Non-parametric tests: definition and types

Non-parametric tests or techniques encompass a series of statistical tests that have in common the absence of assumptions about the law of probability that the population follows from which the sample has been extracted. Thus, these techniques are applied when we do not know if the population from which the sample is drawn is normal or approximately normal.

These non-parametric techniques are frequently used, since there are many variables that do not follow the conditions of parametricity. These are: the use of continuous quantitative variables, normal distribution of samples, similar variances and balanced samples.

When these prerequisites are not met or there are serious doubts that they are met, the non-parametric or distribution-free tests. Thus, non-parametric tests bring together the following characteristics:

They are used much less than would be recommended (they are less known by researchers). They are applicable to hierarchical data. They can be used when two series of observations come from different populations (populations in which the variable is not equally distributed). They are the only realistic alternative when the sample size is small.

Classification of non-parametric tests

In this classification of non-parametric tests, there is a lack of consensus when grouping them. The authors Berlanga and Rubio (2012) made a summary of the main parametric tests.

One Sample Nonparametric Tests

Pearson Chi-square test

It is a widely used test when the researcher wants analyze the relationship between two variables that are quantitative. It is also widely used to evaluate to what extent the data collected in a categorical variable (empirical distribution) fits or does not (resemble or not) a certain theoretical distribution (uniform, binomial, multinomial, etc.).

Binomial Proof

This test It allows you to find out whether or not a dichotomous variable follows a certain probability model. It allows us to test the hypothesis that the observed proportion of correct answers fits the theoretical proportion of a binomial distribution.

Streak Test

It is a test that allows you to determine if the number of streaks (R) observed in a sample size n is large enough or small enough to reject the independence hypothesis (or randomness) between observations.

A streak is a sequence of observations of the same attribute or quality.. That there are more or fewer streaks than expected by chance in a series of data can be an indicator that there is an important variable that is conditioning the results and that we are not taking into account.

Kolmogorov-Smirnov (KS) test

This test is used to test the null hypothesis about what the distribution of a variable fits a certain theoretical distribution probability (normal, exponential or Poisson). The fact that the data distribution fits or does not fit a certain distribution will suggest some data analysis techniques versus others.

Nonparametric tests for two related samples

McNemar test

The McNemar test is used to test hypotheses about equality of proportions. It is used when there is a situation where each subject’s measurements are repeated. Thus, the response of each of them is obtained twice: once before and once after a specific event.

Test of the Signs

It allows test the hypothesis of equality between two population medians. It can be used to find out if one variable tends to be greater than another. Also to test the trend that a series of positive variables follow.

Wilcoxon test

It allows test the hypothesis of equality between two population medians.

Non-parametric tests for K-related samples

Friedman test

It is a Wilcoxon test extension. Thus, it is used to include data recorded in more than two time periods or groups of three or more subjects, with one subject from each group having been randomly assigned to one of the three or more conditions.

Cochran test

It is identical to the previous one, but applies when all responses are binary. Cochran’s Q approves the hypothesis that several dichotomous variables that are related to each other have the same average.

Kendall’s W coefficient of agreement

It has the same indications as the Friedman test. However, its use in research has been mainly for know the agreement between ranges.

Non-parametric tests for two independent samples

Mann-Whitney U test

It is equivalent to the Wilcoxon rank sum test and also to the Kruskal-Wallis two-group test.

Kolmogorov-Smirnov test

This test is used to test the hypothesis that two samples come from the same population.

Wald-Wolfowitz Streak Test

Contrast if two samples with independent data come from populations with the same distribution.

Moses Extreme Reactions Test

Serves for study if there is a difference in the degree of dispersion or variability of two distributions. It focuses on the distribution of the control group and is a measure of how many extreme values ​​of the experimental group influence the distribution when combined with the control group.

Non-parametric tests for K-independent samples

Median Test

Contrast differences between two or more groups in relation to their median. Means are not used, either because they do not meet the conditions of normality or because the variable is discrete quantitative. It is similar to the Chi-square test.

Jonckheere-Terpstra test

It is the most powerful when it comes to analyze an ascending or descending ordering of the K populations from which the samples are extracted.

Kruskal-Wallis H test

Lastly, the Kruskal-Wallis H test It is an extension of the Mann-Whitney U and represents an excellent alternative to the one-way ANOVA.

So, These tests are used when the data distribution is not normal. We can turn to them when we have data that is not on a ratio scale or when, being so, we have doubts about whether the distribution of any of the variables fits the normal curve. On the other hand, It is true that many parametric tests are relatively robust to violating assumptions; However, if there are better tests, why not use them?

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