In statistics, Gtests are likelihoodratio or maximum likelihood statistical significance tests that are increasingly being used in situations where chisquared tests were previously recommended.^{[1]}
The general formula for G is

G = 2\sum_{i} {O_{i} \cdot \ln\left(\frac{O_i}{E_i}\right)},
where O_{i} is the observed frequency in a cell, E_{i} is the expected frequency under the null hypothesis, ln denotes the natural logarithm, and the sum is taken over all nonempty cells.
Gtests have been recommended at least since the 1981 edition of the popular statistics textbook by Robert R. Sokal and F. James Rohlf.^{[2]}
Contents

Distribution and usage 1

Relation to the chisquared test 2

Relation to Kullback–Leibler divergence 3

Relation to mutual information 4

Application 5

Statistical software 6

References 7

External links 8
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 chisquared distribution, with the same number of degrees of freedom as in the corresponding chisquared 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 Gtest .
Relation to the chisquared test
The commonly used chisquared tests for goodness of fit to a distribution and for independence in contingency tables are in fact approximations of the loglikelihood ratio on which the Gtests are based. The general formula for Pearson's chisquared test statistic is

\chi^2 = \sum_{i} {\frac{\left(O_i  E_i\right)^2}{E_i}} .
The approximation of G by chi squared is obtained by a second order Taylor expansion of the natural logarithm around 1. This approximation was developed by Karl Pearson because at the time it was unduly laborious to calculate loglikelihood ratios. With the advent of electronic calculators and personal computers, this is no longer a problem. A derivation of how the chisquared test is related to the Gtest and likelihood ratios, including to a full Bayesian solution is provided in Hoey (2012).^{[3]}
For samples of a reasonable size, the Gtest and the chisquared test will lead to the same conclusions. However, the approximation to the theoretical chisquared distribution for the Gtest is better than for the Pearson's chisquared test.^{[4]} In cases where O_i >2 \cdot E_i for some cell case the Gtest is always better than the chisquared test.
For testing goodnessoffit the Gtest 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 Hodge and Lehman.^{[5]}^{[6]}
Relation to Kullback–Leibler divergence
The Gtest quantity is proportional to the Kullback–Leibler divergence of the empirical distribution from the theoretical distribution.
Relation to mutual information
For analysis of contingency tables the value of G can also be expressed in terms of mutual information.
Let

N = \sum_{ij}{O_{ij}} \; , \; \pi_{ij} = \frac{O_{ij}}{N} \; , \; \pi_{i.} = \frac{\sum_j O_{ij}}{N} \; , and \; \pi_{. j} = \frac{\sum_i O_{ij}}{N} \;.
Then G can be expressed in several alternative forms:

G = 2 \cdot N \cdot \sum_{ij}{\pi_{ij} \left( \ln(\pi_{ij})\ln(\pi_{i.})\ln(\pi_{.j}) \right)} ,

G = 2 \cdot N \cdot \left[ H(r) + H(c)  H(r,c) \right] ,

G = 2 \cdot N \cdot MI(r,c) \, ,
where the entropy of a discrete random variable X \, is defined as

H(X) =  {\sum_{x \in \text{Supp}(X)} p(x) \log p(x)} \, ,
and where

MI(r,c)= H(r) + H(c)  H(r,c) \,
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
Statistical software

The R programming language has the likelihood.test function in the Deducer package.

In SAS, one can conduct Gtest by applying the
/chisq
option after the proc freq
.^{[8]}

In Stata, one can conduct a Gtest by applying the
lr
option after the tabulate
command.

Fisher's Gtest in the GeneCycle Package of the R programming language (fisher.g.test) does not implement the Gtest as described in this article, but rather Fisher's exact test of Gaussian whitenoise in a time series.^{[9]}
References

^ McDonald, J.H. (2014). "G–test of goodnessoffit". Handbook of Biological Statistics (Third ed.). Baltimore, Maryland: Sparky House Publishing. pp. 53–58.

^ Sokal, R. R.; Rohlf, F. J. (1981). Biometry: The Principles and Practice of Statistics in Biological Research (Second ed.). New York: Freeman.

^ Hoey, J. (2012). "The TwoWay Likelihood Ratio (G) Test and Comparison to TwoWay ChiSquared Test".

^ Harremoës, P.; Tusnády, G. (2012). "Information divergence is more chi squared distributed than the chi squared statistic". Proceedings ISIT 2012. pp. 538–543.

^ Quine, M. P.; Robinson, J. (1985). "Efficiencies of chisquare and likelihood ratio goodnessoffit tests".

^ Harremoës, P.; Vajda, I. (2008). "On the Bahadurefficient testing of uniformity by means of the entropy".

^ Dunning, Ted (1993). "Accurate Methods for the Statistics of Surprise and Coincidence", Computational Linguistics, Volume 19, issue 1 (March, 1993).

^ Gtest of independence, Gtest for goodnessoffit in Handbook of Biological Statistics, University of Delaware. (pp. 46–51, 64–69 in: McDonald, J. H. (2009) Handbook of Biological Statistics (2nd ed.). Sparky House Publishing, Baltimore, Maryland.)

^ Fisher, R. A. (1929), "Tests of significance in harmonic analysis", Proceedings of the Royal Society of London: Series A, Volume 125, Issue 796, pp. 54–59.
External links

/Loglikelihood calculator^{2}G
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