Instatistics,G-tests arelikelihood-ratio ormaximum likelihoodstatistical significance tests that are increasingly being used in situations wherechi-squared tests were previously recommended.^[1]

The general formula forG is

${\displaystyle G=2\sum _i{O}_i\cdot \ln \left(\frac{O_i}{E_i}\right),}$

where$O_i\ge 0$ is the observed count in a cell,$E_i>0$ is the expected count under thenull hypothesis,$\ln$ denotes thenatural logarithm, and the sum is taken over all non-empty cells. The resulting$G$ ischi-squared distributed.

Furthermore, the total observed count should be equal to the total expected count:$${\displaystyle \sum _i{O}_i=\sum _i{E}_i=N}$$where$N$ is the total number of observations.

We can derive the value of theG-test from thelog-likelihood ratio test where the underlying model is a multinomial model.

Suppose we had a sample$x=(x_1,\dots ,x_m)$ where each$x_i$ is the number of times that an object of type$i$ was observed. Furthermore, let$n=\sum _{i=1}^m{x}_i$ be the total number of objects observed. If we assume that the underlying model is multinomial, then the test statistic is defined by$${\displaystyle \ln \left(\frac{L(\tilde{\theta }|x)}{L(\hat{\theta }|x)}\right)=\ln \left(\frac{\prod _{i=1}^m{\tilde{\theta }}_i^{x_i}}{\prod _{i=1}^m{\hat{\theta }}_i^{x_i}}\right)}$$where$\tilde{\theta }$ is the null hypothesis and${\displaystyle \hat{\theta }}$ is themaximum likelihood estimate (MLE) of the parameters given the data. Recall that for the multinomial model, the MLE of${\hat{\theta }}_i$ given some data is defined by$${\displaystyle {\hat{\theta }}_i=\frac{x_i}{n}}$$Furthermore, we may represent each null hypothesis parameter${\displaystyle {\tilde{\theta }}_i}$ as$${\displaystyle {\tilde{\theta }}_i=\frac{e_i}{n}}$$Thus, by substituting the representations of$\tilde{\theta }$ and$\hat{\theta }$ in the log-likelihood ratio, the equation simplifies toRelabel the variables$e_i$ with$E_i$ and$x_i$ with$O_i$. Finally, multiply by a factor of$-2$ (used to make the G test formulaasymptotically equivalent to the Pearson's chi-squared test formula) to achieve the form

Heuristically, one can imagine${\displaystyle O_i}$ as continuous and approaching zero, in which case${\displaystyle O_i\ln O_i\to 0,}$ and terms with zero observations can simply be dropped. However theexpected count in each cell must be strictly greater than zero for each cell (${\displaystyle E_i>0\mathrm{\forall }i}$) to apply the method.

Given the null hypothesis that the observed frequencies result from random sampling from a distribution with the given expected frequencies, thedistribution ofG is approximately achi-squared distribution, with the same number ofdegrees of freedom as in the corresponding chi-squared test.

For very small samples themultinomial test for goodness of fit, andFisher's exact test for contingency tables, or even Bayesian hypothesis selection are preferable to theG-test.^[2] McDonald recommends to always use an exact test (exact test of goodness-of-fit,Fisher's exact test) if the total sample size is less than 1 000 .

There is nothing magical about a sample size of 1 000, it's just a nice round number that is well within the range where an exact test, chi-square test, andG–test will give almost identicalp values. Spreadsheets, web-page calculators, andSAS shouldn't have any problem doing an exact test on a sample size of 1 000 .

— John H. McDonald (2014)^[2]

G-tests have been recommended at least since the 1981 edition ofBiometry, a statistics textbook byRobert R. Sokal andF. James Rohlf.^[3]

The commonly usedchi-squared tests for goodness of fit to a distribution and for independence incontingency tables are in fact approximations of thelog-likelihood ratio on which theG-tests are based.^[4]

The general formula for Pearson's chi-squared test statistic is

${\displaystyle {\chi }^2=\sum _i\frac{{\left(O_i-E_i\right)}^2}{E_i}.}$

The approximation ofG by chi squared is obtained by a second orderTaylor expansion of the natural logarithm around 1 (see#Derivation (chi-squared) below).We have${\displaystyle G\approx {\chi }^2}$ when the observed counts${\displaystyle O_i}$ are close to the expected counts${\displaystyle E_i.}$ When this difference is large, however, the${\displaystyle {\chi }^2}$ approximation begins to break down. Here, the effects of outliers in data will be more pronounced, and this explains the why${\displaystyle {\chi }^2}$ tests fail in situations with little data.

For samples of a reasonable size, theG-test and the chi-squared test will lead to the same conclusions. However, the approximation to the theoretical chi-squared distribution for theG-test is better than for thePearson's chi-squared test.^[5] In cases where${\displaystyle O_i>2\cdot E_i}$ for some cell case theG-test is always better than the chi-squared test.^{[citation needed]}

For testing goodness-of-fit theG-test is infinitely moreefficient 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.^[6][7]

Consider

${\displaystyle G=2\sum _i{O}_i\ln \left(\frac{O_i}{E_i}\right),}$

and let${\displaystyle O_i=E_i+{\delta }_i}$ with${\displaystyle \sum _i{\delta }_i=0,}$ so that the total number of counts remains the same. Upon substitution we find,

${\displaystyle G=2\sum _i(E_i+{\delta }_i)\ln \left(1+\frac{{\delta }_i}{E_i}\right).}$

A Taylor expansion around${\displaystyle 1+\frac{{\delta }_i}{E_i}}$ can be performed using${\displaystyle \ln (1+x)=x-\frac{1}{2}{x}^2+\mathcal{O}(x^3)}$. The result is

${\displaystyle G=2\sum _i(E_i+{\delta }_i)\left(\frac{{\delta }_i}{E_i}-\frac{1}{2}\frac{{\delta }_i^2}{E_i^2}+\mathcal{O}\left({\delta }_i^3\right)\right),}$ and distributing terms we find,

${\displaystyle G=2\sum _i{\delta }_i+\frac{1}{2}\frac{{\delta }_i^2}{E_i}+\mathcal{O}\left({\delta }_i^3\right).}$

Now, using the fact that${\displaystyle \sum _i{\delta }_i=0}$ and${\displaystyle {\delta }_i=O_i-E_i,}$ we can write the result,

${\displaystyle G\approx \sum _i\frac{{\left(O_i-E_i\right)}^2}{E_i}.}$

TheG-test statistic is proportional to theKullback–Leibler divergence of the theoretical distribution from the empirical distribution:

whereN is the total number of observations and${\displaystyle o_i}$ and${\displaystyle e_i}$ are the empirical and theoretical frequencies, respectively.

For analysis ofcontingency tables the value ofG can also be expressed in terms ofmutual information.

Let

${\displaystyle N=\sum _{ij}{O}_{ij}}$ ,${\displaystyle {\pi }_{ij}=\frac{O_{ij}}{N}}$ ,${\displaystyle {\pi }_{i.}=\frac{\sum _j{O}_{ij}}{N}}$, and${\displaystyle {\pi }_{.j}=\frac{\sum _i{O}_{ij}}{N}}$.

ThenG can be expressed in several alternative forms:

${\displaystyle G=2\cdot N\cdot \sum _{ij}{\pi }_{ij}\left(\ln ({\pi }_{ij})-\ln ({\pi }_{i.})-\ln ({\pi }_{.j})\right),}$

${\displaystyle G=2\cdot N\cdot \left[H(r)+H(c)-H(r,c)\right],}$

${\displaystyle G=2\cdot N\cdot \mathrm{MI}(r,c),}$

where theentropy of a discrete random variable${\displaystyle X}$ is defined as

${\displaystyle H(X)=-\sum _{x\in \text{Supp}(X)}p(x)\log p(x),}$

and where

${\displaystyle \mathrm{MI}(r,c)=H(r)+H(c)-H(r,c)}$

is themutual information between the row vectorr and the column vectorc of the contingency table.

It can also be shown^{[citation needed]} that the inverse document frequency weighting commonly used for text retrieval is an approximation ofG 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 theG statistic.^{[citation needed]}

• TheMcDonald–Kreitman test instatistical genetics is an application of theG-test.
• Dunning^[8] introduced the test to thecomputational linguistics community where it is now widely used.
• The R-scape program (used byRfam) uses G-test to detect co-variation between RNA sequence alignment positions.^[9]

• InR fast implementations can be found in theAMR andRfast packages. For the AMR package, the command isg.test which works exactly likechisq.test from base R. R also has thelikelihood.testArchived 2013-12-16 at theWayback Machine function in theDeducerArchived 2012-03-09 at theWayback Machine package.Note: Fisher'sG-test in theGeneCycle Package of theR programming language (fisher.g.test) does not implement theG-test as described in this article, but rather Fisher's exact test of Gaussian white-noise in a time series.^[10]
• AnotherR implementation to compute the G statistic and corresponding p-values is provided by the R packageentropy. The commands areGstat for the standard G statistic and the associated p-value andGstatindep for the G statistic applied to comparing joint and product distributions to test independence.
• InSAS, one can conductG-test by applying the/chisq option after theproc freq.^[11]
• InStata, one can conduct aG-test by applying thelr option after thetabulate command.
• InJava, useorg.apache.commons.math3.stat.inference.GTest.^[12]
• InPython, usescipy.stats.power_divergence withlambda_=0.^[13]

• G²/Log-likelihood calculator

• Statistical tests for contingency tables

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