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.[1]

Formulation

The general formula for test statistics of the G-test is

G = 2\sum_{i} {O_{i} \cdot \ln\left(\frac{O_i}{E_i}\right)},

where O_i \geq 0 is the observed count in a cell, E_i > 0 is the expected count under the null hypothesis, \ln denotes the natural logarithm, and the sum is taken over all non-empty cells. The resulting G is asymptotically chi-squared distributed as the total number of observations tends to infinity (convergence in distribution[2]).

Furthermore, the total observed count must be equal to the total expected count:

\sum_i O_i = \sum_i E_i = N,

where N is the total number of observations.

Both, the G-test statistics G and the chi-square test statistics \chi^2 are special cases of a general family of power divergence statistics by Cressie and Read[2]. For \lambda\notin\{0,-1\} set

\operatorname{CR}_\lambda = \frac{2}{\lambda(\lambda+1)} \sum_i O_i \left( \left( \frac{O_i}{E_i} \right)^\lambda - 1 \right).

Then,

G = \lim_{\lambda\to0} \operatorname{CR}_\lambda, \qquad \chi^2 = \operatorname{CR}_1.

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 O=(O_1,\ldots,O_m) where each O_i is the number of times that an object of type i was observed. Furthermore, let N = \sum_{i=1}^m O_i be the total number of observations. If we assume that the underlying model is multinomial, then the test statistic is defined by

\ln \left( \frac{L(\tilde p|O)}{L(\hat p|O)} \right) = \ln \left( \frac{\prod_{i=1}^m \tilde p_i^{O_i}}{\prod_{i=1}^m \hat{p}_i^{O_i}} \right),

where \tilde p=(\tilde p_1,\ldots,\tilde p_m) is the null hypothesis and \hat p=(\hat p_1,\ldots,\hat p_m) is the maximum likelihood estimate (MLE) of the parameters given the data. Recall that for the multinomial model, the MLE of \hat p_i given some data is given by

\hat p_i = \frac{O_i}{N} \,.

Furthermore, we may represent each null hypothesis parameter \tilde p_i as

\tilde p_i = \frac{E_i}{N} \,,

where E_i is the expected count of objects of type i under the null hypothesis. Thus, by substituting the representations of \tilde p_i and \hat p_i in the log-likelihood ratio, the equation simplifies to

\ln \left( \frac{L(\tilde p|O)}{L(\hat p|O)} \right) = \ln \left( \prod_{i=1}^m \left(\frac{E_i}{O_i}\right)^{O_i} \right) = \sum_{i=1}^m O_i \ln\left(\frac{E_i}{O_i}\right)

Finally, multiply by a factor of -2 (used to make the G-test formula asymptotically equivalent to the Pearson's chi-squared test statistics) to achieve the form

G = -2 \sum_{i=1}^m O_i \ln\left(\frac{E_i}{O_i}\right) = 2 \sum_{i=1}^m O_i \ln\left(\frac{O_i}{E_i}\right)

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

Distribution and use

Given the null hypothesis that the observed frequencies result from random sampling from a distribution with the given expected frequencies, the distribution of the test statistics 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.[3] 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, and G–test will give almost identical p values. Spreadsheets, web-page calculators, and SAS shouldn't have any problem doing an exact test on a sample size of 1 000 .
— John H. McDonald (2014)[3]

G-tests have been recommended at least since the 1981 edition of Biometry, a statistics textbook by Robert R. Sokal and F. James Rohlf.[4]

Relation to other metrics

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.[5]

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

\chi^2 = \sum_{i} {\frac{\left(O_i - E_i\right)^2}{E_i}}.

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

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.[6] In cases where O_i > 2\cdot E_i for some cell case the G-test is always better than the chi-squared test.[citation needed]

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.[7][8]

Derivation (chi-squared)

Consider

G = 2\sum_i {O_i \ln\left(\frac{O_i}{E_i}\right)},

and let O_i = E_i + \delta_i with \textstyle\sum_i \delta_i = 0, so that the total number of counts remains the same. Assume that \delta_i=O_i-E_i is small in comparison to E_i for all i. To be more precise, notice that E_i = \Theta(n) using big Θ notation. If O_i = E_i + \mathcal{O}(n^{1/2}) using big O notation for large n, which should be true under the null hypothesis because of the central limit theorem, then \delta_i = \mathcal{O}(n^{1/2}) and

\frac{\delta_i^3}{E_i^2} = \mathcal{O}\left(\frac{n^{3/2}}{n^2}\right) = \mathcal{O}(n^{-1/2})

follow, which will be used later.

Upon substitution we find,

G = 2\sum_i (E_i + \delta_i) \ln \left(1+\frac{\delta_i}{E_i}\right).

Using the Taylor expansion \ln(1 + x) = x - \tfrac{1}{2}x^2 + \mathcal{O}(x^3) yields

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(\frac{\delta_i^3}{E_i^3}\right) \right),

and distributing terms we find,

G = 2\sum_i \left( \delta_i + \frac{1}{2}\frac{\delta_i^2}{E_i} + \mathcal{O}\left(\frac{\delta_i^3}{E_i^2}\right) \right).

Now, using \textstyle\sum_i \delta_i = 0 and \delta_i = O_i - E_i and \mathcal{O}(\delta_i^3/E_i^2)=\mathcal{O}(n^{-1/2}) for large n, we can write the result,

G \approx \sum_{i} \frac{\left(O_i-E_i\right)^2}{E_i}.

Relation to Kullback–Leibler divergence

The G-test statistic is proportional to the Kullback–Leibler divergence of the theoretical distribution \tilde p=(\tilde p_1,\ldots,\tilde p_m) of the null hypothesis from the empirical distribution \hat p=(\hat p_1,\ldots,\hat p_m) of the observed data:

\begin{align} G &= 2\sum_i {O_i \cdot \ln\left(\frac{O_i}{E_i}\right)} = 2 N \sum_i {\hat p_i \cdot \ln\left(\frac{\hat p_i}{\tilde p_i}\right)} \\ &= 2 N \, D_{\mathrm{KL}}(\hat p\|\tilde p), \end{align}

where N is the total number of observations and \tilde p_i = \tfrac{E_i}{N} and \hat p_i = \tfrac{O_i}{N} are the theoretical and empirical probabilities of objects of type i, respectively.

Relation to mutual information

For analysis of contingency tables the value of the G-test statistics can also be expressed in terms of mutual information.

In this case objects with two-dimensional types (i,j) are considered. Let O_{ij} be the count of objects of type (i,j), i.e., O_{ij} is the entry in the contingency table in row i and column j. Set

N = \sum_{ij} O_{ij}, \qquad \hat p_{ij} = \frac{O_{ij}}{N} \,, \qquad \hat p_{i \bullet} = \frac{\sum_j O_{ij}}{N} \,, \qquad \hat p_{\bullet j} = \frac{\sum_i O_{ij}}{N} \,.

Then the estimated expected count of objects of type (i,j) assuming independence is given by

E_{ij} = N \hat p_{i \bullet} \hat p_{\bullet j}.

Finally, the G-test statistics in this case is given by

G = 2 \sum_{ij} O_{ij} \ln\left(\frac{O_{ij}}{E_{ij}}\right)

Let X,Y be random variables with joint distribution given by the empirical distribution \hat p_{ij} of the contingency table, i.e.,

P(X=i, Y=j) = \hat p_{ij}, \qquad P(X=i) = \hat p_{i \bullet}, \qquad P(Y=j) = \hat p_{\bullet j}.

Then the G-test statistics can be expressed in several alternative forms:

\begin{align} G &= 2N \cdot \sum_{ij}{\hat p_{ij} \left( \ln(\hat p_{ij})-\ln(\hat p_{i \bullet})-\ln(\hat p_{\bullet j}) \right)} \\ &= 2N \cdot \Bigl( H(X) + H(Y) - H(X,Y) \Bigr) \\ &= 2N \cdot \operatorname{MI}(X,Y), \end{align}

where the entropies H(X) and H(Y) are given

H(X) = -\sum_i \hat p_{i \bullet} \ln(\hat p_{i \bullet}), \qquad H(Y) = -\sum_j \hat p_{\bullet j} \ln(\hat p_{\bullet j})

and the joint entropy H(X,Y) is given by

H(X,Y) = -\sum_{ij} \hat p_{ij} \ln(\hat p_{ij})

and the mutual information of X and Y is

\operatorname{MI}(X,Y) = H(X) + H(Y) - H(X,Y).

It can also be shown[citation needed] 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-test statistic.[citation needed]

Application

Statistical software

  • In R fast implementations can be found in the AMR and Rfast packages. For the AMR package, the command is g.test which works exactly like chisq.test from base R. R also has the likelihood.test Archived 2013-12-16 at the Wayback Machine function in the Deducer Archived 2012-03-09 at the Wayback Machine package. Note: Fisher's G-test in the GeneCycle Package of the R programming language (fisher.g.test) does not implement the G-test as described in this article, but rather Fisher's exact test of Gaussian white-noise in a time series.[11]
  • Another R implementation to compute the G-test statistic and corresponding p-values is provided by the R package entropy. The commands are Gstat for the standard G statistic and the associated p-value and Gstatindep for the G statistic applied to comparing joint and product distributions to test independence.
  • In SAS, one can conduct G-test by applying the /chisq option after the proc freq.[12]
  • In Stata, one can conduct a G-test by applying the lr option after the tabulate command.
  • In Java, use org.apache.commons.math3.stat.inference.GTest.[13]
  • In Python, use scipy.stats.power_divergence with lambda_=0.[14]

References

  1. ^ McDonald, J.H. (2014). "G–test of goodness-of-fit". Handbook of Biological Statistics. Third ed. Baltimore, Maryland: Sparky House Publishing. pp. 53–58.
  2. ^ Cressie, Noel & Read, Timothy R. C. (1984). "Multinomial goodness-of-fit tests". Journal of the Royal Statistical Society. Series B (Methodological). 46 (3): 440–464. doi:10.1111/j.2517-6161.1984.tb01318.x. JSTOR 2345686. Retrieved 14 January 2026.
  3. ^ McDonald, John H. (2014). "Small numbers in chi-square and G–tests". Handbook of Biological Statistics. 3rd ed. Baltimore, MD: Sparky House Publishing. pp. 86–89.
  4. ^ Sokal, R. R. & Rohlf, F. J. (1981). Biometry: The Principles and Practice of Statistics in Biological Research. Second ed. New York: Freeman. ISBN 978-0-7167-2411-7.
  5. ^ Hoey, J. (2012). "The Two-Way Likelihood Ratio (G) Test and Comparison to Two-Way Chi-Squared Test". arXiv:1206.4881
  6. ^ Harremoës, P. & Tusnády, G. (2012). "Information divergence is more chi squared distributed than the chi squared statistic". Proceedings ISIT 2012. pp. 538–543. arXiv:1202.1125. Bibcode:2012arXiv1202.1125H
  7. ^ Quine, M. P. & Robinson, J. (1985). "Efficiencies of chi-square and likelihood ratio goodness-of-fit tests". Annals of Statistics. 13 (2): 727–742. doi:10.1214/aos/1176349550
  8. ^ Harremoës, P. & Vajda, I. (2008). "On the Bahadur-efficient testing of uniformity by means of the entropy". IEEE Transactions on Information Theory. 54 (1): 321–331. Bibcode:2008ITIT...54..321H. CiteSeerX 10.1.1.226.8051. doi:10.1109/tit.2007.911155. S2CID 2258586
  9. ^ Dunning, Ted (1993). "Accurate Methods for the Statistics of Surprise and Coincidence". Computational Linguistics. 19 (1): 61–74.
  10. ^ Rivas, Elena (30 October 2020). "RNA structure prediction using positive and negative evolutionary information". PLOS Computational Biology. 16 (10). Bibcode:2020PLSCB..16E8387R. doi:10.1371/journal.pcbi.1008387. PMC 7657543. PMID 33125376
  11. ^ Fisher, R. A. (1929). "Tests of significance in harmonic analysis". Proceedings of the Royal Society of London A. 125 (796): 54–59. Bibcode:1929RSPSA.125...54F. doi:10.1098/rspa.1929.0151. hdl:2440/15201
  12. ^ G-test of independence, G-test for goodness-of-fit 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.)
  13. ^ "org.apache.commons.math3.stat.inference.GTest". Archived 2018-07-26 at the Wayback Machine. Retrieved 2018-07-11.
  14. ^ "Scipy.stats.power_divergence — SciPy v1.7.1 Manual"