The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g. to test for normalityofresiduals, to test whether two samples are drawn from identical distributions (see Kolmogorov–Smirnov test), or whether outcome frequencies follow a specified distribution (see Pearson's chi-square test). In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares.
Pearson's chi-square test uses a measure of goodness of fit which is the sum of differences between observed and expected outcome frequencies (that is, counts of observations), each squared and divided by the expectation:
where:
Oi = an observed count for bin i
Ei = an expected count for bin i, asserted by the null hypothesis.
The resulting value can be compared with a chi-square distribution to determine the goodness of fit. The chi-square distribution has (k − c) degrees of freedom, where k is the number of non-empty cells and c is the number of estimated parameters (including location and scale parameters and shape parameters) for the distribution plus one. For example, for a 3-parameter Weibull distribution, c = 4.
A binomial experiment is a sequence of independent trials in which the trials can result in one of two outcomes, success or failure. There are n trials each with probability of success, denoted by p. Provided that npi ≫ 1 for every i (where i = 1, 2, ..., k), then
This has approximately a chi-square distribution with k − 1 degrees of freedom. The fact that there are k − 1 degrees of freedom is a consequence of the restriction . We know there are k observed cell counts, however, once any k − 1 are known, the remaining one is uniquely determined. Basically, one can say, there are only k − 1 freely determined cell counts, thus k − 1 degrees of freedom.
where and are the same as for the chi-square test, denotes the natural logarithm, and the sum is taken over all non-empty cells. Furthermore, the total observed count should be equal to the total expected count:
where is the total number of observations.
G-tests have been recommended at least since the 1981 edition of the popular statistics textbook by Robert R. Sokal and F. James Rohlf.[8]
^Berk, Robert H.; Jones, Douglas H. (1979). "Goodness-of-fit test statistics that dominate the Kolmogorov statistics". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 47 (1): 47–59. doi:10.1007/BF00533250.
^Moscovich, Amit; Nadler, Boaz; Spiegelman, Clifford (2016). "On the exact Berk-Jones statistics and their p-value calculation". Electronic Journal of Statistics. 10 (2). arXiv:1311.3190. doi:10.1214/16-EJS1172.
^Liu, Qiang; Lee, Jason; Jordan, Michael (20 June 2016). "A Kernelized Stein Discrepancy for Goodness-of-fit Tests". Proceedings of the 33rd International Conference on Machine Learning. The 33rd International Conference on Machine Learning. New York, New York, USA: Proceedings of Machine Learning Research. pp. 276–284.
^Chwialkowski, Kacper; Strathmann, Heiko; Gretton, Arthur (20 June 2016). "A Kernel Test of Goodness of Fit". Proceedings of the 33rd International Conference on Machine Learning. The 33rd International Conference on Machine Learning. New York, New York, USA: Proceedings of Machine Learning Research. pp. 2606–2615.
^Vexler, Albert; Gurevich, Gregory (2010). "Empirical Likelihood Ratios Applied to Goodness-of-Fit Tests Based on Sample Entropy". Computational Statistics and Data Analysis. 54 (2): 531–545. doi:10.1016/j.csda.2009.09.025.
^McDonald, J.H. (2014). "G–test of goodness-of-fit". Handbook of Biological Statistics (Third ed.). Baltimore, Maryland: Sparky House Publishing. pp. 53–58.
where:
where:
where 
is the total number of observations.
