G-test

G-test

The G-test, also known as the likelihood-ratio test, is a statistical test used to determine whether there is a significant association between two categorical variables. It is an alternative to the more commonly used Chi-squared test, and is particularly useful when dealing with small sample sizes or when the data do not meet the assumptions of the chi-squared test.

History

The G-test was developed as part of the likelihood-ratio tests introduced by Ronald A. Fisher in the early 20th century. It gained popularity in the mid-20th century as computational methods improved, allowing for more complex calculations that the G-test requires.

Mathematical Foundation

The G-test is based on the concept of likelihood, which measures the probability of observing the given data under different hypotheses. The test statistic, denoted as G, is calculated as follows:

$G=2\sum _i{O}_i\ln \left(\frac{O_i}{E_i}\right)$

where:

• O_i is the observed frequency for category i.
• E_i is the expected frequency for category i under the null hypothesis.

The G statistic follows a chi-squared distribution with degrees of freedom equal to the number of categories minus one.

Applications

The G-test is widely used in genetics, ecology, and other fields where categorical data are analyzed. It is particularly useful in cases where the assumptions of the chi-squared test are violated, such as when expected frequencies are low.

Advantages and Disadvantages

Advantages

• The G-test is more flexible than the chi-squared test and can be used in a wider range of situations.
• It is more accurate for small sample sizes.

Disadvantages

• The G-test is computationally more intensive than the chi-squared test.
• It may not be as intuitive to interpret as the chi-squared test.

Comparison with Chi-squared Test

While both the G-test and the chi-squared test are used to test for independence in contingency tables, they have different underlying assumptions and calculations. The chi-squared test is based on the squared differences between observed and expected frequencies, while the G-test uses the likelihood ratio.

See Also

• Chi-squared test
• Likelihood-ratio test
• Statistical hypothesis testing

References

• Sokal, R. R., & Rohlf, F. J. (1995). Biometry: The Principles and Practice of Statistics in Biological Research. W. H. Freeman and Company.
• McDonald, J. H. (2014). Handbook of Biological Statistics (3rd ed.). Sparky House Publishing.