Chapter 17 Analyzing Data Using Nonparametric Statistics Parametric vs. Nonparametric Statistics
Parametric Statistics are statistical techniques based on assumptions about the population from
which the sample data are collected.
Assumption that data being analyzed are randomly selected from a normally distributed population. Requires quantitative measurement that yield interval or ratio level data.
Nonparametric Statistics are based on fewer assumptions about the population and the parameters. Sometimes called “distribution-free” statistics.
A variety of nonparametric statistics are available for use with nominal or ordinal data.
Advantages of Nonparametric Techniques
Sometimes there is no parametric alternative to the use of nonparametric statistics. Certain nonparametric test can be used to analyze nominal data. Certain nonparametric test can be used to analyze ordinal data.
The computations on nonparametric statistics are usually less complicated than those for parametric statistics, particularly for small samples.
Probability statements obtained from most nonparametric tests are exact probabilities. Disadvantages of Nonparametric Statistics
Nonparametric tests can be wasteful of data if parametric tests are available for use with the data. Nonparametric tests are usually not as widely available and well know as parametric tests. For large samples, the calculations for many nonparametric statistics can be tedious. Runs Test
Test for randomness - is the order or sequence of observations in a sample random or not Each sample item possesses one of two possible characteristics Run - a succession of observations which possess the same characteristic Example with two runs: F, F, F, F, F, F, F, F, M, M, M, M, M, M, M Example with fifteen runs: F, M, F, M, F, M, F, M, F, M, F, M, F, M, F Runs Test: Sample Size Consideration
Sample size: n
Number of sample member possessing the first characteristic: n1 Number of sample members possessing the second characteristic: n2 n = n1 + n2
If both n1 and n2 are ￡ 20, the small sample runs test is appropriate. Runs Test: Large Sample
If either n1 or n2 is > 20, the sampling distribution of R is approximately normal.
?R?n?n12n1n22?1?R?2n1n2(2n1n2?n1?n2)(n1?n2)2?(n1?n2?1)Z?R???RR Mann-Whitney U Test
Mann-Whitney U test - a nonparametric counterpart of the t test used to compare the means of two independent populations. Nonparametric counterpart of the t test for independent samples Does not require normally distributed populations
May be applied to ordinal data Assumptions
Independent Samples
At Least Ordinal Data
Mann-Whitney U Test: Sample Size Consideration
Size of sample one: n1 Size of sample two: n2
If both n1 and n2 are ≤ 10, the small sample procedure is appropriate.
If either n1 or n2 is greater than 10, the large sample procedure is appropriate. Mann-Whitney U Test: Small Sample Example
Mann-Whitney U Test: Formulas for Large Sample Case
nU1?n1n2?1(n1?1)2?W1?W2nU2?n1n2?2(n2?1)2W12where: n1? number in group 111U?n1n2??1??nn???U??n?n12n?n?n?n12122?1?UnW21? number in group 2? sum or the ranks of values in group 1Z?12U???UU Wilcoxon Matched-Pairs Signed Rank Test
Mann-Whitney U test is a nonparametric alternative to the t test for two independent samples. If the two samples are related, the U test is not applicable. Handle related data
Serves as a nonparametric alternative to the t test for two related samples A nonparametric alternative to the t test for related samples Before and After studies
Studies in which measures are taken on the same person or object under different conditions Studies or twins or other relatives
Differences of the scores of the two matched samples Differences are ranked, ignoring the sign Ranks are given the sign of the difference Positive ranks are summed
Negative ranks are summed T is the smaller sum of ranks
Wilcoxon Matched-Pairs Signed Rank Test: Sample Size Consideration
n is the number of matched pairs
If n > 15, T is approximately normally distributed, and a Z test is used. If n ￡ 15, a special “small sample” procedure is followed. The paired data are randomly selected.
The underlying distributions are symmetrical.
Wilcoxon Matched-Pairs Signed Rank Test: Small Sample Example
Wilcoxon Matched-Pairs Signed Rank Test: Large Sample Formulas
??T???n??n?1?4n?n?1??2n?1?24T??Twhere: n = number of pairsT = total ranks for either + or - differences, whichever is less in magnitude TZ??T Kruskal-Wallis K Statistic 2?C Kruskal-Wallis Test 12?Tj???3?n?1?K?? A nonparametric alternative to one-way analysis of variance n?n?1??j?1nj??? May used to analyze ordinal data where: C = number of groups No assumed population shape n = total number of items Assumes that the C groups are independent ? total of ranks in a groupTj Assumes random selection of individual items nj = number of items in a group Friedman Test 2K?χ, with df = C -1 A nonparametric alternative to the randomized block design Assumptions
The blocks are independent.
There is no interaction between blocks and treatments.
C2122 Observations within each block can be ranked. ?3b(C?1)?r?bC(C?1)?Rjj?1 Hypotheses
Ho: The treatment populations are equal Ha: At least one treatment population yields larger values than at least one other treatment population
where: C? number of treatment levels (columns)b = number of blocks (rows)Rj= total ranks for a particular treatment levelj = particular treatment level Spearman’s Rank Correlation
Analyze the degreeof association of two variables Applicable to ordinal level data (ranks)
?2r??2, with df = C - 1rs?1?6?d2nn?1?2?where: n = number of pairs being correlatedd = the difference in the ranks of each pair
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