A brief description of a public health example (from the literature)
In the course text, Essentials of Biostatistics In Public Health, second edition by Lisa M. Sullivan, example 10.9 is of a personal trainer interested in testing and comparing the anaerobic thresholds (the point at which muscles cannot get more oxygen to sustain activity) of the elite athletes performing in the different field of athletics. Measurements of the anaerobic thresholds also related to the maximum heart rate are taken from distance runners, distance cyclists, distance swimmers, and cross-country skiers.
The samples used in the experiment are from the four independent groups of athletes. Therefore, to test and validate the null hypothesis that the population medians are equal across all the independent groups, an appropriate test should be carried out. For the public health example, the Kruskal-Wallis test is appropriate to use. This is because it is an appropriate test to use given that the small sample size is from four independent groups of athletes with no underlying assumptions made regarding their population distribution.
In order to test a given null hypothesis, use a parametric or nonparametric test. Parametric tests are appropriate when there are underlying assumptions regarding the distribution of the population used to make an inference.
However, when there are few or no assumptions regarding the distribution of the population, use nonparametric tests. They are usually more robust than the parametric tests in such situations.
The Kruskal-Wallis test is an example of a nonparametric test. A chi-square statistic is used to evaluate the differences in the independent sample groups where the mean ranks of more than two are used in the experiment. The test is to assess whether the medians of all groups from which samples are drawn are equal.
Therefore, the Kruskal-Wallis test is appropriate to use when testing inferences made regarding populations with the unknown or unassumed underlying distribution. The test assesses the means and medians of the independent sample groups. Consequently, it employs evaluation methods to test the null hypothesis. The medians across all the independent sample groups used in the experiment are equal.
An assessment of the assumptions required to use the parametric form of the Kruskal-Wallis test
A Parametric form of the Kruskal-Wallis test can be used when using small sample size, of more than two independent sample groups, from a population with known or assumed distribution. The Kruskal-Wallis test can aid in the assessment and evaluation of the independent samples from the different groups of the population, and test the null hypothesis that the medians across all the independent groups are equal.
Suppose, you take a small random sample of people from a given population with the assumed or known underlying distribution. Subdivide the sample further into three groups and take the sample heights of people in the respective groups. On further assessment and evaluation of the data, you establish that the mean heights of the respective groups are almost similar. However, the median heights of the respective independent groups vary widely. Hence, a parametric form of the Kruskal-Wallis test will be crucial in testing the null hypothesis that the median heights across all groups are equal.
The Kruskal-Wallis test is more robust when assumptions of ANOVA are not made. It is classified as a distribution free test and is robust in cases where assumptions of population distribution of the independent samples, are lacking.
Lisa, M.S. (2011). Essentials of Biostatistics In Public Health, Jones & Bartlett Learning.
Conover, W.J. (2005). Practical Nonparametric Statistics, New York: Wiley & Sons.
Rosner, B. (2000). Fundamentals of Biostatistics, California: Duxbury Press.
Motulsky, H. (1995). Intuitive Biostatistics, New York: Oxford University Press.