13-1 Basics of Nonparametric Tests

• Parametric tests have requirements about the distribution of the populations involved

• Nonparametric (or distribution-free) tests do not require that samples come from populations with normal distributions or any other particular distributions

• Advantages of Nonparametric tests:

  • they can be applied to a wider variety of situations because they have less rigid requirements

  • they can be applied to more data types

• Disadvantages of Nonparametric tests:

  • they tend to waste information because exact numerical data are often reduced to a qualitative form

  • they are not as efficient as parametric tests

• Data are sorted when they are arranged according to some criterion, such as smallest to largest or best to worst. A rank is a number assigned to an individual sample item according to its order in the sorted list.

13-2 Sign Test

• A sign test is a nonparametric test that uses positive and negative signs to test different claims, including claims involving matched pairs of sample data, claims involving nominal data with two categories, claims about the median of a single population

• If the two sets of data have equal medians, the number of positive signs should be approximately equal to the number of negative signs

13-3 Wilcoxon Signed-Ranks Test for Matched Pairs

• The Wilcoxon signed-ranks testis a nonparametric test that uses ranks for these applications:

  • Testing a claim that a population of matched pairs has the property that the matched pairs have differences with a median equal to zero

  • Testing a claim that a single population of individual values has a median equal to some claimed value

• When testing a claim about the median of a single population, create matched pairs by pairing each sample value with the claimed value of the median. The procedure can then be used.

13-4 Wilcoxon Rank-Sum Test for 2 Independent Samples

• The Wilcoxon rank-sum test is a nonparametric test that uses ranks of sample data from 2 independent populations to test this null hypothesis: H0: two independent samples come from populations with equal medians. The alternative hypothesis can be that the 2 populations have different means, or the first population has a median greater than the median of the 2nd population, or the first population has a median less than the median of the 2nd population

• Do not confuse the Wilcoxon rank-sum test for 2 independent samples with the Wilcoxon signed-ranks test for matched pairs

13-5 Kruskal-Wallis Test for Three or More Samples

• The Kruskal-Wallis test (also called the H Test) is a nonparametric test that uses ranks of combined simple random samples from 3 or more independent populations to test the null hypothesis that the populations have the same median

• The test statistic H has a distribution that can be approximated by the chi-square distribution provided that each sample has at least 5 observations

13-6 Rank Correlation

• The rank correlation test (or Spearman's rank correlation test) is a nonparametric test that uses ranks of sample data consisting of matched pairs. It is used to test for an association between two variables

• r subscript s is often called the Spearman's rank correlation coefficient

• Advantages: rank correlation can be used with paired data that are ranks or can be converted to ranks, and can be used to detect some (not all) relationships that are not linear

• Disadvantages: it has an efficiency rating of 0.91

13-7 Runs Test for Randomness

• After characterizing each data value as 1 of 2 separate categories, a run is a sequence of data having the same characteristic, the sequence is preceded and followed by data with a different characteristic or by no data at all

• The runs test uses the number of runs in a sequence of sample data to test for randomness in the order of the data