Likelihood Ratio Tests

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Likelihood Ratio Tests

• Vasileios Hatzivassiloglou
• University of Texas at Dallas

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Aligned French-English words

• Uses an aligned corpus at the sentence level

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Chi-square test results

• X2 for new companies example 1.55
• X2 for vache cow example 456,400
• Critical values
• 3.84 for 5 significance
• 6.63 for 1 significance
• 10.83 for 0.1 significance

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Goodness-of-fit test

• We can also apply Pearsons chi-square as a
  goodness-of-fit test
• X and Y are two random variables with the same
  range of categorical values
• Then we construct a 2n table (n is the number of
  possible values) and compare if Xi and Yi have
  similar values

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Application corpus comparison

• We use as Xi and Yi the word counts of different
  words i in two different corpora
• Then the chi-square test can be used to determine
  if the corpora are similar
• We usually limit the words i to those with
  reasonable overall frequency

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Likelihood ratio tests

• The likelihood ratio of two hypotheses is the
  ratio of the probabilities that we would have
  seen the observed data under each of the
  hypotheses

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Likelihood ratio advantages

• It is interpretable directly as the odds in favor
  of H1 versus H2
• It is a more robust statistic than ?2, and
  therefore more reliable when the assumptions for
  the chi-square test are not met (e.g., rare
  events and low expected cell counts)

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Likelihood ratio for association

• We are testing if two words w1 and w2 are
  associated (e.g., in a collocation)
• Hypothesis 1 No association
• P(w2 w1) P(w2 w1) p
• Hypothesis 2 Baseline / no assumption
• No particular relation between p1 P(w2 w1)
  and p2 P(w2 w1)

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Estimating the parameters

• We use maximum likelihood estimates
• Then

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Calculating the likelihoods

• The distribution is binomial (with different
  parameters in each case)
• P(data) P(w2 after w1 data) P(w2 after w1
  data)
• We observed w2 after w1 c(w1w2) times out of
  c(w1)
• We observed w2 after something other than w1
  c(w2)-c(w1w2) times out of N-c(w1) times

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Component likelihoods

• Under hypothesis 1,
• P(w2 after w1 data) b(c12 c1, p)
• P(w2 after w1 data) b(c2-c12 N-c1, p)
• Under hypothesis 2,
• P(w2 after w1 data) b(c12 c1, p1)
• P(w2 after w1 data) b(c2-c12 N-c1, p2)
• where

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Likelihood ratio formula
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Reading

• Section 5.3.4 on log-likelihood tests