G-Test

See: Likelihood Ratio, Maximum Likelihood.

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References

2011

• http://en.wikipedia.org/wiki/G-test
  • QUOTE: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 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. This approximation was developed by Karl Pearson because at the time it was unduly laborious to calculate log-likelihood ratios. With the advent of electronic calculators and personal computers, this is no longer a problem. G-tests are coming into increasing use, particularly since they were recommended at least since the 1981 edition of the popular statistics textbook by Sokal and Rohlf. Dunning introduced the test to the computational linguistics community where it is now widely used.

    The general formula for Pearson's chi-squared test statistic is [math]\displaystyle{ \Chi^2 = \sum_{ij} {(O_{ij} - E_{ij})^2 \over E_{ij}} , }[/math] where O_i is the frequency observed in a cell, E is the frequency expected on the null hypothesis, and the sum is taken across all cells. The corresponding general formula for G is [math]\displaystyle{ G = 2\sum_{ij} {O_{ij} \cdot \ln(O_{ij}/E_{ij}) }, }[/math] where ln denotes the natural logarithm (log to the base e) and the sum is again taken over all non-empty cells.

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