# NONPARAMETRIC HYPOTHESIS TESTS

Nonparametric techniques make no assumptions about the specific shape or distribution from which the sample is drawn. This lack of assumptions is different from the other hypotheses tests such as ANOVA or t-tests (parametric tests) where the sample is assumed to be drawn from a population that is normally or approximately normally distributed. If normality is assumed, the power of the test is higher due to this normality restriction. However, if flexibility on distributional requirements is needed, then nonparametric techniques are superior. In general, nonparametric methodologies provide the following advantages over parametric tests:

• Normality or approximate normality does not have to be assumed.
• Fewer assumptions about the population are required and nonparametric tests don’t require population assumption of any specific distribution.
• Smaller sample sizes can be analyzed.
• Compared to parametric tests, nonparametric tests use data less efficiently.
• The power of the test is lower than that of the parametric tests.
• Samples with nominal and ordinal scales of measurement can be tested.
• Sample variances do not have to be equal (required in parametric tests).
• Nonparametric Chi-Square Goodness-of-Fit for Normality (Grouped Data).The chi-square test for goodness-of-fit is used to examine if a sample dataset could have been drawn from a population having a specified probability distribution. The probability distribution tested here is the normal distribution. The null hypothesis tested is such that the sample is randomly drawn from the normal distribution.
• Short Tip: Nonparametric test on normality (H0: the dataset is normally distributed).
• Model Input: Data Type B. Two input variables are required with at least 3 rows of data each.
• Upper Limit of Data Category, Frequency within that Category, Mean, Standard Deviation:
• >VAR1
• >VAR2
• >945
• >145
• Nonparametric Chi-Square Independence. The chi-square test for independence examines two variables to see if there is some statistical relationship between them. This test is not used to find the exact nature of the relationship between the two variables, but to simply test if the variables could be independent of each other. The null hypothesis tested is such that the variables are independent of each other.
• Short Tip: Nonparametric test on independence between two variables (H0: the variables are independent and have no effects on one another).
• Model Input: Data Type B. Two input variables are required with at least 3 rows of data each.
• Variables:
• > VAR1; VAR2
• Nonparametric Chi-Square Population Variance. The chi-square test for population variance is used for hypothesis testing and confidence interval estimation for a population variance. The population variance of a sample is typically unknown, and hence the need for quantifying this confidence interval. The population is assumed to be normally distributed.
• Short Tip: Nonparametric test on the sample variance compared to a hypothesized variance (H0: sample variance equals hypothesized variance).
• Model Input: Data Type A. Manually entered input variables are required.
• Hypothesized Variance, Sample Variance, Sample Size:
• >4
• >5
• >20
• Nonparametric: Cochran’s Q (Binary Repeated Measures). Runs the Cochran’s Test, which is the nonparametric equivalent of an ANOVA with repeated measures but where the values are binary 0 and 1. The null hypothesis tested is that the proportions of 1s and 0s for all variables are equivalent.
• Short Tip: Nonparametric equivalent of ANOVAwith multiple treatments (H0: the binary proportions are equivalent for all variables).
• Model Input: Data Type C. Two or more input variables are required. Different variables are arranged in columns and all variables must have at least 5 data points each, with the same number of total data points or rows per variable. Data must be 0 or 1 binary values only.
• Variables:
• >VAR1; VAR2; VAR3; …
• Nonparametric: D’Agostino–Pearson Normality Test.Runs the D’Agostino–Pearson test for normality to test the null hypothesis if the data is normally distributed.
• Short Tip: Tests for normality (H0: the data is normally distributed).
• Model Input: Data Type A. One input variable is required with at least 5 rows of data.
• Variable:
• >VAR1
• Nonparametric Friedman’s Test. The Friedman test is the extension of the Wilcoxon Signed-Rank test for paired samples. The corresponding parametric test is the Randomized Block Multiple Treatment ANOVA, but, unlike the ANOVA, the Friedman test does not require that the dataset be randomly sampled from normally distributed populations with equal variances. The Friedman test uses a two-tailed hypothesis test where the null hypothesis is such that the population medians of each treatment are statistically identical to the rest of the group; that is, there is no effect among the different treatment groups.
• Short Tip: Runs the nonparametric Freidman’s test, an equivalent of ANOVA with blocking variables.
• Model Input: Data Type C. Two or more input variables are required. Different variables are arranged in columns and all variables must have at least 3 data points each, with the same number of total data points or rows per variable.
• Variable:
• >VAR1; VAR2; VAR3; …
• Nonparametric Kruskal–Wallis Test. The Kruskal–Wallis test is the extension of the Wilcoxon Signed-Rank test by comparing more than two independent samples. The corresponding parametrictest is the One-Way ANOVA, but unlike the ANOVA, the Kruskal–Wallis does not require that the dataset be randomly sampled from normally distributed populations with equal variances. The Kruskal–Wallis test is a two-tailed hypothesis test where the null hypothesis is such that the population medians of each treatment are statistically identical to the rest of the group; that is, there is no effect among the different treatment groups.
• Short Tip: Runs the nonparametric Kruskal–Wallis test, an equivalent of ANOVA with Multiple Treatments.
• Model Input: Data Type C. Two or more input variables are required. Different variables are arranged in columns and all variables must have at least 3 data points each, with the same number of total data points or rows per variable.
• Variables:
• >VAR1; VAR2; VAR3; …
• Nonparametric Lilliefors Test for Normality. The Lilliefors test evaluates the null hypothesis of whether the data sample was drawn from a normally distributed population, versus an alternate hypothesis that the data sample is not normally distributed. If the calculated p-value is less than or equal to the alpha significance value, then reject the null hypothesis and accept the alternate hypothesis. Otherwise, if the p-value is higher than the alpha significance value, do not reject the null hypothesis. This test relies on two cumulative frequencies: one derived from the sample dataset and one from a theoretical distribution based on the mean and standard deviation of the sample data. An alternative to this test is the chi-square test for normality. The chi-square test requires more data points to run compared to the Lilliefors test.
• Short Tip: Runs nonparametric Lilliefors test for normality of your data (H0: the data is assumed to be normally distributed).
• Model Input: Data Type A. One input variable is required with at least 5 data points or rows of data.
• Variable:
• >VAR1
• Nonparametric: Mann–Whitney Test (Two Var).Runs the nonparametric Mann–Whitney Test for two independent samples (related to Wilcoxon Signed-Rank Test) and is the nonparametric equivalent of the Two-Sample T-Test for Independent Variables. The null hypothesis tested is that there is zero difference between the two variables.
• Short Tip: Nonparametric test on two variables (H0: no difference between the two medians).
• Model Input: Data Type B. Two input variables are required with at least 3 rows of data each. The two variables do not need to have the same number of rows.
• Variables:
• >VAR1; VAR2
• Nonparametric: Mood’s Multivariate Median Test. Returns the Nonparametric Mood’s test for medians of multiple variables simultaneously. It is an extension of the Nonparametric Wilcoxon Signed-Rank test for two variables extended to multiple variables. Mood’s test is related to the parametric ANOVA with Multiple Treatments and its Nonparametric Kruskal–Wallis Test equivalence.
• Short Tip: Nonparametrically tests if the medians from various variables are similar (H0: all medians are equal or homogeneous), related to Wilcoxon and Kruskal–Wallis tests.
• Model Input: Data Type C. Two or more input variables are required. Different variables are arranged in columns and all variables must have at least 3 data points each. Different numbers of total data points or rows per variable are allowed.
• Variables:
• >VAR1; VAR2; VAR3; …
• Nonparametric Runs Test for Randomness. The Runs test evaluates the randomness of a series of observations by analyzing the number of runs it contains. A run is a consecutive appearance of one or more observations that are similar. The null hypothesis tested is whether the data sequence is random, versus the alternate hypothesis that the data sequence is not random.
• Short Tip: Runs nonparametric Runs test for randomness of the data (H0: the data is random).
• Model Input: Data Type A. One input variable is required with at least 5 data points or rows of data.
• Variable:
• >VAR1
• Nonparametric: Shapiro–Wilk–Royston Normality Test. Runs the Shapiro–Wilk test for normality using the Royston algorithm to test the null hypothesis if the data is normally distributed.
• Short Tip: Tests for normality of your data (H0: the data is assumed to be normally distributed).
• Model Input: Data Type A. One input variable is required with at least 3 rows of data.
• Variable:
• >VAR1
• Nonparametric Wilcoxon Signed-Rank Test (One Var). The single-variable Wilcoxon Signed-Rank test looks at whether a sample dataset could have been randomly drawn from a population whose median is being hypothesized. The corresponding parametric test is the one-sample t-test, which should be used if the underlying population is assumed to be normal, providing a higher power on the test.
• Short Tip: Runs a nonparametric Wilcoxon test for one variable (H0: the median is equivalent to zero).
• Model Input: Data Type A. One input variable is required with at least 3 rows of data.
• Variable:
• >VAR1
• Nonparametric Wilcoxon Signed-Rank Test (Two Var). The Wilcoxon Signed-Rank test for paired variables looks at whether the median of the differences between the two paired variables are equal. This test is specifically formulated for testing the same or similar samples before and after an event (e.g., measurements taken before a medical treatment are compared against those measurements taken after the treatment to see if there is a difference). The corresponding parametric test is the two-sample t-test with dependent means, which should be used if the underlying population is assumed to be normal, providing a higher power on the test.
• Short Tip: Nonparametric test on equality of medians (H0: the two variables have equal medians).
• Model Input: Data Type B. Two input variables are required with at least 3 rows of data each.
• Variable 1, Variable 2, Hypothesized Median
• >VAR1; VAR2
• >0
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