# G-test - Relation To The Chi-squared Test

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

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.

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 for some cell case the G-test is always better than the chi-squared test.

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.

### Famous quotes containing the words relation to the, relation to, test and/or relation:

Concord is just as idiotic as ever in relation to the spirits and their knockings. Most people here believe in a spiritual world ... in spirits which the very bullfrogs in our meadows would blackball. Their evil genius is seeing how low it can degrade them. The hooting of owls, the croaking of frogs, is celestial wisdom in comparison.
Henry David Thoreau (1817–1862)

We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.
—D.H. (David Herbert)

A test of what is real is that it is hard and rough. Joys are found in it, not pleasure. What is pleasant belongs to dreams.
Simone Weil (1909–1943)

We shall never resolve the enigma of the relation between the negative foundations of greatness and that greatness itself.
Jean Baudrillard (b. 1929)