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Statistics for Linguists with R a SIGIL course Unit 8: Non-Randomness of Corpus Data & Generalised Linear Models Marco Baroni 1 & Stefan Evert 2 http://purl.org/stefan.evert/SIGIL 1 Center for Mind/Brain Sciences, University of Trento

  1. 10 random sample percentage of samples with X=k n = 100 8 6 4 2 0 0 5 10 15 20 25 30 35 40 45 50 value k of observed frequency X 10 pooled data percentage of samples with X=k 2 × n = 50 8 6 4 2 0 0 5 10 15 20 25 30 35 40 45 50 17 value k of observed frequency X

  2. 10 random sample percentage of samples with X=k n = 100 8 6 4 2 0 0 5 10 15 20 25 30 35 40 45 50 value k of observed frequency X 10 pooled data percentage of samples with X=k 2 × n = 50 8 6 4 2 0 0 5 10 15 20 25 30 35 40 45 50 17 value k of observed frequency X

  3. Duplicates 18

  4. Duplicates ◆ Duplication = extreme form of non-randomness • Did you know the British National Corpus contains duplicates of entire texts (under different names)? 18

  5. Duplicates ◆ Duplication = extreme form of non-randomness • Did you know the British National Corpus contains duplicates of entire texts (under different names)? ◆ Duplicates can appear at any level • The use of keys to move between fields is fully described in Section 2 and summarised in Appendix A 18

  6. Duplicates ◆ Duplication = extreme form of non-randomness • Did you know the British National Corpus contains duplicates of entire texts (under different names)? ◆ Duplicates can appear at any level • The use of keys to move between fields is fully described in Section 2 and summarised in Appendix A • 117 (!) occurrences in BNC, all in file HWX • very difficult to detect automatically 18

  7. Duplicates ◆ Duplication = extreme form of non-randomness • Did you know the British National Corpus contains duplicates of entire texts (under different names)? ◆ Duplicates can appear at any level • The use of keys to move between fields is fully described in Section 2 and summarised in Appendix A • 117 (!) occurrences in BNC, all in file HWX • very difficult to detect automatically ◆ Even worse for newspapers & Web corpora • see Evert (2004) for examples 18

  8. 3 Measuring non-randomness 19

  9. A sample of random samples is a random sample ◆ Larger unit of sampling is not the original cause of non-randomness • if each text in a corpus is a genuinely random sample from the same population, then the pooled data also form a random sample • we can illustrate this with a thought experiment 20

  10. The random library 21

  11. The random library ◆ Suppose there’s a vandal in the library 21

  12. The random library ◆ Suppose there’s a vandal in the library • who cuts up all books into single sentences and leaves them in a big heap on the floor 21

  13. The random library ◆ Suppose there’s a vandal in the library • who cuts up all books into single sentences and leaves them in a big heap on the floor • the next morning, the librarian takes a handful of sentences from the heap, fills them into a book-sized box, and puts the box on one of the shelves 21

  14. The random library ◆ Suppose there’s a vandal in the library • who cuts up all books into single sentences and leaves them in a big heap on the floor • the next morning, the librarian takes a handful of sentences from the heap, fills them into a book-sized box, and puts the box on one of the shelves • repeat until the heap of sentences is gone ➡ library of random samples 21

  15. The random library ◆ Suppose there’s a vandal in the library • who cuts up all books into single sentences and leaves them in a big heap on the floor • the next morning, the librarian takes a handful of sentences from the heap, fills them into a book-sized box, and puts the box on one of the shelves • repeat until the heap of sentences is gone ➡ library of random samples ◆ Pooled data from 2 (or more) boxes form a perfectly random sample of sentences from the original library! 21

  16. A sample of random samples is a random sample 22

  17. A sample of random samples is a random sample ◆ The true cause of non-randomness • discrepancy between unit of sampling and unit of measurement only leads to non-randomness if the sampling units (i.e. the corpus texts) are not random samples themselves (from same population) • with respect to specific phenomenon of interest 22

  18. A sample of random samples is a random sample ◆ The true cause of non-randomness • discrepancy between unit of sampling and unit of measurement only leads to non-randomness if the sampling units (i.e. the corpus texts) are not random samples themselves (from same population) • with respect to specific phenomenon of interest ◆ No we know how to measure non-randomness • find out if corpus texts are random samples • i.e., if they follow a binomial sampling distribution ➡ tabulate observed frequencies across corpus texts 22

  19. Measuring non-randomness ◆ Tabulate number of texts with k passives • illustrated for subsets of Brown/LOB (310 texts each) • meaningful because all texts have the same length ◆ Compare with binomial distribution • for population proportion H 0 : π = 21.1% (Brown) and π = 22.2% (LOB); approx. n = 100 sentences per text • estimated from full corpus ➞ best possible fit ◆ Non-randomness ➞ larger sampling variation 23

  20. Passives in the Brown corpus 35 AmE 30 binomial 25 number of texts 20 15 10 5 0 0 5 10 15 20 25 30 35 40 45 50 55 60 observed frequency k of passives 24

  21. Passives in the Brown corpus 35 35 AmE AmE 30 30 binomial binomial 25 25 number of texts number of texts 20 20 AmE binomial 15 15 10 10 5 5 0 0 0 0 5 5 10 10 15 15 20 20 25 25 30 30 35 35 40 40 45 45 50 50 55 55 60 60 observed frequency k of passives observed frequency k of passives 24

  22. Passives in the LOB corpus 35 BrE 30 binomial 25 number of texts 20 BrE binomial 15 10 5 0 0 5 10 15 20 25 30 35 40 45 50 55 60 observed frequency k of passives 25

  23. number of chunks 0 200 400 600 800 1000 0 5 10 chunk frequency Tag 15 20 25 30 Data from Frankfurter Rundschau corpus, divided into 10,000 equally-sized chunks 26

  24. number of chunks 0 200 400 600 800 1000 0 5 10 chunk frequency Tag 15 20 25 30 number of chunks 0 200 400 600 800 1000 0 5 10 chunk frequency Zeit 15 20 25 30 Data from Frankfurter Rundschau corpus, divided into 10,000 equally-sized chunks 26

  25. number of chunks number of chunks 0 200 400 600 800 1000 0 200 400 600 800 1000 0 0 5 5 10 10 chunk frequency chunk frequency Polizei Tag 15 15 20 20 25 25 30 30 number of chunks 0 200 400 600 800 1000 0 5 10 chunk frequency Zeit 15 20 25 30 Data from Frankfurter Rundschau corpus, divided into 10,000 equally-sized chunks 26

  26. number of chunks number of chunks 0 200 400 600 800 1000 0 200 400 600 800 1000 0 0 5 5 10 10 chunk frequency chunk frequency Polizei Tag 15 15 20 20 25 25 30 30 number of chunks number of chunks 0 200 400 600 800 1000 0 200 400 600 800 1000 0 0 5 50 10 chunk frequency chunk frequency Uhr Zeit 100 15 20 150 25 200 30 Data from Frankfurter Rundschau corpus, divided into 10,000 equally-sized chunks 26

  27. 4 Consequences 27

  28. Consequences of non- randomness 28

  29. Consequences of non- randomness ◆ Accept that corpus is a sample of texts • data cannot be pooled into random sample of tokens • results in much smaller sample size … (BNC: 4,048 texts rather than 6,023,627 sentences) • … but more informative measurements (relative frequencies on interval rather than nominal scale) 28

  30. Consequences of non- randomness ◆ Accept that corpus is a sample of texts • data cannot be pooled into random sample of tokens • results in much smaller sample size … (BNC: 4,048 texts rather than 6,023,627 sentences) • … but more informative measurements (relative frequencies on interval rather than nominal scale) ◆ Use statistical techniques that account for the overdispersion of relative frequencies • Gaussian distribution allows us to estimate spread (variance) independently from location • Standard technique: Student’s t-test 28

  31. A case study: Passives in AmE and BrE 29

  32. A case study: Passives in AmE and BrE ◆ Are there more passives in BrE than in AmE? • based on data from subsets of Brown and LOB - 9 categories: press reports, editorials, skills & hobbies, misc., learned, fiction, science fiction, adventure, romance - ca. 310 texts / 31,000 sentences / 720,000 words each 29

  33. A case study: Passives in AmE and BrE ◆ Are there more passives in BrE than in AmE? • based on data from subsets of Brown and LOB - 9 categories: press reports, editorials, skills & hobbies, misc., learned, fiction, science fiction, adventure, romance - ca. 310 texts / 31,000 sentences / 720,000 words each ◆ Pooled data (random sample of sentences) • AmE: 6584 out of 31,173 sentences = 21.1% • BrE: 7091 out of 31,887 sentences = 22.2% 29

  34. A case study: Passives in AmE and BrE ◆ Are there more passives in BrE than in AmE? • based on data from subsets of Brown and LOB - 9 categories: press reports, editorials, skills & hobbies, misc., learned, fiction, science fiction, adventure, romance - ca. 310 texts / 31,000 sentences / 720,000 words each ◆ Pooled data (random sample of sentences) • AmE: 6584 out of 31,173 sentences = 21.1% • BrE: 7091 out of 31,887 sentences = 22.2% ◆ Chi-squared test ( ➞ pooled data, binomial) vs. t-test ( ➞ sample of texts, Gaussian) 29

  35. Let’s do that in R … # passive counts for each text in Brown and LOB corpus > Passives <- read.delim("passives_by_text.tbl") # display 10 random rows to get an idea of the table layout > Passives[sample(nrow(Passives), 10), ] # add relative frequency of passives in each file (as percentage) > Passives <- transform(Passives, relfreq = 100 * passive / n_s) # split into separate data frames for Brown and LOB texts > Brown <- subset(Passives, lang=="AmE") > LOB <- subset(Passives, lang=="BrE") 30

  36. A case study: Passives in AmE and BrE 31

  37. A case study: Passives in AmE and BrE ◆ Chi-squared test: highly significant • p-value: .00069 < .001 • confidence interval for difference: 0.5% – 1.8% • large sample ➞ large amount of evidence 31

  38. A case study: Passives in AmE and BrE ◆ Chi-squared test: highly significant • p-value: .00069 < .001 • confidence interval for difference: 0.5% – 1.8% • large sample ➞ large amount of evidence ◆ R code: pooled counts + proportions test > passives.B <- sum(Brown$passive) > n_s.B <- sum(Brown$n_s) > passives.L <- sum(LOB$passive) > n_s.L <- sum(LOB$n_s) > prop.test(c(passives.L, passives.B), c(n_s.L, n_s.B)) 31

  39. A case study: Passives in AmE and BrE 32

  40. A case study: Passives in AmE and BrE ◆ t-test: not significant • p-value: .1340 > .05 ( t =1.50, df=619.96) • confidence interval for difference: -0.6% – +4.9% • H 0 : same average relative frequency in AmE and BrE 32

  41. A case study: Passives in AmE and BrE ◆ t-test: not significant • p-value: .1340 > .05 ( t =1.50, df=619.96) • confidence interval for difference: -0.6% – +4.9% • H 0 : same average relative frequency in AmE and BrE ◆ R code: apply t.test() function > t.test(LOB$relfreq, Brown$relfreq) # alternative syntax: “formula” interface > t.test(relfreq ~ lang, data=Passives) 32

  42. 33

  43. What are we really testing? 34

  44. What are we really testing? ◆ Are population proportions meaningful? • corpus should be balanced and representative (broad coverage of genres, … in appropriate proportions) • average frequency depends on composition of corpus • e.g. 18% passives in written BrE / 4% in spoken BrE 34

  45. What are we really testing? ◆ Are population proportions meaningful? • corpus should be balanced and representative (broad coverage of genres, … in appropriate proportions) • average frequency depends on composition of corpus • e.g. 18% passives in written BrE / 4% in spoken BrE ◆ How many passives are there in English? 34

  46. What are we really testing? ◆ Are population proportions meaningful? • corpus should be balanced and representative (broad coverage of genres, … in appropriate proportions) • average frequency depends on composition of corpus • e.g. 18% passives in written BrE / 4% in spoken BrE ◆ How many passives are there in English? • 50% written / 50% spoken: π = 13.0% 34

  47. What are we really testing? ◆ Are population proportions meaningful? • corpus should be balanced and representative (broad coverage of genres, … in appropriate proportions) • average frequency depends on composition of corpus • e.g. 18% passives in written BrE / 4% in spoken BrE ◆ How many passives are there in English? • 50% written / 50% spoken: π = 13.0% • 90% written / 10% spoken: π = 16.6% 34

  48. What are we really testing? ◆ Are population proportions meaningful? • corpus should be balanced and representative (broad coverage of genres, … in appropriate proportions) • average frequency depends on composition of corpus • e.g. 18% passives in written BrE / 4% in spoken BrE ◆ How many passives are there in English? • 50% written / 50% spoken: π = 13.0% • 90% written / 10% spoken: π = 16.6% • 20% written / 80% spoken: π = 6.8% 34

  49. Average relative frequency? press reportage press editorial skills / hobbies miscellaneous learned general fiction science fiction adventure romance 80 ● relative frequency of passives (%) ● 60 ● ● ● ● ● 40 ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE 35

  50. Average relative frequency? > library(lattice) > bwplot(relfreq ~ lang | genre, press reportage press editorial skills / hobbies miscellaneous learned general fiction science fiction adventure romance data=Passives) # bw = "Box and Whiskers" 80 ● relative frequency of passives (%) ● 60 ● ● ● ● ● 40 ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE 35

  51. Average relative frequency? press reportage press editorial skills / hobbies miscellaneous learned general fiction science fiction adventure romance 80 ● relative frequency of passives (%) ● 60 ● ● ● ● ● 40 ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE 36

  52. Average relative frequency? press reportage press editorial skills / hobbies miscellaneous learned general fiction science fiction adventure romance 80 ● relative frequency of passives (%) ● 60 ● ● ● ● ● 40 ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE 36

  53. Average relative frequency? press reportage press editorial skills / hobbies miscellaneous learned general fiction science fiction adventure romance 80 ● relative frequency of passives (%) ● 60 ● ● ● ● ● 40 ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE 36

  54. Problems with statistical inference statistical inference random population sample library metaphor (extensional def.) corpus linguistic hypothesis data question problem operationalisation 37

  55. Problems with statistical inference statistical inference random population sample library metaphor (extensional def.) corpus linguistic hypothesis data question problem operationalisation 37

  56. 5 Rethinking corpus frequencies 38

  57. Studying variation in language ◆ It seems absurd now to measure & compare relative frequencies in “language” (= library) • proportion π depends more on composition of library than on properties of the language itself ◆ Quantitative corpus analysis has to account for the variation of relative frequencies between individual texts (cf. Gries 2006) • research question ➞ one factor behind this variation 39

  58. Studying variation in language 40

  59. Studying variation in language ◆ Approach 1: restrict study to sublanguage in order to eliminate non-randomness • data from this sublanguage (= single section in library) can be pooled into large random sample 40

  60. Studying variation in language ◆ Approach 1: restrict study to sublanguage in order to eliminate non-randomness • data from this sublanguage (= single section in library) can be pooled into large random sample ◆ Approach 2: goal of quantitative corpus analysis is to explain variation between texts in terms of • random sampling (of tokens within text) • stylistic variation: genre, author, domain, register, … • subject matter of text ➞ term clustering effects • differences between language varieties research question 40

  61. Eliminating non-randomness press reportage press editorial skills / hobbies miscellaneous learned general fiction science fiction adventure romance 80 ● relative frequency of passives (%) ● 60 ● ● ● ● ● 40 ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE 41

  62. Eliminating non-randomness press reportage press editorial skills / hobbies miscellaneous learned general fiction science fiction adventure romance 80 ● relative frequency of passives (%) ● 60 ● ● ● ● ● 40 ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE 41

  63. Eliminating non-randomness press reportage press editorial skills / hobbies miscellaneous learned general fiction science fiction adventure romance 80 ● X 2 = 6.83 ** relative frequency of passives (%) ● 60 ● ● ● ● ● 40 ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE 41

  64. Eliminating non-randomness press reportage press editorial skills / hobbies miscellaneous learned general fiction science fiction adventure romance 80 ● t = 2.38 * t = 2.34 * X 2 = 6.83 ** relative frequency of passives (%) ● 60 ● ● ● ● ● 40 ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE AmE BrE 41

  65. Explaining variation 42

  66. Explaining variation ◆ Statisticians explain variation with the help of linear models (and other statistical models) • linear models predict response (“dependent variable”) from one or more factors (“independent variables”) • simplest model: linear combination of factors 42

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The Redistributive Consequences of Segregation Lisa Windsteiger Max Planck Institute for Tax Law and Public Finance First WID World Conference, December 2017 Lisa Windsteiger (MPI) Segregation and Redistribution December 2017 Introduction

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