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tal number of counts remains the same assume that δ i o i e i displaystyle delta _ i o_ i e_ i is small in comparison to e i displaystyle e_ i for all i displaystyle i to be more precise notice that e i θ n displaystyle e_ i theta n using big θ notation if o i e i o n 1 2 displaystyle o_ i e_ i mathcal o n 1 2 using big o notation for large n displaystyle n which should be true under the null hypothesis because of the central limit theorem then δ i o n 1 2 displaystyle delta _ i mathcal o n 1 2 and δ i 3 e i 2 o n 3 2 n 2 o n 1 2 displaystyle 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 i e i δ i ln 1 δ i e i displaystyle 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 1 2 x 2 o x 3 displaystyle ln 1 x x tfrac 1 2 x 2 mathcal o x 3 yields g 2 i e i δ i δ i e i 1 2 δ i 2 e i 2 o δ i 3 e i 3 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 frac delta _ i 3 e_ i 3 right right and distributing terms we find g 2 i δ i 1 2 δ i 2 e i o δ i 3 e i 2 displaystyle 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 i δ i 0 displaystyle textstyle sum _ i delta _ i 0 and δ i o i e i displaystyle delta _ i o_ i e_ i and o δ i 3 e i 2 o n 1 2 displaystyle mathcal o delta _ i 3 e_ i 2 mathcal o n 1 2 for large n displaystyle n we can write the result g i o i e i 2 e i displaystyle g approx sum _ i frac left o_ i e_ i right 2 e_ i relation to kullback leibler divergence edit the g test statistic is proportional to the kullback leibler divergence of the theoretical distribution p p 1 p m displaystyle tilde p tilde p _ 1 ldots tilde p _ m of the null hypothesis from the empirical distribution p p 1 p m displaystyle hat p hat p _ 1 ldots hat p _ m of the observed data g 2 i o i ln o i e i 2 n i p i ln p i p i 2 n d k l p p displaystyle begin aligned g 2 sum _ i o_ i cdot ln left frac o_ i e_ i right 2n sum _ i hat p _ i cdot ln left frac hat p _ i tilde p _ i right 2n d_ mathrm kl hat p tilde p end aligned where n displaystyle n is the total number of observations and p i e i n displaystyle tilde p _ i tfrac e_ i n and p i o i n displaystyle hat p _ i tfrac o_ i n are the theoretical and empirical probabilities of objects of type i displaystyle i respectively relation to mutual information edit 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 displaystyle i j are considered let o i j displaystyle o_ ij be the count of objects of type i j displaystyle i j i e o i j displaystyle o_ ij is the entry in the contingency table in row i displaystyle i and column j displaystyle j set n i j o i j p i j o i j n p i j o i j n p j i o i j n displaystyle 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 displaystyle i j assuming independence is given by e i j n p i p j displaystyle e_ ij n hat p _ i bullet hat p _ bullet j finally the g test statistics in this case is given by g 2 i j o i j ln o i j e i j displaystyle g 2 sum _ ij o_ ij ln left frac o_ ij e_ ij right let x y displaystyle x y be random variables with joint distribution given by the empirical distribution p i j displaystyle hat p _ ij of the contingency table i e p x i y j p i j p x i p i p y j p j displaystyle 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 g 2 n i j p i j ln p i j ln p i ln p j 2 n h x h y h x y 2 n mi x y displaystyle begin aligned 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 aligned where the entropies h x displaystyle h x and h y displaystyle h y are given h x i p i ln p i h y j p j ln p j displaystyle 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 displaystyle h x y is given by h x y i j p i j ln p i j displaystyle h x y sum _ ij hat p _ ij ln hat p _ ij and the mutual information of x displaystyle x and y displaystyle y is mi x y h x h y h x y displaystyle 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 edit the mcdonald kreitman test in statistical genetics is an application of the g test dunning 9 introduced the test to the computational linguistics community where it is now widely used the r scape program used by rfam uses g test to detect co variation between rna sequence alignment positions 10 statistical software edit 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 edit mcdonald j h 2014 g test of goodness of fit handbook of biological statistics third ed baltimore maryland sparky house publishing pp 53 58 1 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 1 2 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 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 hoey j 2012 the two way likelihood ratio g test and comparison to two way chi squared test arxiv 1206 4881 stat me 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 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 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 cite journal cite uses deprecated parameter citeseerx help dunning ted 1993 accurate methods for the statistics of surprise and coincidence computational linguistics 19 1 61 74 rivas elena 30 october 2020 rna structure prediction using positive and negative evolutionary information plos computational biology 16 10 e1008387 bibcode 2020plscb 16e8387r doi 10 1371 journal pcbi 1008387 pmc 7657543 pmid 33125376 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 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 org apache commons math3 stat inference gtest archived from the original on 2018 07 26 retrieved 2018 07 11 scipy stats power_divergence scipy v1 7 1 manual external links edit g 2 log likelihood calculator v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional study natural experiment quasi experiment statistical inference statistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model model specification l p space parameter location scale shape parametric family likelihood monotone location scale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness frequentist inference point estimation estimating equations maximum likelihood method of moments m estimator minimum distance unbiased estimators mean unbiased minimum variance rao blackwellization lehmann scheffé theorem median unbiased plug in interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2 tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihood ratio score lagrange multiplier wald specific tests z test normal student s t test f test goodness of fit chi squared g test kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk likelihood ratio test model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodges lehmann estimator rank sum mann whitney nonparametric anova 1 way kruskal wallis 2 way friedman ordered alternative jonckheere terpstra van der waerden test bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlation regression analysis correlation pearson product moment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model bayesian regression non standard predictors nonlinear regression nonparametric semiparametric isotonic robust homoscedasticity and heteroscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova anova analysis of covariance multivariate anova degrees of freedom categorical multivariate time series survival analysis categorical cohen s kappa contingency table graphical model log linear model mcnemar s test cochran mantel haenszel statistics multivariate regression manova principal components canonical correlation discriminant analysis cluster analysis classification structural equation model factor analysis multivariate distributions elliptical distributions normal time series general decomposition trend stationarity seasonal adjustment exponential smoothing cointegration structural break granger causality specific tests dickey fuller johansen q statistic ljung box durbin watson breusch godfrey time domain autocorrelation acf partial pacf cross correlation xcf arma model arima model box jenkins autoregressive conditional heteroskedasticity arch vector autoregression var autoregressive model ar frequency domain spectral density estimation fourier analysis least squares spectral analysis wavelet whittle likelihood survival survival function kaplan meier estimator product limit proportional hazards models accelerated failure time aft model first hitting time hazard function nelson aalen estimator test log rank test applications biostatistics bioinformatics clinical trials studies epidemiology medical statistics engineering statistics chemometrics methods engineering probabilistic design process quality control reliability system identification social statistics actuarial science census crime statistics demography econometrics jurimetrics national accounts official 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