NONPARAMETRIC TESTS

Nonparametric Tests



Nonparametric Tests

Chi-Square Test

The chi-square test is a nonparametric test of the statistical significance of a relation between two nominal or ordinal variables. Because a chi-square analyzes grosser data than do parametric tests such as t tests and analyses of variance (ANOVAs), the chi-square test can report only whether groups in a sample are significantly different in some measured attribute or behavior; it does not allow one to generalize from the sample to the population from which it was drawn. Nonetheless, because chi-square is less “demanding” about the data it will accept, it can be used in a wide variety of research contexts. This entry focuses on the application, requirements, computation, and interpretation of the chi-square test, along with its role in determining associations among variables.

A common method for investigating “general” association between two categorical variables is to perform a chi-square test. This method compares the observed number of individuals within cells of a cross-tabulation of the categorical variables with the number of individuals one would expect in the cells if there was no association and the individuals were randomly distributed. If the observed and expected frequencies differ statistically (beyond random chance according to the chi-square distribution), the variables are said to be associated. Statistical procedures for reporting survey data include frequencies, percent, cross-tabulations (cross-tabs), and chi-square statistic. The level of statistical significance that the researcher sets for a study is closely related to hypothesis testing. This is called the alpha level. It is the level of probability that indicates the maximum risk a researcher is willing to take that observed differences are due to chance.

In statistics, there are two types of hypotheses: null hypothesis (H0) and alternative/research/maintained hypothesis (Ha). A null hypothesis (H0) is a falsifiable proposition, which is assumed to be true until it is shown to be false. Hypothesis tests rely on the calculation of a statistic from the observed data and the determination of the distribution of that statistic under the terms of the null hypothesis. Whereas in modern times, a multitude of software packages have been created to perform the calculations required for most hypothesis testing, they have not always been available and they are not necessary to the process. They are merely programmed with the probability distributions of various test statistics under the user-inputted null hypothesis. Regardless of the details of how a test is formed and the means by which it is executed, tests for significance allow a researcher to make a determination regarding statistical significance by means of a p value.

After setting the null and alternative hypotheses, collecting data, choosing the appropriate test, and calculating the appropriate test statistic(s), performing the hypothesis test returns a p value. The p value is a measure of statistical significance. Specifically, the p value is the probability of observing data as extreme or more extreme than what was observed given that the null hypothesis is true. In other words, the p value is a measure of how likely (or unlikely) it is for the ...