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Association Measures

Association Measures. Reminder: Contingency Tables. General Remarks. we will only use data from contingency tables we will consider each pair type on its own, independently from all other pair types (  no distributional information)

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Association Measures

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  1. Association Measures

  2. Reminder: Contingency Tables

  3. General Remarks • we will only use data from contingency tables • we will consider each pair typeon its own, independently from all other pair types( no distributional information) • we won't distinguish between relational and positional cooccurrences

  4. Association Measures (AMs) • goal: assign association score to each pair type = strength of association between components • high score = strong association • association in a statistical sense,but there is no precise definition • positive vs. negative association("colourless green ideas")

  5. Using Association Scores • absolute values (cut-off threshold) • input forhigher-order statistics(AMs are first-order statistics) scores should be meaningful • ranking of collocation candidates only relative scores matter • rank collocates of given base one marginal frequency fixed  only two free parameters

  6. First Steps: Proportions • Workshop on Mechanized Documentation (Washington, 1964)

  7. First Steps: Proportions • proportions between 0 and 1 • high proportion = strong (directional) association • need to combine two proportions into a single association score • average (P1 + P2) / 2 is not useful • f=1, f1=1, f2=1000 avg.=0.5005 • f=50, f1=100, f2=100  avg.=0.5  more "conservative" weighting

  8. First Steps: Proportions • harmonic mean • geometric mean • minimum • Jaccard

  9. First Steps: Proportions • coefficients range from 0 to 1 • 1 = total (positive) association • interpretation of lower scoresis less clear • positive vs. negative association? • which score for no association? • what is "no association"?? random combinations

  10. Expected Frequencies • assume that types u and v cooccur only by chance • f1(u) occs. of u and f2(v) occs. of v spread randomly over N tokens • each instance of u has a chance of f2(v)/N to cooccur with a v  expected # of cooccurrences:

  11. Expected Frequencies • expected frequencies for all cells of the contingency table • assuming random combinations( statistical independence)

  12. Expected Frequencies • comparison of expected against observed frequencies • note that row and column sums are the same for both tables

  13. Mutual Information • compares O11 with E11 • ratio O11/E11 ranges from 0 to  • 1 = no association (O11=E11) • usually logarithmic values • range: - to + • 0 = no assoc., < 0 neg., > 0 pos. • used in English lexicography

  14. Low-Frequency Pairs & Random Variation • large amount of low-frequency data (consequence of Zipf's law) • a simple (invented) example • A:f=50, f1=100, f2=100, N=1000 O11=50, E11=10,MI = log 5 • B:f=1, f1=1, f2=1, N=1000 O11=1, E11=.001, MI = log 1000

  15. Low-Frequency Pairs & Random Variation • three problems with case B • how meaningful is a single example? (not very much, actually) • could well be a spelling mistake or noise from automatic processing • we want to make generalisations (from particular corpus to "language")  this is the domain of statistics:draw inferences about population (=language) from a sample (=corpus)

  16. The Statistical Model:Random Sample • assumption: corpus data is a random sample from the language  base data is a random sample from all coocs. in the language

  17. The Statistical Model:Random Sample • random sample of size N is described by random variablesUi and Vi (i = 1..N), representing the labels of the i-th bigram token • notation: U and V as "prototypes" • for a given pair type (u,v), contingency table can becomputed from Ui and Vi  random variablesX11, X12, X21, X22

  18. The Statistical Model:Random Sample • population parameters11, 12, 21, 22 for pair type (u,v) • observed frequenciesO11, O12, O21, O22 represent one particular realisation of the sample • theory of random samples predicts distribution of X11, X12, X21, X22 from assumptions about the population parameters 11, 12, 21, 22

  19. The Statistical Model:Random Sample

  20. Two Footnotes • vector notation for cont. tables • population  general language • restricted to domain(s), genre(s), ...covered by source corpus • e.g. black box in computer science vs. newspapers vs. cooking

  21. The Sampling Distribution • multinomial sampling distribution • each individual cell count Xij has a binomial distribution (but these are not independent)

  22. The Sampling Distribution • given assumptions about the population parameters, we can compute the likelihood of the observed contingency table • relatively high likelihood= consistent with assumptions • relatively low likelihood= evidence against assumptions(inversely proportional to likelihood)

  23. Adequacy of the Statistical Model • particular sequence of pair tokens is irrelevant, only the overall frequencies matter ( sufficiency) • randomness assumption (random sample from fixed population) • independence of pair tokens • constancy of population parameters • violations problematic only when they affect sampling distribution

  24. Adequacy of the Statistical Model • three causes of non-randomness • local dependencies (e.g. syntax)  usually not problematic • inhomogeneity of source corpus(speakers, domains, topics, ...)  mixture population • repetition / clustering of bigrams  can be a serious problem(does not affect segment-based data if clustered within segments)

  25. Making Assumptions about the Population Parameters • population parameters (, 1, 2) are unknown • best guess from observation: MLE = maximum-likelihood estimate

  26. Making Assumptions about the Population Parameters • conditional probabilities with MLE • Dice coefficient etc. are MLE for population characteristics • MI is MLE for log( /(1  2))  unreliable for small frequencies

  27. The Null Hypothesis • null hypothesis H0: no association= independence of instances, i.e.P(U=u  V=v) = P(U=u)  P(V=v) • not all parameters determined • MLE maximise probability of observed data under H0

  28. Likelihood Measures • probability of observed data under H0 (with MLE) • probability of single cell: X11 should be most "informative"

  29. Likelihood Measures • small likelihood values = strong association • computed probabilities are often extremely small • use negative base-10 logarithm more convenient scale  high scores indicate strong association

  30. Problems of Likelihood Measures • three reasons for low likelihood • observed data is inconsistent with the null hypothesis because of strong association • association may also be negative (fewer coocs. than expected) • observed data is consistent, but probability mass is spread across many similar contingency tables

  31. Problems of Likelihood Measures • high frequency = low likelihood • e.g. binomial likelihood • O11=1, E11=1 L = 0.3679 • O11=1000, E11=1000 L = 0.0126 • O11=4, E11=1 L  0.0126 • need to "normalise" likelihood • NB: likelihood association measures often have good empirical results nonetheless

  32. Likelihood Ratios • simplest normalisation technique • divide maximum probability of data under H0 by unconstrained maximum probability • suggested by Dunning (1993)

  33. Statistical Hypothesis Tests • compute probability of group of outcomes instead of single one • observed contingency table is grouped with all tables that provide at least the same amount of evidence against H0 • total probability is known as the p-value or significance • problem: ranking of cont. tables

  34. Asymptotic Tests • asymptotic tests defined ranking of contingency tables explicitly • compute test statistic from data • higher values = more evidence against H0 • can use test statistic as an AM • theory: approximation of p-value associated with test statistic(accurate in the limit N  )

  35. Asymptotic Tests • standard test for independence is Pearson's chi-squared test • limiting distribution = 2 distribution with df=1 • number of degrees of freedom was subject of a long debate

  36. Two-Sided Tests • chi-squared test is two-sided, i.e. no difference between positive and negative association • ignore small number of pairs with (non-total) negative association • or convert to one-sided test:reject H0 only when O11 > E11 • p-value is usually divided by 2

  37. Yates Continuity Correction • Pearson's chi-squared test approximates discrete binomial distributions of each cell by continuous normal distribution( "normal theory") • estimating probabilities P(Xij  k) from normal distribution introduces systematic errors

  38. Yates' Continuity Correction

  39. Yates' Continuity Correction

  40. Yates' Continuity Correction • generic form of Yates' continuity correction for contingency tables • usefulness is still controversial (criticised as too conservative) • applicability for chi-squared test is generally accepted

  41. Asymptotic Tests • different form of chi-squared test (comparison of two binomials) is equivalent to independence test • special eq. with Yates' correction

  42. Asymptotic Tests • can also use log-likelihood ratio as a test statistic (two-sided) • limiting distribution is found to be 2 distribution with df=1 • more conservative than Pearson's chi-squared test • Dunning (1993) showed that Pearson's test over-estimates evidence against H0 (simulation)

  43. Something I'd Rather Not Mention • Church & Hanks: O11 and E11are both random variables • H0: expected values are equal • assume normal distribution with unknown variance • compare O11 and E11 with Student's t-test, estimating unknown variance from the observed data

  44. Something I'd Rather Not Mention • one-sided test • statistical model is questionable • limiting distribution: t-distribution with df  N • even more conservative than log-likelihood (low-frequency data)

  45. Exact Tests • problem: how to establish ranking of contingency tables • solution: reduce set of alternatives • if we consider only the cell X11,the difference X11 – E11 gives a sensible ranking: binomial test

  46. Exact Tests • another solution: marginal frequencies do not provide evidence for or against H0( "ancillary" statistics) • condition on fixed row and column sums R1, R2, C1, C2 • conditional hypergeometric distribution does not depend on parameters 1 and 2

  47. Exact Tests • X11 is the only free parameter • we can use X11 – E11 for ranking • Fisher's exact test (Pedersen 1996) • computationally expensive • numerical difficulties

  48. Comparing Hypothesis Tests • Fisher's test is now widely accepted as most appropriate • tends to be conservative • log-likelihood gives good approximation of "correct" p-values(slightly less conservative) • chi-squared over-estimates • t-score far too conservative

  49. Other Approaches to Measuring Association • information-theoretic (MI, entropy) equivalent to log-likelihood • combined measures ("boosting") • conservative estimates instead of MLE (confidence intervals) • hypothesis tests with different null hypothesis:  = C  1  2 • mixture of conservative estimates and hypothesis tests?

  50. Implementation • one-sided vs. two-sided tests • need special software to obtain p-values for asymptotic tests • numerical accuracy • beware of zero frequencies!

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