C 3 Sign test

H 0 median -50 -40 -35 -30 -25 -20 -15 -10 -5 0 5

. . . : . . ---+---------+---------+---------+---------+---------+---change -400 -320 -240 -160 -80 0 The

The real definition of 95% confidence interval • is “the set of values of

MTB > sinterval 'change' Sign Confidence Interval Sign confidence interval for median ACHIEVED POSI

Why does Minitab give three confidence intervals for the sign test? • the p-value

Sign vs Wilcoxon Signed Ranks • Less powerful • More powerful

Sign vs Wilcoxon Signed Ranks • Less powerful – Less sensitive – Wider confidence

In these randomisation tests, • there is a simple direct connection between the null

Last remarks • Randomisation tests are powerful tools • All parametric and nonparametric tests

Slides: 68

Download presentation

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 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

after before change 640. 0 1050. 0 -410. 0 70. 0 84. 0 -14. 0 83. 0 77. 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

after before change 640. 0 1050. 0 -410. 0 70. 0 84. 0 -14. 0 83. 0 77. 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

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 >

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 >

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)

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. 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 MEDIAN change 10 8 0 2 0. 1094 -21. 00 MTB >

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 MEDIAN change 10 8 0 2 0. 1094 -21. 00 MTB >

MTB > stest 0 c 3 Sign Test for Median: C 3 Sign test of median = 0. 00000 versus C 3 N 10 Below 8 Equal 0 Above 2 not = 0. 00000 P 0. 1094 Median -21. 00 MTB > stest 10 c 3 Sign Test for Median: C 3 Sign test of median = 10. 00 versus C 3 N 10 Below 10 Equal 0 not = Above 0 10. 00 P 0. 0020 . . . : . . ---+---------+---------+-----+---change -400 -320 -240 -160 -80 0 Median -21. 00

H 0 median -50 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 N Below Equal Above P Median 10 10 10 10 3 4 4 5 6 8 8 8 9 10 10 0 0 1 0 0 0 0 7 6 6 4 4 2 2 2 1 0 0 0. 3438 0. 7539 1. 0000 0. 7539 0. 1094 0. 0215 0. 0020 -21. 00 -21. 00 . . . : . . ---+---------+---------+-----+---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 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 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

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

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 > 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 > sinterval 'change' Sign Confidence Interval Sign confidence interval for median ACHIEVED POSI N MEDIAN 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 INTERVAL change 10 -46 94. 7 ( -218, -8) MTB >

MTB > sinterval 'change' Sign Confidence Interval Sign confidence interval for median ACHIEVED POSI N MEDIAN 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 INTERVAL change 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

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

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.