Euclidean quantum gravity

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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

The general formula for G is

$G=2\sum _i{O}_i\cdot \ln (O_i/E_i),$

where O_i is the observed frequency in a cell, E is the expected frequency on the null hypothesis, where ln denotes the natural logarithm (log to the base e) and the sum is taken over all non-empty cells.

G-tests are coming into increasing use, particularly since they were recommended at least since the 1981 edition of the popular statistics textbook by Sokal and Rohlf.

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 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 .Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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

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

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 log-likelihood ratios. With the advent of electronic calculators and personal computers, this is no longer a problem. A derivation of how the chi-squared test is related to the G-test and likelihood ratios, including to a full Bayesian solution is provided in.

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 chi-squared tests. In cases where $O_i>2\cdot E_i$ for some cell case the G-test is always better than the chi-squared test.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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 Hodge and Lehman.

Relation to Kullback-Leibler divergence

The G-test 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:

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

$G=2\cdot N\cdot [H(row)+H(col)-H(row,col)],$

$G=2\cdot N\cdot MI(row,col),$

where the entropy of a discrete random variable $X$ is defined as

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

and where

$MI(row,col)=H(row)+H(col)-H(row,col)$

is the mutual information between the row vector and the column vector of the contingency table.

It can also be shownPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. 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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Application

• The McDonald–Kreitman test in statistical genetics is an application of the G-test.
• Dunning introduced the test to the computational linguistics community where it is now widely used.

Statistical software

• The R programming language has the likelihood.test function in the Deducer package.
• In SAS, one can conduct G-Test by applying the /chisq option in proc freq.
• 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.

External links

• G²/Log-likelihood calculator