G-test


In statistics, G-tests are likelihood-ratio or maximum likelihood statistical significance tests that are increasingly being used in situations where chi-squared tests were previously recommended.
The general formula for G is
where is the observed count in a cell, is the expected count under the null hypothesis, denotes the natural logarithm, and the sum is taken over all non-empty cells. Furthermore, the total observed count should be equal to the total expected count:where is the total number of observations.
G-tests have been recommended at least since the 1981 edition of Biometry, a statistics textbook by Robert R. Sokal and F. James Rohlf.

Derivation

We can derive the value of the G-test from the log-likelihood ratio test where the underlying model is a multinomial model.
Suppose we had a sample where each is the number of times that an object of type was observed. Furthermore, let be the total number of objects observed. If we assume that the underlying model is multinomial, then the test statistic is defined bywhere is the null hypothesis and is the maximum likelihood estimate of the parameters given the data. Recall that for the multinomial model, the MLE of given some data is defined byFurthermore, we may represent each null hypothesis parameter asThus, by substituting the representations of and in the log-likelihood ratio, the equation simplifies toRelabel the variables with and with. Finally, multiply by a factor of to achieve the form

Distribution and usage

Given the null hypothesis that the observed frequencies result from random sampling from a distribution with the given expected frequencies, the distribution of G is approximately a chi-squared distribution, with the same number of degrees of freedom as in the corresponding chi-squared test.
For very small samples the multinomial test for goodness of fit, and Fisher's exact test for contingency tables, or even Bayesian hypothesis selection are preferable to the G-test. McDonald recommends to always use an exact test if the total sample size is less than 1000.

Relation to the chi-squared test

The commonly used chi-squared tests for goodness of fit to a distribution and for independence in contingency tables are in fact approximations of the log-likelihood ratio on which the G-tests are based. The general formula for Pearson's chi-squared test statistic is
The approximation of G by chi squared is obtained by a second order Taylor expansion of the natural logarithm around 1. With the advent of electronic calculators and personal computers, this is no longer a problem. A derivation of how the chi-squared test is related to the G-test and likelihood ratios, including to a full Bayesian solution is provided in Hoey.
For samples of a reasonable size, the G-test and the chi-squared test will lead to the same conclusions. However, the approximation to the theoretical chi-squared distribution for the G-test is better than for the Pearson's chi-squared test. In cases where for some cell case the G-test is always better than the chi-squared test.
For testing goodness-of-fit the G-test is infinitely more efficient than the chi squared test in the sense of Bahadur, but the two tests are equally efficient in the sense of Pitman or in the sense of Hodges and Lehmann.

Relation to Kullback–Leibler divergence

The G-test statistic is proportional to the Kullback–Leibler divergence of the theoretical distribution from the empirical distribution:
where N is the total number of observations and and are the empirical and theoretical frequencies, respectively.

Relation to mutual information

For analysis of contingency tables the value of G can also be expressed in terms of mutual information.
Let
Then G can be expressed in several alternative forms:
where the entropy of a discrete random variable is defined as
and where
is the mutual information between the row vector r and the column vector c of the contingency table.
It can also be shown that the inverse document frequency weighting commonly used for text retrieval is an approximation of G applicable when the row sum for the query is much smaller than the row sum for the remainder of the corpus. Similarly, the result of Bayesian inference applied to a choice of single multinomial distribution for all rows of the contingency table taken together versus the more general alternative of a separate multinomial per row produces results very similar to the G statistic.

Application