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- 1. Conducting Laboratory Experiments Properly with Statistical Tools: An Easy Hands-on Tutorial Tetsuya Sakai (Waseda University) tetsuyasakai@acm.org 8th July@SIGIR 2018, Ann Arbor Michigan, USA.
- 2. Tutorial materials: If you want a hands-on experience at the tutorial, please do the following BEFORE ATTENDING: - Download http://waseda.box.com/SIGIR2018tutorial - Install R on your laptop. Then install the tidyr (for reformatting data) and pwr libraries (for power calculation) If you have reshape2 instead of tidyr and know how to use melt that’s okay too
- 3. Tutorial Outline • Part I (9:00-10:00) - Preliminaries - Paired and two-sample t-tests, confidence intervals - One-way ANOVA and two-way ANOVA without replication - Familywise error rate [Coffee Break (10:00-10:30)] • Part II (10:30-12:00) - Tukey’s HSD test, simultaneous confidence intervals - Randomisation test and randomised Tukey HSD test - What’s wrong with statistical significance tests? - Effect sizes, statistical power - Topic set size design and power analysis - Summary: how to report your results IF YOU WANT MORE, PLEASE READ: Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes, and statistical power, to appear, Springer, 2018.
- 4. Which search engine is better? (paired data) 0.4 0.4 0.8 0.6 0.7 0.5 Some evaluation measure score Sample size n = 3
- 5. Which search engine is better? (unpaired data) 0.4 0.8 0.7 1.0 0.8 0.1 0.5 n1 = 3 n2 = 4
- 6. Statistical significance testing [Robertson81, p.23] “More particularly, having performed a comparison of two systems on specific samples of documents and requests, we may be interested in the statistical significance of the difference, that is in whether the difference we observe could be simply an accidental property of the sample or can be assumed to represent a genuine characteristic of the populations.” “Document collection as a sample” [Robertson12] not discussed in this tutorial.
- 7. Statistically significant result may not be practically significant (and vice versa) “It must nevertheless be admitted that the basis for applying significance tests to retrieval results is not well established, and it should also be noted that statistically significant performance differences may be too small to be of much operational interest.” [SparckJones81, Chapter 12, p.243] Karen Sparck Jones 1935-2007 Roger Needham 1935-2003
- 8. What do samples tell us about population means? Are they the same?
- 9. Parametric tests for comparing means • In IR experiments, we often compare sample means to guess if the population means are different. • We often employ parametric tests (assume specific population distributions, e.g., normal) - paired and two-sample t-tests (Are the m(=2) population means equal?) - ANOVA (Are the m(>2) population means equal?) - Tukey HSD test for m(m-1)/2 system pairs scores EXAMPLE (paired data) n topics m systems Sample mean for a system
- 10. Null hypothesis, test statistic, p-value • H0: tentative assumption that all population means are equal • test statistic: what you compute from observed data – under H0, this should obey a known distribution (e.g. t-distribution) • p-value: probability of observing what you have observed (or something more extreme) assuming H0 is true Null hypothesis test statistic t0
- 11. Type I error and statistical power Reject H0 if p-value <= α test statistic t0 tinv(φ; α) Can’t reject H0 Reject H0 H0 is true systems are equivalent Correct conclusion (1-α) Type I error α H0 is false systems are different Type II error β Correct conclusion (1-β) α/2 α/2 Statistical power: ability to detect real differences
- 12. Type II error Can’t reject H0 if p-value > α test statistic t0 tinv(φ; α) Can’t reject H0 Reject H0 H0 is true systems are equivalent Correct conclusion (1-α) Type I error α H0 is false systems are different Type II error β Correct conclusion (1-β) α/2 α/2
- 13. Cohen’s five-eighty convention Can’t reject H0 Reject H0 H0 is true systems are equivalent Correct conclusion (1-α) Type I error α H0 is false systems are different Type II error β Correct conclusion (1-β) Statistical power: ability to detect real differencesCohen’s five-eighty convention: α=5%, 1-β=80% (β=20%) Type I errors 4 times as serious as Type II errors The ratio may be set depending on specific situations
- 14. Population means and variances x: a random variable f(x): probability density function (pdf) of x The expectation of any function g(x) is: Population mean: E(x) i.e. expectation of g(x)=x Population variance: How does x move around the population mean?
- 15. Law of large numbers ANY distribution is okay! If want a good estimate of the population mean, just take a large sample and compute the sample mean.
- 16. Normal distribution For given μ and σ (>0), the pdf of a normal distribution is given by: where The distribution is denoted by .
- 17. Standard normal distribution
- 18. Upper 100P% z-value For any u ～ that satisfies zinv(0.05) 5%
- 19. Properties of the normal distribution [Sakai18book]
- 20. Central Limit Theorem [Sakai18book] ANY distribution is okay!
- 21. Does CLT really hold? Test it with uniform distributions (1) n = 1
- 22. Does CLT really hold? Test it with uniform distributions (2) n = 2
- 23. Does CLT really hold? Test it with uniform distributions (3) n = 4
- 24. Does CLT really hold? Test it with uniform distributions (4) n = 8 Already looking quite like a normal distribution Variance getting smaller
- 25. Sample variance V is an unbiased estimator of the population variance (just as is an unbiased estimator of the population mean : )
- 26. Chi-square distribution The distribution is denoted by φ=5
- 27. t distribution The distribution is denoted by t(φ). φ=5
- 28. Two-sided 100P% t-value For any t ～ t(φ) that satisfies 2.5% 2.5% tinv(5; 0.05) qt returns one-sided t-values
- 29. Basis of the paired t-test [Sakai18book]
- 30. t-distribution is like a standard normal distribution with uncertainty Compare this with: population variance sample variance
- 31. Basis of the two-sample t-test [Sakai18book] Pooled variance homoscedasticity (equal variance)
- 32. F distribution The distribution is denoted by φ1 =10 φ2 =20
- 33. Upper 100P% F-value For any F ～ F(φ1 , φ2 ) that satisfies 5% Finv(10, 20; 0.05)
- 34. Basis of ANOVA See “Basis of the two-sample t-test
- 35. Tutorial Outline • Part I (9:00-10:00) - Preliminaries - Paired and two-sample t-tests, confidence intervals - One-way ANOVA and two-way ANOVA without replication - Familywise error rate [Coffee Break (10:00-10:30)] • Part II (10:30-12:00) - Tukey’s HSD test, simultaneous confidence intervals - Randomisation test and randomised Tukey HSD test - What’s wrong with statistical significance tests? - Effect sizes, statistical power - Topic set size design and power analysis - Summary: how to report your results IF YOU WANT MORE, PLEASE READ: Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes, and statistical power, to appear, Springer, 2018.
- 36. Historically, IR people were hesitant about the t-test (1) “since this normality is generally hard to prove for statistics derived from a request-document correlation process, the sign test probabilities may provide a better indicator of system performance than the t-test” [Salton+68] (p.15)
- 37. Historically, IR people were hesitant about the t-test (2) “Parametric tests are inappropriate because we do not know the form of the underlying distribution” [VanRijsbergen79] (p.136) “Since the form of the population distributions underlying the observed performance values is not known, only weak tests can be applied; for example, the sign test.” [SparckJones+97]
- 38. But actually the t-test is robust to minor assumption violations This assumes normality BUT CLT says ANYTHING can be viewed as normally distributed once averaged over a large sample. The robustness of the t-test can also be demonstrated using the randomisation test.
- 39. Which search engine is better? (paired data) 0.4 0.4 0.8 0.6 0.7 0.5 Some evaluation measure score Sample size n = 3
- 40. Paired t-test (1) x1j : nDCG of System 1 for the j-th topic x2j: nDCG of System 2 for the j-th topic Assume that the scores are independent and that Then for per-topic differences
- 41. Paired t-test (2) Therefore (from Corollary 5) where Sample mean Sample variance
- 42. Paired t-test (3) Two-sided test: H0 : μ1 = μ2 H1 : μ1 ≠ μ2 Under H0 the following should hold: Two-sided vs one-sided tests: See [Sakai18book] Ch.1
- 43. Paired t-test (4) Under H0 , should hold. So reject H0 iff The difference is statistically significant at the significance criterion α
- 44. http://www.f.waseda.jp/tetsuya/20topics3runs.mat.csv n=20 topics m=3 systems
- 45. Lazy paired t-test with Excel p-value two-sided test paired t-test
- 46. Loading 20topics3runs.mat.csv to R sample means
- 47. Paired t-test with R Compare with the Excel case Two-sided test
- 48. Which search engine is better? (unpaired data) 0.4 0.8 0.7 1.0 0.8 0.1 0.5 n1 = 3 n2 = 4
- 49. Two-sample t-test (1) x1j : nDCG of System 1 for the j-th topic (n1 topics) x2j: nDCG of System 2 for the j-th topic (n2 topics) Assume that the scores are independent and that Homoscedasticity (equal variance) assumption. But the t-test is actually quite robust to the assumption violation. For a discussion on Student’s and Welch’s t-tests, see [Sakai16SIGIRshort, Sakai18book]
- 50. Two-sample t-test (2) (From Corollary 6) Pooled variance Sample means
- 51. Two-sample t-test (3) H0 : μ1 = μ2 H1 : μ1 ≠ μ2 Under H0 the following should hold: So reject H0 iff
- 52. Lazy two-sample t-test with Excel p-value two-sided test Student’s t-test Two sets of nDCG scores treated as unpaired data ⇒ a much larger p-value In practice, if the scores are paired, use a paired test.
- 53. Two-sample (Student’s) t-test with R Two-sided test Compare with the Excel case
- 54. Confidence intervals for the difference in population means - paired data (1) From the paired t-test, therefore
- 55. Confidence intervals for the difference in population means - paired data (2) hence where 100(1-α)% CI: Margin of error Difference in population means
- 56. What does a 95% CI mean? Difference in population means (a constant) : 0.05 If you compute 100 different CIs from 100 different samples (i.e. topic sets) … About 95 of the CIs will capture the true difference
- 57. CIs for the difference in population means - unpaired data From the two-sample t-test, we obtain the following in a similar manner: where
- 58. Computing 95% CIs in practice (paired data) We had it already! To compute with Excel, use T.INV.2T(α, n-1) with R: qt(α/2, n-1, lower.tail=FALSE)
- 59. Computing 95% CIs in practice (unpaired data) We had it already! To compute with Excel, use T.INV.2T(α, n1 + n2 - 2) with R: qt(α/2, n1+n2-2, lower.tail=FALSE)
- 60. Tutorial Outline • Part I (9:00-10:00) - Preliminaries - Paired and two-sample t-tests, confidence intervals - One-way ANOVA and two-way ANOVA without replication - Familywise error rate [Coffee Break (10:00-10:30)] • Part II (10:30-12:00) - Tukey’s HSD test, simultaneous confidence intervals - Randomisation test and randomised Tukey HSD test - What’s wrong with statistical significance tests? - Effect sizes, statistical power - Topic set size design and power analysis - Summary: how to report your results IF YOU WANT MORE, PLEASE READ: Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes, and statistical power, to appear, Springer, 2018.
- 61. Analysis of Variance • A typical question ANOVA addresses: Given observed scores for m systems, are the m population means all equal or not? • ANOVA does NOT tell you which system means are different from others. • If you are interested in the difference between every system pair (i.e. obtaining m(m-1)/2 p-values), conduct an appropriate multiple comparison procedure, e.g. Tukey HSD test. No need to do ANOVA before Tukey HSD.
- 62. One-way ANOVA, equal group sizes (1) • Data format: • Basic assumption: or • Question: Are the m population means equal? unpaired data, but equal group sizes (e.g. #topics) homoscedasticity Generalises the two-sample t-test, and can handle unequal group sizes as well. See [Sakai18book] population mean for System i
- 63. One-way ANOVA, equal group sizes (2) Let Null hypothesis: ⇔ μ2 = μ3 = 0.2 μ = 0.3 μ1 = 0.5 a1 = 0.2 a2 = -0.1 a3 = -0.1 population grand mean i-th system effect All population means are equal (to μ) m=3
- 64. One-way ANOVA, equal group sizes (3) Let Clearly, sample grand mean System i’s sample mean Diff between an individual score and the grand mean can be broken down into… Diff between the system mean and the grand mean and… Diff between the individual score and the grand mean
- 65. One-way ANOVA, equal group sizes (4) Interestingly, this also holds: Between-system sum of squares Within-system sum of squares Total sum of squares (variations)
- 66. One-way ANOVA, equal group sizes (5) From a property of the chi-square distribution, hence
- 67. One-way ANOVA, equal group sizes (6) As for SA , since Corollary 1 gives us Hence, under H0 , From a property of the chi-square distribution, under H0.
- 68. One-way ANOVA, equal group sizes (7) Hence, by definition, under H0 , Under H0 :
- 69. One-way ANOVA, equal group sizes (8) Under H0 , so reject H0 iff Conclusion: probably not all population means are equal
- 70. One-way ANOVA with R (1) Here, just as an exercise, treat the matrix as if it’s unpaired data (i.e., sample sizes equal but no common topic set) The sample means (mean nDGG scores) suggest System1 > System2 > System3. But is the system effect statistically significant?
- 71. One-way ANOVA with R (2) mat is a 20x3 topic-by-run matrix: Let’s convert the format for convenience…
- 72. One-way ANOVA with R (3) A 60x2 data.frame Gather all columns of mat
- 73. One-way ANOVA with R (4) • φA = m-1 = 3-1 = 2 • φE1 = m(n-1) = 3(20-1) = 57 The system effect is statistically significant at α = 0.05 p-value The three systems are probably not all equally effective, but we don’t know where the difference lies.
- 74. Two-way ANOVA without replication (1) • Data format: • Basic assumption: i-th system effect j-th topic effect A common topic set for all m systems (paired data)
- 75. Two-way ANOVA without replication (2) Clearly, sample grand mean System i’s sample mean Topic j’s sample mean
- 76. Two-way ANOVA without replication (3) Similarly: Between-topic sum of squaresFrom one-way ANOVA
- 77. Two-way ANOVA without replication (4) It can be shown that under H0, so reject H0 iff The system effect is statistically significant at α All population system means are equal
- 78. Two-way ANOVA without replication (5) If also interested in the topic effect, under H0 so reject H0 iff The topic effect is statistically significant at α All population topic means are equal
- 79. Two-way ANOVA without replication with R (1) Just inserting a column for topic IDs
- 80. Two-way ANOVA without replication with R (2) Just converting the data format Gather all columns of mat except Topic A 60x3 data.frame
- 81. Two-way ANOVA without replication with R (3) • φA = 3-1 = 2 • φB = 20-1 = 19 • φE1 = (3-1)*(20-1)= 38 The system effect is statistically highly significant (so is the topic effect) The three systems are probably not all equally effective, but we don’t know where the difference lies.
- 82. Tutorial Outline • Part I (9:00-10:00) - Preliminaries - Paired and two-sample t-tests, confidence intervals - One-way ANOVA and two-way ANOVA without replication - Familywise error rate [Coffee Break (10:00-10:30)] • Part II (10:30-12:00) - Tukey’s HSD test, simultaneous confidence intervals - Randomisation test and randomised Tukey HSD test - What’s wrong with statistical significance tests? - Effect sizes, statistical power - Topic set size design and power analysis - Summary: how to report your results IF YOU WANT MORE, PLEASE READ: Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes, and statistical power, to appear, Springer, 2018.
- 83. Interested in the differences for all system pairs. So just repeat t-tests m(m-1)/2 times? (1) The following is the same as repeating t.test with paired=TRUE for every system pair... Compare with the Paired t-test with R slide ... but is NOT the right thing to do.
- 84. Interested in the differences for all system pairs. So just repeat t-tests m(m-1)/2 times? (2) The following is the same as repeating t.test with var.equal=TRUE for every system pair... Compare with the Two-sample (Student’s) t-test with R slide This means using Vp rather than VE1 from one-way ANOVA ... but is NOT the right thing to do.
- 85. Don’t repeat a regular t-test m(m-1)/2 times! Why? Suppose a restaurant has a wine cellar. It is known that one in every twenty bottles is sour. Pick a bottle; the probability that it is sour is 1/20 = 0.05 (Assume that we have an infinite number of bottles) VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN
- 86. A customer orders one bottle A bottle of red please The probability that sour wine is served to him is 0.05 VIN
- 87. A customer orders two bottles Two bottles please The probability that sour wine is served to him is 1-P(both bottles are good) = 1- 0.95^2 = 0.0975 VIN VIN
- 88. A customer orders three bottles Three bottles please The probability that sour wine is served to him is 1-P(all bottles are good) = 1- 0.95^3 = 0.1426 VIN VIN VIN
- 89. Comparisonwise vs Familywise error rate (restaurant owner) • The restaurant is worried not about the probability of each bottle being sour, but about the probability of accidentally serving sour wine to the customer who orders k bottles. The latter probability should be no larger than (say) 5%. YOU SERVED ME SOUR WINE I’M GONNA TWEET ABOUT IT
- 90. Comparisonwise vs Familywise error rate (researcher) • We should be worried not about the comparisonwise Type I error rate, but about the familywise error rate – the probability of making at least one Type I error among the k=m(m-1)/2 tests. • Just repeating a t-test k times gives us a familywise error rate of 1-(1-α)^k if the tests are independent. e.g. α=0.05, k=10 ⇒ familywise error rate = 40%!
- 91. Multiple comparison procedures [Carterette12][Nagata+97][Sakai18book] • Make sure that the familywise error rate is no more than α. • Stepwise methods: outcome of one hypothesis test determines what to do next • Single step methods: test all hypotheses at the same time – we discuss these only. - Bonferroni correction (considered obsolete) - Tukey’s Honestly Significant Difference (HSD) test - others (e.g. those available in pairwise.t.test)
- 92. Bonferroni correction [Crawley15 ](pp.17-18) “The old fashioned approach was to use Bonferroni’s correction: in looking up a value for Student’s t, you divide your α value by the number of comparisons you have done. […] Bonferroni’s correction is very harsh and will often throw out the baby with the bathwater. […] The modern approach is […] to use the wonderfully named Tukey’s honestly significant differences” Or, equivalently, multiply each p-value by k
- 93. Come back by 10:30!
- 94. Tutorial Outline • Part I (9:00-10:00) - Preliminaries - Paired and two-sample t-tests, confidence intervals - One-way ANOVA and two-way ANOVA without replication - Familywise error rate [Coffee Break (10:00-10:30)] • Part II (10:30-12:00) - Tukey’s HSD test, simultaneous confidence intervals - Randomisation test and randomised Tukey HSD test - What’s wrong with statistical significance tests? - Effect sizes, statistical power - Topic set size design and power analysis - Summary: how to report your results IF YOU WANT MORE, PLEASE READ: Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes, and statistical power, to appear, Springer, 2018.
- 95. • Instead of conducting a t-test k = m(m-1)/2 times, consider the maximum difference (best system – worst system) among the k differences. • The distribution that the max difference obeys is called a studentised range distribution. Its upper 100P% value is denoted by • We compare the k differences against the above distribution. By construction, if the maximum is not statistically significant, the other differences are not statistically significant either. Hence the familywise error rate can be controlled. How Tukey HSD works qtukey(P, m, φ, lower.tail=FALSE) in R
- 96. Tukey HSD with equal group sizes (1) Data structure: same as one-way ANOVA with equal group sizes Tukey HSD can handle unequal group sizes as well. See [Sakai18book] sample mean for System i unpaired data
- 97. Null hypothesis : the population means for systems i and i’ are equal Test statistic: Reject iff Tukey HSD with equal group sizes (2)
- 98. R: Tukey HSD with equal group sizes The data.frame we made for one-way ANOVA Only the diff between Systems 1 and 3 statistically significant at α=0.05
- 99. Tukey HSD with paired observations (1) Data structure: same as two-way ANOVA without replication sample mean for System i sample mean for Topic j paired data
- 100. Null hypothesis : the population means for systems i and i’ are equal Test statistic: Reject iff Tukey HSD with paired observations (2)
- 101. R: Tukey HSD with paired observations The data.frame we made for two-way ANOVA without replication The difference between Systems 1 and 3 and that between Systems 2 and 3 are statistically highly significant
- 102. If you want a 95%CI for the diff between every system pair… • If you use the t-test-based MOE for every system pair, this approach has the same problem as repeating t-tests multiple times. • Use a Tukey-based MOE instead to construct a simultaneous CI – to capture all true means at the same time, not individually.
- 103. Computing the MOE for simultaneous CIs From Tukey HSD with equal group sizes (unpaired data) From Tukey HSD with paired observations Apply the above MOE to each of the k differences
- 104. R: Simultaneous 95%CI, equal group sizes (unpaired data) MOE = 0.081314 Upper limit = diff + MOELower limit = diff - MOE
- 105. R: Simultaneous 95%CI, paired observations MOE = 0.033342 (CIs are narrower than the unpaired case) Upper limit = diff + MOELower limit = diff - MOE
- 106. Tutorial Outline • Part I (9:00-10:00) - Preliminaries - Paired and two-sample t-tests, confidence intervals - One-way ANOVA and two-way ANOVA without replication - Familywise error rate [Coffee Break (10:00-10:30)] • Part II (10:30-12:00) - Tukey’s HSD test, simultaneous confidence intervals - Randomisation test and randomised Tukey HSD test - What’s wrong with statistical significance tests? - Effect sizes, statistical power - Topic set size design and power analysis - Summary: how to report your results IF YOU WANT MORE, PLEASE READ: Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes, and statistical power, to appear, Springer, 2018.
- 107. Computer-based tests • Unlike classical significance tests, do not require assumptions about the underlying distribution • Bootstrap test [Sakai06SIGIR][Savoy97] – assumes the observed data are a random sample from the population. Samples with replacement from the observed data. • Randomisation test [Smucker+07] – no random sampling assumption. Permutes the observed data.
- 108. Randomisation test for paired data (1) Suppose we have an nDCG matrix for two systems with n topics. Are these systems equally effective?
- 109. Randomisation test for paired data (2) Suppose we have an nDCG matrix for two systems with n topics. Are these systems equally effective? Let’s assume there is a single hidden system. For each topic, it generates two nDCG scores. They are randomly assigned to the two systems.
- 110. Randomisation test for paired data (3) If H0 is right, these alternative matrices (obtained by randomly permuting each row of U) could also have occurred
- 111. Randomisation test for paired data (4) There are 2 possible matrices, but let’s just consider B of them (e.g. 10000 trials) n
- 112. Randomisation test for paired data (5) How likely is the observed difference (or something even more extreme) under H0? → p-value
- 113. Randomisation test for paired data - pseudocode The exact p-value changes slightly depending on B.
- 114. Random-test in Discpower [Sakai14PROMISE] http://research.nii.ac.jp/ntcir/tools/discpower-en.html Contains a tool for conducting a randomisation test or randomised Tukey HSD test
- 115. http://www.f.waseda.jp/tetsuya/20topics2runs.scorematrix Mean difference and p-value (compare with paired t-test) Paired randomisation test, B=5000 trials A 20x2 matrix, white-space-separated
- 116. Randomised Tukey HSD test for paired data (1) Suppose we have an nDCG matrix for more than two systems with n topics. Are these systems equally effective?
- 117. Suppose we have an nDCG matrix for more than two systems with n topics. Which system pairs are really different? Randomised Tukey HSD test for paired data (2) Let’s assume there is a single hidden system. For each topic, it generates m nDCG scores. They are randomly assigned to the m systems.
- 118. If H0 is right, these alternative matrices (obtained by randomly permuting each row of U) could also have occurred Randomised Tukey HSD test for paired data (3)
- 119. Randomised Tukey HSD test for paired data (4) There are (m!) possible matrices, but let’s just consider B of them (e.g. 10000 trials) n
- 120. How likely are the observed differences given the null distribution of the maximum differences? → Tukey HSD p-value Randomised Tukey HSD test for paired data (5)
- 121. Randomised Tukey HSD – pseudocode (adapted from [Carterette12]) The exact p-value changes slightly depending on B.
- 122. http://www.f.waseda.jp/tetsuya/20topics3runs.scorematrix Randomised Tukey HSD test, B=5000 trials Compare the p-values with those of the Tukey HSD test with R (paired data) A 20x3 matrix, white-space-separated
- 123. Tutorial Outline • Part I (9:00-10:00) - Preliminaries - Paired and two-sample t-tests, confidence intervals - One-way ANOVA and two-way ANOVA without replication - Familywise error rate [Coffee Break (10:00-10:30)] • Part II (10:30-12:00) - Tukey’s HSD test, simultaneous confidence intervals - Randomisation test and randomised Tukey HSD test - What’s wrong with statistical significance tests? - Effect sizes, statistical power - Topic set size design and power analysis - Summary: how to report your results IF YOU WANT MORE, PLEASE READ: Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes, and statistical power, to appear, Springer, 2018.
- 124. [Bakan66] “The test of significance does not provide the information concerning psychological phenomena characteristically attributed to it; and a great deal of mischief has been associated with its use.”
- 125. [Deming75] “Little advancement in the teaching of statistics is possible, and little hope for statistical methods to be useful in the frightful problems that face man today, until the literature and classroom be rid of terms so deadening to scientific enquiry as null hypothesis, population (in place of frame), true value, level of significance for comparison of treatments, representative sample.”
- 126. [Loftus91] “Despite the stranglehold that hypothesis testing has on experimental psychology, I find it difficult to imagine a less insightful means of transiting from data to conclusions.”
- 127. [Cohen94] (1) “And we, as teachers, consultants, authors, and otherwise perpetrators of quantitative methods, are responsible for the ritualization of null hypothesis significance testing (NHST; I resisted the temptation to call it statistical hypothesis inference testing) to the point of meaninglessness and beyond. I argue herein that NHST has not only failed to support the advances of psychology as a science but also has seriously impeded it.”
- 128. [Cohen94] (2) “What’s wrong with NHST? Well, among many other things, it does not tell us what we want to know, and we so much want to know what we want to know that, out of desperation, we nevertheless believe that it does! What we want to know is “Given these data, what is the probability that H0 is true?” But as most of us know, what it tells us is “Given that H0 is true, what is the probability of these (or more extreme) data?”” p-value = Pr(D+| H) not Pr(H|D)! See also [Carterette15] [Sakai17SIGIR]
- 129. [Rothman98] “When writing for Epidemiology, you can also enhance your prospects if you omit tests of statistical significance. Despite a wide spread belief that many journals require significance tests for publication, […] every worthwhile journal will accept papers that omit them entirely. In Epidemiology, we do not publish them at all. Not only do we eschew publishing claims of the presence or absence of statistical significance, we discourage the use of this type of thinking in the data analysis [….]”
- 130. [Ziliak+08] “Statistical significance is surely not the only error in modern science, although it has been, as we will show, an exceptionally damaging one.” “Most important is to minimise Error of the Third Kind, “the error of undue inattention,” which is caused by trying to solve a scientific problem using statistical significance or insignificance only.”
- 131. [Harlow+16] “The main opposition to NHST then and now is the tendency for researchers to narrowly focus on making a dichotomous decision to retain or reject a null hypothesis, which is usually not very informative to current or future research[…] Although there is still not a universally agreed upon set of practices regarding statistical inference, there does seem to be more consistency in agreeing on the need to move away from an exclusive focus on NHST […]”
- 132. Statistical significance: Problem 1 Many misinterpret and/or misuse significance tests. American Statistical Association statement (March 2016): • P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone. • A p-value, or statistical significance, does not measure the size of an effect or the importance of a result. p-value = Pr(D+|H) not Pr(H|D)!
- 133. Statistical significance: Problem 2 Dichotomous thinking: statistically significant or not? (More important questions: - How much is the difference? - What does a difference of that magnitude mean to us?) p=0.049 ⇒ statistically significant at α=0.05 p=0.051 ⇒ NOT statistically significant at α=0.05 Reporting the exact p-value is more informative than saying “p<0.05” BUT THIS IS STILL NOT SUFFICIENT.
- 134. Statistical significance: Problem 3 P-value = f(sample_size, effect_size) - A large effect size ⇒ a small p-value - A large sample size ⇒ a small p-value For example, consider: From the paired t-test Magnitude of the difference A large effect size (standardised mean difference) ⇒ a large t-value ⇒ a small p-value A large sample size (topic set size) ⇒ a large t-value ⇒ a small p-value Anything can be made statistically significant by making n large enough!
- 135. Tutorial Outline • Part I (9:00-10:00) - Preliminaries - Paired and two-sample t-tests, confidence intervals - One-way ANOVA and two-way ANOVA without replication - Familywise error rate [Coffee Break (10:00-10:30)] • Part II (10:30-12:00) - Tukey’s HSD test, simultaneous confidence intervals - Randomisation test and randomised Tukey HSD test - What’s wrong with statistical significance tests? - Effect sizes, statistical power - Topic set size design and power analysis - Summary: how to report your results IF YOU WANT MORE, PLEASE READ: Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes, and statistical power, to appear, Springer, 2018.
- 136. Effect size definition [Cohen88] “it is convenient to use the phrase “effect size” to mean “the degree to which the phenomenon is present in the population,” or “the degree to which the null hypothesis is false.” Whatever the manner of representation of a phenomenon in a particular research in the present treatment, the null hypothesis always means that the effect size is zero.”
- 137. Effect size definition [Olejnik+03] “An effect-size measure is a standardized index and estimates a parameter that is independent of sample size and quantifies the magnitude of the difference between populations or the relationship between explanatory and response variable.”
- 138. Effect size definition [Kelley+12] “Effect size is defined as a quantitative reflection of the magnitude of some phenomenon that is used for the purpose of addressing a question of interest.”
- 139. Various effect sizes • Effect sizes for t-tests (and Tukey HSD): standardized mean differences (covered in this lecture) • Effect sizes for ANOVAs: contribution to the overall variance: see [Olejnik+03][Sakai18book] etc. for details • Other forms of effect sizes: see [Cohen88] etc.
- 140. Paired t-test effect size Standardised mean difference (diff measured in standard deviation units) Note that sample size effect size Reporting dpaired along with the p-value is more informative than just reporting the p-value. But dpaired uses the standard deviation of the differences – works only with the paired t-test.
- 141. Hedge’s g (1) Standardised mean difference for the two-sample case: Hedge’s g estimates the above: Common standard deviation Pooled variance
- 142. Hedge’s g (2) From Student’s t-test, so, since , note that Reporting Hedge’s g along with the p-value is more informative than just reporting the p-value. See [Sakai18book] for bias correction. See [Sakai18book] for Cohen’s d
- 143. Glass’s Δ (my favourite) • No homoscedasticity assumption! • Works for both paired and unpaired data! • A bias-corrected Δ: “How much is the difference, when measured in standard deviation units computed from the control group (i.e., baseline) data that we are familiar with?”
- 144. Effect sizes for (randomised) Tukey HSD If homoscedasticity is assumed (as in classical Tukey HSD): If not, and if there is a common baseline, use Glass’ Δ by using the standard deviation of that baseline. Bottom line: report effect sizes as standardised mean differences, along with p-values. See “One-way ANOVA, equal group sizes (8)” See “Two-way ANOVA without replication (4)”
- 145. Statistical power: ability to detect real differences Given α and an effect size that you are interested in (e.g. standardized mean difference >=0.2), increasing the sample size n improves statistical power (1-β). - An overpowered experiment: n larger than necessary - An underpowered experiment: n smaller than necessary (cannot detect real differences – a waste of research effort!) Can’t reject H0 Reject H0 H0 is true systems are equivalent Correct conclusion (1-α) Type I error α H0 is false systems are different Type II error β Correct conclusion (1-β)
- 146. Tutorial Outline • Part I (9:00-10:00) - Preliminaries - Paired and two-sample t-tests, confidence intervals - One-way ANOVA and two-way ANOVA without replication - Familywise error rate [Coffee Break (10:00-10:30)] • Part II (10:30-12:00) - Tukey’s HSD test, simultaneous confidence intervals - Randomisation test and randomised Tukey HSD test - What’s wrong with statistical significance tests? - Effect sizes, statistical power - Topic set size design and power analysis - Summary: how to report your results IF YOU WANT MORE, PLEASE READ: Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes, and statistical power, to appear, Springer, 2018.
- 147. http://sigir.org/files/museum/pub-14/pub_14.pdf [SparckJones+75]
- 148. On TREC topic set sizes [Voorhees09] “Fifty-topic sets are clearly too small to have confidence intervals in a conclusion when using a measure as unstable as P(10). Even for stable measures, researchers should remain skeptical of conclusions demonstrated on only a single test collection.” TREC 2007 Million Query track [Allan+08] had “sparsely-judged” 1,800 topics, but this was an exception…
- 149. Deciding on the number of topics to create based on statistical requirements • Desired statistical power [Webber+08][Sakai16IRJ] • A cap on the confidence interval width for the mean difference [Sakai16IRJ] • Sakai’s Excel tools based on [Nagata03]: samplesizeTTEST2.xlsx (paired t-test power) samplesize2SAMPLET.xlsx (two-sample t-test power) samplesizeANOVA2.xlsx (one-way ANOVA power) samplesizeCI2.xlsx (paired data CI width) samplesize2SAMPLECI (two-sample CI width)
- 150. Recommendations on topic set size design tools • If you’re interested in the statistical power of the paired t-test, two-sample t-test, or one-way ANOVA, use samplesizeANOVA2. • If you’re interested in the CI width of the mean difference for paired or two-sample data, use samplesize2SAMPLECI. • … unless you have an accurate estimate of the population variance of the score differences which the paired-data tools require. See “Paired t-test (1)” samplesizeTTEST2 samplesizeCI2
- 151. samplesizeANOVA2 input α: Type I Error probability β: Type II Error probability, i.e., you want 100(1-β)% power (see below) m: number of systems to be compared in one-way ANOVA minD: minimum detectable range, i.e., whenever the true difference D between the best and the worst systems is minD or larger, you want to guarantee 100(1-β)% power : variance estimate for a particular evaluation measure (under the homoscedasticity assumption) μbest μworst D m system means
- 152. samplesizeANOVA2: “alpha=.05, beta=.20” sheet (1) Enter values in the orange cells (α=5%, β=20%): m=10, minD=0.1, =0.1 To ensure 80% power (at α=5%) for one-way ANOVA with any m=10 systems with a minimum detectable range of 0.1 in terms of a measure whose variance is 0.1, we need n=312 topics. μbest μworst D>= 0.1 m system means
- 153. samplesizeANOVA2: “alpha=.05, beta=.20” sheet (2) Enter values in the orange cells (α=5%, β=20%): m=2, minD=0.1, =0.1 To ensure 80% power (at α=5%) for one-way ANOVA with any m=2 systems with a minimum detectable difference of 0.1 in terms of a measure whose variance is 0.1, we need n=154 topics. μbest μworst D>= 0.1 Two system means
- 154. samplesizeANOVA2: “alpha=.05, beta=.20” sheet (3) Since one-way ANOVA with m=2 systems is strictly equivalent to the two-sample t-test [Sakai18book], To ensure 80% power (at α=5%) for the two-sample t-test with a minimum detectable difference of 0.1 in terms of a measure whose variance is 0.1, we need n=154 topics. μbest μworst D>= 0.1 Two system meansThis n can also be regarded as a pessimistic estimate for the paired data case.
- 155. samplesizeANOVA2: how does it work? (1) [Nagata03][Sakai16IRJ][Sakai18book] Remember what we do in one-way ANOVA: we reject H0 (all m means are equal) iff See “One-way ANOVA, equal group sizes (8)”
- 156. samplesizeANOVA2: how does it work? (2) So, whether H0 is true or not, the probability of rejecting H0 is: If H0 is true, then F0 ～ F(φA, φE1) and the above = α Probability of incorrectly concluding that the system means are different
- 157. samplesizeANOVA2: how does it work? (3) So, whether H0 is true or not, the probability of rejecting H0 is: If H0 is false, then F0 ～ F’(φA, φE1, λ) and the above = 1-β statistical power: probability of correctly concluding that the two system means are different A noncentral F distribution with a noncentrality parameter λ Accumulates squared system effects
- 158. samplesizeANOVA2: how does it work? (4) If H0 is false, the power can be approximated as: where Let’s call it Formula P
- 159. samplesizeANOVA2: how does it work? (5) Let’s ensure 100(1-β)% power whenever D >= minD (e.g., 0.1 in mean nDCG). To do this, we define: so that Δ >= minΔ holds. μbest μworst D m system means minΔ is the worst-case Δ for topic set sizes
- 160. samplesizeANOVA2: how does it work? (6) • λ=nΔ so the worst-case topic set size n can be estimated very roughly as: where λ can be approximated using a linear function of φA for given (α, β). • Having thus obtained an n, check with Formula P to see if the desired power is really achieved. If not, n++. If the power is too high, n--, etc. • The excel tool does just that.
- 161. samplesize2SAMPLECI input α: Type I Error probability for 100(1-α)% CIs δ: cap on the CI width for the difference between two systems (two-sample data). That is, you want the width of any CI to be δ or smaller. : variance estimate for a particular evaluation measure (under the homoscedasticity assumption) The n returned by samplesize2SAMPLECI can also be regarded as a pessimistic estimate for the paired data case. Difference in population means (a constant) : Width of this CI
- 162. Enter values in the orange cells: α=5%, δ=0.1, =0.1 To ensure the CI width of any between-system difference to be 0.1 or smaller, we need n=309 topics. samplesize2SAMPLECI: “approximation” sheet
- 163. samplesize2SAMPLECI: how does it work? (1) [Nagata03][Sakai16IRJ][Sakai18book] From “CIs for the difference in population means – unpaird data” we have: and the CI width is twice the MOE. We want 2MOE <= δ, but since Vp is a random variable, we use E(Vp) instead:
- 164. samplesize2SAMPLECI: how does it work? (2) Consider a balanced design, n = n1 = n2 . Then the above can be approximated as: Let’s call it Inequality C
- 165. samplesize2SAMPLECI: how does it work? (3) To obtain an initial estimate of n that satisfies the above, consider the CI for an ideal case where σ is known: cf. Inequality C Remember, the t- distribution is like a standard normal distribution with uncertainty
- 166. samplesize2SAMPLECI: how does it work? (4) Hence • This gives us an optimistic estimate of n, so check with Inequality C. If the condition is not satisfied, n++, etc. • The excel tool does just that.
- 167. Estimating the common variance If you have a topic-by-system score matrix or two from some pilot data, an unbiased estimator can be obtained as: Pooled estimate Estimate from one matrix See “Two-way ANOVA without replication (4)” A score matrix from test collection C
- 168. Some real estimates based on TREC data (using VE1 rather than VE2) See “One-way ANOVA, equal group sizes (8)” Some measures are less stable ⇒ require larger topic set sizes under the same requirement
- 169. Some topic set size design results [Sakai18book] The paired t-test tool does not return tight estimates due to my crude estimate (covariance not considered)
- 170. So, to build a test collection… 1. Build a small data set first (or borrow one from a past task similar to your own). 2. Decide on a primary evaluation measure, and create a small topic-by-system score matrix with the small data set. 3. Compute as VE1 or VE2 and use a topic set size design tool to decide on n. 4. You can advertise your test collection as follows: “We created 70 topics, which, according to topic set size design with = 0.044, is more than sufficient for achieving 80% power with a (paired) t-test whenever the true difference in Mean nDCG@10 is 0.10 or larger.” See previous two slides
- 171. Power analysis with R scripts [Sakai16SIGIR] (adapted from [Toyoda09]) • Given an adequately reported significance test result in a paper, - compute the effect size and the achieved power in that experiment. - propose a new sample size to achieve a desired power. Relies on the pwr library of R
- 172. The five R power analysis scripts [Sakai16SIGIR] • future.sample.paired (for paired t-tests) • future.sample.unpairedt (for two-sample t-tests) • future.sample.1wayanova (for one-way ANOVAs) • future.sample.2waynorep (for two-way ANOVAs without replication) • future.sample.2wayanova2 (for two-way ANOVAs)
- 173. future.sample.pairedt Basically just enter t0 and the actual sample size OUTPUT: - Effect size dpaired - Achieved power of the experiment - future sample size for achieving 80% power
- 174. future.sample.pairedt: an actual example from a SIGIR paper A highly underpowered experiment. In future, use 244 topics, not 28, to achieve 80% power for this small effect (dpaired = 0.18). Only 15% power! Underpowered experiments can be a waste of research effort: there’s a high chance that you will miss a true difference! about 85%
- 175. future.sample.2waynorep Basically just enter F0, the number of systems m and the actual sample size OUTPUT: - A partial effect size [Sakai18book] - Achieved power of the experiment - future sample size for achieving 80% power
- 176. future.sample.2waynorep: an actual example from a SIGIR paper A highly underpowered experiment. In future, use 75 topics, not 17, to achieve 80% power for this small effect. Only 18% power! Underpowered experiments can be a waste of research effort: there’s a high chance that you will miss a true difference! about 82%
- 177. From [Sakai16SIGIR]
- 178. Underpowered/Overpowered experiments: t-tests
- 179. Underpowered/Overpowered experiments: ANOVAs
- 180. Tutorial Outline • Part I (9:00-10:00) - Preliminaries - Paired and two-sample t-tests, confidence intervals - One-way ANOVA and two-way ANOVA without replication - Familywise error rate [Coffee Break (10:00-10:30)] • Part II (10:30-12:00) - Tukey’s HSD test, simultaneous confidence intervals - Randomisation test and randomised Tukey HSD test - What’s wrong with statistical significance tests? - Effect sizes, statistical power - Topic set size design and power analysis - Summary: how to report your results IF YOU WANT MORE, PLEASE READ: Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes, and statistical power, to appear, Springer, 2018.
- 181. Reporting a paired t-test result (1) Clearly state the sample size within the table! Clearly indicate what the numbers are! Which evaluation measure? Are they mean values?
- 182. Reporting a paired t-test result (2) “We conducted a paired t-test for the difference between our proposed system and the baseline in terms of mean nDCG over n=20 topics. The difference is not statistically significant (t(19) = 1.3101, p=0.2058, 95%CI[-0.0134, 0.0581]), with the effect size dpaired = t0/√n = 0.2929.” Provide as much information as possible for future researchers. There are several types of effect sizes. Clarify which one you are talking about.
- 183. Reporting a two-sample t-test result (1) Clearly indicate what the numbers are! Which evaluation measure? Are they mean values? Clearly state the sample sizes within the table!
- 184. Reporting a two-sample t-test result (2) “We conducted a Student’s t-test for the difference between the our proposed system (sample size n1 = 20) and the baseline (sample size n2 = 20) in terms of mean nDCG. The difference is not statistically significant (t(38) = 0.6338, p=0.53, 95%CI[-0.0491, 0.0938]), with Hedge’s g = 0.2004.” Note that If you used Welch’s t-test [Sakai16SIGIRshort], you consciously avoided the homoscedasticity assumption, so I recommend using Glass’ Δ, not Hedge’s d.
- 185. Reporting a (randomised) Tukey HSD result (1)
- 186. Reporting a (randomised) Tukey HSD result (2) “We conducted a randomised Tukey HSD test with B=5,000 trials to compare every system pair […] It can be observed that System 3 statistically significantly underperforms Systems 1 (p≒0.0000) and 2 (p=0.0024). Moreover, we computed an effect size ESE2 [Sakai18book] for each system pair. It can be observed that Systems 1 and 3 are almost two standard deviations apart from each other.” Also visualise each system’s performance using graphs with error bars, or boxplots! State clearly what the error bars represent (e.g., 95% CIs)!
- 187. Summary • To design a test collection, use some pilot data to estimate the variance of a particular evaluation measure for sample size considerations. • To design an experiment, use a pilot or existing study for sample size considerations to ensure sufficient statistical power. Underpowered experiments can be a waste of research effort. • When reporting on statistical significance test results, report the sample sizes, test statistics, degrees of freedom, p-values (not the stars */**/***), and effect sizes! Despite what R outputs, it is wrong to decide on α AFTER looking at the results (e.g. “this difference is statistically significant at α=1% while that is significant at α=5%”)
- 188. [Sakai18book]
- 189. Thank you for staying until the end!
- 190. References: A-B [Allan+08] Allan, J., Carterette, B., Aslam, J.A., Pavlu, V., Dachev, B., and Kanoulas, E.: Million Query Track 2007 Overview, Proceedings of TREC 2007, 2008. [Bakan66] Bakan, D.: The Test of Significance in Psychological Research, Psychological Bulletin, 66(6), pp.423-437, 1966.
- 191. References: C-D [Carterette12] Carterette, B.: Multiple Testing in Statistical Analysis of Systems-based Information Retrieval Experiments, ACM TOIS 30(1), 2012. [Carterette15] Carterette, B.: Bayesian Inference for Information Retrieval Evaluation, ACM ICTIR 2015, pp.31-40, 2015. [Cohen88] Cohen, J.: Statistical Power Analysis for the Behavioral Sciences (Second Edition), Psychology Press, 1988. [Cohen94] Cohen, J.: The Earth is Round (p<.05), American Psychologist, 49(12), pp.997-1003, 1994. [Crawley15] Crawly, M.J.: Statistics: An Introduction Using R (Second Edition), Wiley, 2015. [Deming75] Deming, W.E., On Probability as a Basic for Action, American Statistician, 29(4), pp.146-152, 1975.
- 192. References: G-H [Good05] Good, P.: Permutation, Parametric, and Bootstrap Tests of Hypothesis (Third Edition), Springer, 2005. [Harlow+16] Harlow, L.L., Mulaik, S.A., and Steiger, J.H.: What If There Were No Significance Tests? (Classic Edition), Routledge, 2016.
- 193. References: K-L [Kelley+12] Kelly, K. and Preacher, K.J.: On Effect Size, Psychological Methods, 17(2), pp.137-152, 2012. [Loftus91] On the Tyranny of Hypothesis Testing in the Social Sciences, Contemporary Psychology, 36(2), pp.102-105, 1991.
- 194. References: N-O [Nagata+97] Nagata, Y. and Yoshida, M.: Introduction to Multiple Comparison Procedures (in Japanese), Scientist press, 1997. [Nagata03] Nagata, Y.: How to Design the Sample Size (in Japanese), Asakura Shoten, 2003. [Olejnik+03] Olejnik, S. and Algina, J.: Generalized Eta and Omega Squared Statistics: Measures of Effect Size for Some Common Research Designs, Psychological Research, 8(4), pp.434-447, 2003.
- 195. References: R [Robertson81] Robertson, S.E.: The Methodology of Information Retrieval Experiment, In Sparck Jones, K. (ed.): Information Retrieval Experiment, Chapter 1, Butterworths, 1981. [Robertson12] Robertson, S.E., Kanoulas, E.: On Per- topic Variance in IR Evaluation, ACM SIGIR 2012, pp.891-900, 2012. [Rothman98] Rothman, K.J.: Writing for Epidemiology, Epidemiology, 9(3), pp.333-337, 1998.
- 196. References: S [Sakai06SIGIR] Sakai, T.: Evaluating Evaluation Metrics based on the Bootstrap, ACM SIGIR 2006, pp.525-532, 2006. [Sakai14PROMISE] Sakai, T.: Metrics, Statistics, Tests, PROMISE Winter School 2013: Bridging between Information Retrieval and Databases (LNCS 8173), Springer, 2014. [Sakai16IRJ] Topic Set Size Design, Information Retrieval, 19(3), pp.256-283, 2016. [Sakai16SIGIR] Sakai, T.: Statistical Significance, Power, and Sample Sizes: A Systematic Review of SIGIR and TOIS, 2006-2015, Proceedings of ACM SIGIR 2016, pp.5-14, 2016. [Sakai16SIGIRshort] Sakai, T.: Two Sample T-tests for IR Evaluation: Student or Welch?, Proceedings of ACM SIGIR 2016, pp.1045-1048, 2016. [Sakai17SIGIR] Sakai, T.: The Probability That Your Hypothesis Is Correct, Credible Intervals, and Effect Sizes for IR Evaluation, Proceedings of ACM SIGIR 2017, pp.25-34, 2017. [Sakai18book] Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes, and statistical power, to appear, Springer, 2018. [Salton+68] Salton, G. and Lesk, M.E.: Computer Evaluation of indexing and text processing, Journal of the ACM, 15, pp.8-36, 1968.
- 197. References: S (continued) [Savoy97] Savoy, J.: Statistical Inference in Retrieval Effectiveness Evaluation, Information Processing and Management, 33(4), pp.495-512, 1997. [Smucker+07] Smucker, M.D., Allan, J. and Carterette, B.: A Comparison of Statistical Significance Tests for Information Retrieval Evaluation, CIKM 2007, pp.623-632, 2007. [SparckJones+75] Sparck Jones, K. and van Rijsbergen, C.J.: Report on the Need for and Provision of an ‘Ideal’ Information Retrieval Test Collection, Computer Laboratory, University of Cambridge, British Library Research and Development Report No.5266, 1975. [SparckJones81] Sparck Jones, K. (ed.): Information Retrieval Experiment, Butterworths, 1981. [SparckJones+97] Sparck Jones, K. and Willet, P.: Readings in Information Retrieval, Morgan Kaufmann, 1997.
- 198. References: T [Toyoda09] Toyoda, H.: Introduction to Statistical Power Analysis: A Tutorial with R (in Japanese), Toyo Tosyo, 2009.
- 199. References: V-W [VanRijsbergen79] Van Rijsbergen, C.J., Information Retrieval, Chapter 7, Butterworths, 1979. [Voorhees09] Voorhees, E.M.: Topic Set Size Redux, Proceedings of ACM SIGIR 2009, pp.806-807, 2009. [Webber+08] Webber, W., Moffat, A., and Zobel, J.: Statistical Power in Retrieval Experimentation, Proceedings of CIKM 2008, pp.571-580, 2008.
- 200. References: Z [Ziliak+08] The Cult of Statistical Significance: How the Standard Error Costs Us Jobs, Justice, and Lives, The University of Michigan Press, 2008.

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