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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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General remarks
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


Association measures ams
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")


Using association scores
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


First steps proportions
First Steps: Proportions

  • Workshop on Mechanized Documentation (Washington, 1964)


First steps proportions1
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


First steps proportions2
First Steps: Proportions

  • harmonic mean

  • geometric mean

  • minimum

  • Jaccard


First steps proportions3
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


Expected frequencies
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:


Expected frequencies1
Expected Frequencies

  • expected frequencies for all cells of the contingency table

  • assuming random combinations( statistical independence)


Expected frequencies2
Expected Frequencies

  • comparison of expected against observed frequencies

  • note that row and column sums are the same for both tables


Mutual information
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


Low frequency pairs random variation
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


Low frequency pairs random variation1
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)


The statistical model random sample
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


The statistical model random sample1
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


The statistical model random sample2
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



Two footnotes
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


The sampling distribution
The Sampling Distribution

  • multinomial sampling distribution

  • each individual cell count Xij has a binomial distribution (but these are not independent)


The sampling distribution1
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)


Adequacy of the statistical model
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


Adequacy of the statistical model1
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)


Making assumptions about the population parameters
Making Assumptions about the Population Parameters

  • population parameters (, 1, 2) are unknown

  • best guess from observation: MLE = maximum-likelihood estimate


Making assumptions about the population parameters1
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


The null hypothesis
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


Likelihood measures
Likelihood Measures

  • probability of observed data under H0 (with MLE)

  • probability of single cell: X11 should be most "informative"


Likelihood measures1
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


Problems of likelihood measures
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


Problems of likelihood measures1
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


Likelihood ratios
Likelihood Ratios

  • simplest normalisation technique

  • divide maximum probability of data under H0 by unconstrained maximum probability

  • suggested by Dunning (1993)


Statistical hypothesis tests
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


Asymptotic tests
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  )


Asymptotic tests1
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


Two sided tests
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


Yates continuity correction
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




Yates continuity correction3
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


Asymptotic tests2
Asymptotic Tests

  • different form of chi-squared test (comparison of two binomials) is equivalent to independence test

  • special eq. with Yates' correction


Asymptotic tests3
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)


Something i d rather not mention
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


Something i d rather not mention1
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)


Exact tests
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


Exact tests1
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


Exact tests2
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


Comparing hypothesis tests
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


Other approaches to measuring association
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?


Implementation
Implementation

  • one-sided vs. two-sided tests

  • need special software to obtain p-values for asymptotic tests

  • numerical accuracy

  • beware of zero frequencies!


Errr help software
Errr.... Help!? Software?

  • Ted Pedersen's N-gram Statistics Package (NSP)[Perl, portable, easy to use]

  • UCS Toolkit will be available soon from www.collocations.de[Perl/Linux, some prerequisites, for the more ambitious :o) ]


More association measures
More Association Measures

  • lots of association measures

  • will be updated

  • references

  • slides from this course

  • under construction


Comparing association measures
Comparing Association Measures

  • mathematical discussion

    • very complex

    • results only for special cases

  • numerical simulation

    • computationally expensive

    • Dunning (1993, 1998)

  • lazy man's approach

    • construct mock data set where frequencies vary systematically













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