NON-PARAMETRIC TESTS

 Non-parametric tests are used when the independent variables are non-metric.  Like parametric tests, nonparametric tests are available for testing variables from one sample, two independent samples or two related samples.

 One Sample

Sometimes the researcher wants to test whether the observations for a particular variable could reasonably have come from a particular distribution, such as the normal, uniform, or Poisson distribution.  Knowledge of the distribution is necessary for finding probabilities corresponding to know values of the variable. 

The Komogorov-Smirnov (K-S) one-sample test is one such goodness-of-fit test.  The K-S compares the cumulative distribution function for a variable with a specified distribution.

The chi-square test can also be performed on a single variable from one sample.  In this context, the chi-square serves as goodness-of-fit test.  It tests whether a significant difference exits between the observed number of cases in each category and the expected number.  Other one-sample nonparametric tests include the runs test and the binomial test.  The runs test is a test of randomness for the dichotomous test and the binomial test.  This test is conducted by determining whether the order or sequence in which observations are obtained is random.  The binomial test is also a goodness-of-fit test for dichotomous variables.  It tests the goodness of fit of the observed number of observations in each category to the number expected under a specified binomial distribution. 

Two Independent Samples

When the difference in the location of two populations is to be compared based on observations from two independent samples, and the variables is measured on an ordinal scale, the Mann-Whitney U test can be used.  This test corresponds to the two-independent-sample t test for interval scale variables, when the variances of the two populations are assumed equal.  In the Mann-Whitney U test, the two samples are combined and the cases are ranked in order of increasing size.  The test statistic, U, is computed as the number of times a score from sample 1 or group 1 precedes a score from group 2.  If the samples are from the same population, the distribution of scores from the two groups in the rank list should be random.  An extreme value of U would indicate a nonrandom pattern, pointing to the inequality of the two groups.  For samples of less than 30, the exact significance level for U is computed.  For larger samples, U is transformed into a normally distributed z statistic. 

We examine again the difference in the Internet usage of males and females.  This time, though, the Mann-Whitney U test is used.  Again, a significant difference is found between the two groups, corroborating the results of the two-independent-samples t test. Because the ranks are assigned from the smallest observation to the largest, the higher mean rank of males indicates that they use the Internet to a greater extent than females.

Researchers often wish to test for a significant difference in proportions obtained from two independent samples.  As an alternative to the parametric z test, one could also use the cross-tabulation procedure to conduct a chi-square test.  In this case, we will have a 2 x 2 table.  One variable will be used to denote the sample and will assume the value 1 for sample 1 and the value of 2 for sample 2.  The other variable will be the binary variable of interest.

Paired Samples

An important nonparametric test for examining differences in the location of two populations based on paired observations is the Wilcoxon matched-pairs signed-ranks test.  This test analyzes the differences between the paired observations, taking into account the magnitude of the differences.  It computes the differences between the pairs of variables and ranks the absolute differences.  The next step is to sum the positive and negative ranks.  The test statistics, z, is computed from the positive and negative and negative rank sums.  Under the null hypothesis of no difference, z is a standard normal variable with mean 0 and variance 1 for large samples.  This test corresponds to the paired t test.

            In summary, this discussion has focused on hypothesis testing, descriptive vs. inferential analysis, null and alternative hypotheses, type I and type II errors, significance level, decision rule, one tailed vs. two tailed tests, steps in conducting a hypothesis test, specific hypothesis tests, cross tabulation, parametric tests, non-parametric tests and paired samples.