Relationship among Statistical Distributions

Summary

Background: Statistical analyses of clinical data are increasingly complex. They often involve multiple groups and measures. Such data can not be assessed simply by differences between means but rather by comparing variances.

Objective: To focus on the Chi-square (X ²)-test as a method to assess variances and test differences between variances. To give examples of clinical data where the emphasis is on variance. To assess interrelation between Chi-square and other statistical methods like normal-test (Z-test), T-test and Analysis-Of-Variance (ANOVA).

Results: A Chi-square-distribution is nothing else than the distribution of square values of a normal-distribution. Null-hypothesis-testing-of-variances is much similar to null-hypothesis-testing-of-means. With the latter we reject the null-hypothesis of no effect if our mean is more than 1.96 SEMs (standard errors of the mean) distant from zero. With the latter we reject the null-hypothesis of no effect if our standardized variance is more than 1.96² SEMs² distant from zero. Because variances are squared and, thus, non-negative values, the Chi-square approach can be extended to test hypotheses about many samples. When variances or add-up variances of many samples are larger than allowed for by the Chi-square-distribution-graphs, we reject the probability that our results are from normal distributions, and conclude that our results are significantly different from zero. The Chi-square test is not only adequate to test multiple samples simultaneously, but is also the basis of analysis of variance (ANOVA).

Conclusions: The Chi-square-distribution focused on in this paper is just another approach of the bell-shape-like normal-distribution and is also the basic element of the F-distribution as used in ANOVA. Having some idea about interrelations between these distributions will be of help in understanding benefits and limitations of Chi-square-statistic and its many extensions for the analysis of experimental clinical data.