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Likelihood measures belong to the **significance of association** group.
They equate the amount of evidence against the null hypothesis of independence
with the probability (or **likelihood**) of the observed cooccurrence frequencies
under the null hypothesis. The smaller this probability, the more unusal the
observed outcome and, consequently, the more evidence there is against H_{0}.
Most measures in this group are based on the point null hypothesis so that the
sampling distribution is uniquely defined. Recall that normally the
**negative base 10 logarithm** of the likelihood is used as an association
score, while the equations given below compute the non-logarithmic value.
All likelihood measures are **two-sided**, i.e. high scores may indicate
either positive or negative association.

The multinomial-likelihood measure computes the probability of the observed contingency table under the point null hypothesis. This value depends on all four cells of the table.

The joint frequency O_{11} provides the most direct evidence
for the association of a pair type. The binomial-likelihood measure
computes the probability of the observed joint frequency *regardless* of the
other cells of the contingency table.^{(1)}

Since the cooccurrence probability E_{11} ⁄ N under the
point null hypothesis is usually small, the binomial distribution of
X_{11} can be approximated by a Poisson distribution, which
is both mathematically and numerically more convenient. This leads to the
Poisson-likelihood measure.

Quasthoff & Wolff (2002) describe an approximation to the negative logarithm of Poisson-likelihood using Stirling's formula. This leads to the Poisson-Stirling measure which has to be scaled appropriately (divide the value shown below by log(10)) in order to allow direct comparison with the other likelihood measures (using the negative-base-10-logarithm convention).

It is possible to avoid the use of maximum-likelihood estimates and the point null hypothesis
by conditioning the sampling distribution on the observed row and column totals. The resulting
hypergeometric distribution under the general null hypothesis H_{0}
does not depend on the particular values of the "nuisance" parameters
π_{1} and π_{2}.
The conditional probability of the observed contingency table defines the
hypergeometric-likelihood association measure.

None of the likelihood measures have found widespread use, with the possible exception of Poisson-Stirling,

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