### Nonparametric Tests

Nonparametric tests are statistical tests used to analyse data for which an underlying distribution is not assumed. They are advantages over their parametric counterparts because they have fewer underlying assumptions (e.g. data normality, equal variance, etc). However, iif the data’s underlying distribution is known, one should use the appropriate parametric test.

The non-parametric tests have a corresponding parametric test. The purpose is the same only in this case, we are testing the median instead of the mean.

#### 1-SAMPLE SIGN

The 1-Sample Sign is a test to find out if the sample median is equal to a hypothesised value. It is the nonparametric alternative to the 1-Sample t-test. The logic of the Sign test is simple. Since under H0, η = η0, then you would expect half the data points of the sample size to be greater than the hypothesized value, η0. The test simply determines whether there is a significant deviation from this assumption. The 1-Sample Sign can handle data that is non-symmetric or skewed. However, it is less powerful than the 1-Sample Wilcoxon if the distribution is symmetric.

Hypothesis: H0: η = η0 H1 : η > η0 or η < η0 or η ≠ η0

Assumptions: This test makes no assumption about the shape of the population distribution, therefore this test can handle a data set that is non-symmetric, that is skewed either to the left or the right.

#### 1-SAMPLE WILCOXON

The 1-Sample Wilcoxon Test (also called Wilcoxon Signed Rank Test) is a test to find out if the sample median is equal to a hypothesised value. It is the nonparametric alternative to the 1-Sample t-test. The test determines is H0 is true which is whether the observations are symmetrically distributed about the hypothesised median. It is similar to the Sign Test but the Wilcoxon Test takes the magnitude of the scores is taken into account. The 1-Sample Wilcoxon Test is more powerful than 1-Sample Sign Test if the distribution is symmetric

Assumptions: Sample is random and from a continuous and symmetric population

#### MANN-WHITNEY

The Mann-Whitney test (also call 2-Sample Rank Test) is a nonparametric test used to compare the difference between two population medians. It basically tests if one variable tends to have values higher than the other. It is the nonparametric alternative to the Paired t-test. The U test is performed as a two-tailed test. If the sample size is large, the Z-test can be used for a one-sided test.

Hypothesis: H0: η = η0 H1 : η ≠ η0

Assumptions:

- Data are independent random samples from two populations that have the same shape and a scale that is continuous or ordinal if discrete
- Data are measured at an interval level of measurement

#### KRUSKAL-WALLIS

The Kruskal-Wallis test is a nonparametric test for the comparison of two or more iindependent samples to determine if the samples have come from different populations. It is the nonparametric equivalent to One-Way ANOVA but because it is performed on ranked data, it is less powerful than ANOVA.

Hypothesis: H0: η1 = η2 = … = ηk H1 : η1 ≠ η2 ≠ … ≠ ηk for k samples

Assumptions:

- data points must be independent from each other
- ideally have more than five data points per sample
- all individuals must be selected at random from the population
- all individuals must have equal chance of being selected
- sample sizes should be as equal as possible but some differences are allowed
- identically-shaped distribution for each group

#### MOOD’S MEDIAN TEST

The Mood's median Test (also called median test or sign scores test) is a nonparametric test for the comparison of two or more independent samples to determine if the samples have come from different populations. It is the nonparametric alternative to the One-Way ANOVA. The test is robust against outliers and errors in data and is particularly appropriate in the preliminary stages of analysis. Mood's median test is more robust than is the Kruskal-Wallis test against outliers, but is less powerful for data from many distributions, including the normal. It is also less powerful for moderate to large sample sizes.

Hypothesis: H0: η1 = η2 = … = ηk H1 : η1 ≠ η2 ≠ … ≠ ηk for k samples

Assumptions: Data from each population are independent random samples and the population distributions have the same shape.

Note: The test is a special case of Pearson's chi-square test. The data in each sample are assigned to two groups, one consisting of data whose values are higher than the median value in the two groups combined, and the other consisting of data whose values are at the median or below. A Pearson's chi-square test is then used to determine whether the observed frequencies in each group differ from expected frequencies derived from a distribution combining the two groups.

#### FRIEDMAN

Friedman test is a nonparametric analysis of a randomized block experiment and it is the nonparametric alternative to the Two-Way ANOVA. The test is a generalization of the paired sign test and it is used to detect differences in treatments across multiple test attempts. The procedure involves ranking each row (or block) together, then considering the values of ranks by columns. The test statistic for the Friedman's test is a Chi-square with k-1 degrees of freedom, where k is the number of repeated measures.

Hypothesis:

- H0: The distributions are the same across repeated measures (All treatment effects are zero)
- H1: The distributions across repeated measures are different (Not all treatment effects are zero)

Assumptions: This test makes no assumptions about the distribution of the data (e.g., normality) but the data must be balanced.