Pearson-Fisher Chi-Square Statistic Revisited

Abstract

The Chi-Square test (χ2 test) is a family of tests based on a series of assumptions and is frequently used in the statistical analysis of experimental data. The aim of our paper was to present solutions to common problems when applying the Chi-square tests for testing goodness-of-fit, homogeneity and independence. The main characteristics of these three tests are presented along with various problems related to their application. The main problems identified in the application of the goodness-of-fit test were as follows: defining the frequency classes, calculating the X2 statistic, and applying the χ2 test. Several solutions were identified, presented and analyzed. Three different equations were identified as being able to determine the contribution of each factor on three hypothesizes (minimization of variance, minimization of square coefficient of variation and minimization of X2 statistic) in the application of the Chi-square test of homogeneity. The best solution was directly related to the distribution of the experimental error. The Fisher exact test proved to be the “golden test” in analyzing the independence while the Yates and Mantel-Haenszel corrections could be applied as alternative tests.

Keywords

• Chi-square statistics
• Fisher exact test
• Chi-square distribution
• 2 × 2 contingency table