RankBased NonParametric Tests


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1 RankBased NonParametric Tests
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3 Parametric and Nonparametric Tests NonParametric Tests Most of the statistical tests that we have used throughout the semester have relied on certain specific assumptions about the distribution of the involved variables and/or their means, and have been set up to test hypotheses about specific population parameters Such tests (ztests, ttests, ANOVAs, Pearson s correlation) are called parametric tests
4 Parametric and Nonparametric Tests NonParametric Tests Though these parametric tests are robust to minor violations of their assumptions, they can lead to gross systematic errors when the data are strongly violate the underlying assumptions and can even be undefined for certain types of data (e.g., nominal or nonnumerical data). Certain tests do not rely on specific distributional assumptions or test hypotheses about particular population parameters. These tests are generally called nonparametric tests The chisquare tests introduced in the last lecture and Spearman s rank correlation coefficient test were examples of nonparametric tests.
5 Parametric versus Nonparametric Tests The advantages of parametric tests are that NonParametric Tests They are more powerful (i.e., you can detect smaller effect sizes with smaller samples) than comparable nonparametric tests when the parametric assumptions are correct (or approximately correct). The hypothesis tests are more specific and easier to interpret. The advantage of nonparametric tests are that They can be used when the distribution of the population is completely unknown They tend to be more robust to illbehaved (e.g., nonnormal, heteroscedastic, & multimodal) data They are less sensitive to outliers They can be used with nominal and ordinal data
6 Nonparametric tests In the last 70 years or so, statisticians have developed many different nonparametric tests. Those most widely used in the behavioral sciences tend to be rank randomization tests Rank randomization (or rankpermutation) tests are hypothesis tests based on the theoretical distribution of randomly assigned ranks. As a first step, they all require the conversion of raw scores to ordinal ranks This makes them obvious candidates for ordinal data, though these data usually still need to be modified
7 Rank Randomization Tests Advantages of rankbased tests: 1. Ranks are simpler, and rankbased tests are easier to compute 2. They are largely insensitive to the particular form of the population distributions and differences between the distributions underlying the scores in different samples 3. They tend to minimize the effects of large sample variances 4. They are insensitive to outlier scores and make it easier to deal with undetermined scores (e.g., time to task completion) 5. The distribution of randomly assigned ranks can be computed exactly
8 720 permutations Sample 1 Ranks Sample 2 Ranks
9 720 permutations Sample 1 Ranks R 1 = Σr Sample 2 Ranks R 2 = Σr W s = min(r 1,R 2 )
10 NonParametric Tests
11 NonParametric Tests
12 Rank Randomization Tests Popular rankbased tests: 1. The MannWhitney U (or Wilcoxon ranksum) test Nonparametric analogue to the independentsamples ttest 2. The Wilcoxon signedrank test Nonparametric analogue to the matchedsamples ttest 3. The KruskalWallis test Nonparametric analogue to the oneway ANOVA (independent meas.) 4. The Friedman test Nonparametric analogue to the repeatedmeasures ANOVA
13 A Note about Computing Ranks All of the rankbased tests will require that you compute ranks based on the total number of scores and from lowest to highest I.e., if you have 3 samples with 5 scores each, the lowest overall score should be assigned the rank 1 and the highest overall score should be assigned the rank 15 In case of ties, each tied score should be assigned the mean of the tied ranks I.e., if the 3 rd and 4 th lowest scores have the same value then you should assign them each a rank of 3.5, with the next highest value receiving a rank of 5. If the 7 th, 8 th, and 9 th scores are all tied, then you should assign them each a rank of 8, with the next highest value receiving a rank of 10.
14 Rank Randomization Tests Aside from converting the raw scores to ranks, the logical steps are similar to those for the parametric hypothesis tests: 1. State the null and alternative hypotheses about the population. 2. Use the null hypotheses to predict the characteristics that the sample ranks should have. 3. Use the samples to compute the test statistic 4. Compare the test statistic with the hypothesis prediction
15 The MannWhitney Test This is a test for independentmeasures (between subjects) research designs with two groups and is thus an alternative to the independentmeasures ttest The basic intuition behind the test is that: A real difference between the two treatments should cause the scores in one sample to be generally larger than the scores in the other sample If all the scores are ranked, the larger ranks should be concentrated in one sample and the smaller ranks should be concentrated in the other sample.
16 The MannWhitney Test The null and alternative hypotheses are a bit more vague than for the ttest, but still test for some sort of difference in central tendency H 0 : There is no difference between treatments. Therefore, there is no tendency for ranks in one sample to be systematically higher or lower than in the other sample H 1 : There is a difference between treatments. Therefore, the ranks in one sample should be systematically higher or lower than in the other sample
17 The MannWhitney Test When comparing two samples, the MannWhitney U statistic for sample 1 represents the sum of the number of scores in sample 2 outranked by scores in sample 1 Treatment A Raw Scores Ranks Treatment B Raw Scores Ranks The smaller of the U values is looked up in the table
18 Example Treatment A Raw Scores Ranks Points Treatment B Raw Scores Ranks Points
19 The MannWhitney Test: Steps In practice, the steps for computing the test statistic (U) are: 1. Rank all the observations from smallest to largest 2. Compute the sum of the ranks in each sample, using the following formulas to compute U statistics from the ranks R: U n ( n 1 ), R1 U n ( n R ) 3. U is the smaller of the sums of these counts
20 Example Treatment A Raw Scores Ranks ΣR 1 27 Treatment B Raw Scores Ranks ΣR 2 51 U n1( n1 1) R 2 6(7)
21 U 6 n 6 n In this case, U crit = 5, so we retain the null hypothesis Note: Most tables for rank permutation statistics are set up so that you reject the null hypothesis if the value is smaller than the tabled value.
22 The MannWhitney Test: Normal Approximation When n 1 and n 2 are sufficiently large (e.g., n 1,n 2 10) the distribution of rank sums becomes roughly normal and you can use a normal approximation, evaluated against a critical z value, to test for significance. U U nn n n N U U zu U
23 Wilcoxon s Signed Ranks Test This is a test for repeatedmeasures (withinsubjects) research designs with two treatment conditions and is thus an alternative to the repeatedmeasures ttest The basic intuition behind the test is that: A real difference between the two treatments should cause the difference scores to be generally positive or negative If all the difference scores are ranked and signed (according to whether they represent increases + or decreases ), the ranks should be concentrated in either the positive or negative set.
24 Wilcoxon s Signed Ranks Test Again, the null and alternative hypotheses are a bit more vague than for the repeated measures ttest, but test for some sort of difference in central tendency H 0 : There is no difference between treatments. Therefore, there is no tendency for the ranks of difference scores to be generally positive or negative H 1 : There is a difference between treatments. Therefore, the ranks of the difference scores should be systematically positive or negative
25 Wilcoxon s Signed Rank Test: Steps The steps for computing the test statistic (T) are: 1. Compute the difference scores NonParametric Tests 2. Rank all the difference scores from smallest to largest absolute value and assign them positive or negative signs based on whether they represent an increment or decrement 3. Compute separate sums for the positively and negatively signed sets 4. T is the smaller of the resulting signedranksums
26 Example (Suddath et al., 1990)
27 Example T = 9 N = 15 T crit is 25, which is greater than our test statistic, so we would reject the null hypothesis
28 Wilcoxon s Signed Ranks Test: Normal Approximation Again, when n is sufficiently large (e.g., n 20) the distribution of T becomes roughly normal and you can use a normal approximation, evaluated against a critical z value, to test for significance. T T n n1 4 n( n 1)(2n 2) 24 zt T T T n n1 T 4 n( n 1)(2n 1) 24
29 The KruskalWallis One Way ANOVA NonParametric Tests This is a test for independentmeasures research designs with more than two groups. As its name suggests, it is a nonparametric alternative to the parametric oneway ANOVA The basic intuition behind the test is analogous to that for the parametric oneway ANOVA: A real difference among treatments should cause the variability of scores between groups to be greater than the variability of scores within groups If all the scores are ranked the variability of ranksums between groups should be greater than the variability of ranksums within groups
30 The KruskalWallis One Way ANOVA NonParametric Tests The null and alternative hypotheses are very similar to those in the parametric oneway ANOVA. H 0 : There is no difference between treatments. There is no tendency for ranks in any sample to be systematically higher or lower than in any other condition. H 1 : There are differences between treatments. The ranks in at least one condition are systematically higher or lower than in another treatment condition
31 NonParametric Tests k i k n i j i i T ij i n M M F C x M k T i k n T i i i j j i r r r n r H C Parametric ANOVA: KruskalWallis: 1 1 within between df N k C df k 2 1 C N Comparison of Conceptual Formulas for the Parametric oneway ANOVA and KruskalWallis Test:
32 The KruskalWallis Test: Steps The steps for computing the test statistic (H) are: 1. Rank all scores from lowest to highest across all samples 2. Compute R, the sum of ranks in each sample 3. Plug into the following formula to solve for H: H 2 12 R i 3 N 1 N N n 1 i i
33 The KruskalWallis Test: Normal Approximation As in the other rankrandomization tests, when N is sufficiently large, the distribution of ranksums becomes approximately normal H is a linear combination (a weighted sum) of squaredrank sums, which means that it can be approximated by the distribution of a sum of squared normal variables For this reason, the significance of H is usually evaluated using a chisquared distribution with k1 degrees of freedom.
34 Friedman s Rank Test This is a test for repeatedmeasures research designs with more than two groups. It is the nonparametric analogue to a oneway repeatedmeasures ANOVA Just as the repeated measures ANOVA tests for consistent changes between individuals across treatment groups, Friedman s test looks for consistent rankings between individuals across treatment groups It can be used with any repeatedmeasures data, but is especially useful for measuring interrater agreement for rankings
35 The Friedman Test The null and alternative hypotheses are identical to those for the KruskalWallis Test. H 0 : There is no difference between treatments. There is no tendency for ranks in any sample to be systematically higher or lower than in any other condition. H 1 : There are differences between treatments. The ranks in at least one condition are systematically higher or lower than in another treatment condition
36 The Friedman Test: Steps 2 The steps for computing the test statistic ( R ) are: 1. Rank scores across each treatment group for each individual 2. Compute R, the sum of ranks for each group 3. Plug into the following formula to compute the test statistic: 12 ( k 1) R 3 n( k 1) 2 k 2 R i nk( k 1) i
37 Friedman Test Example Sommelier Wine 1 Wine 2 Wine 3 A B C D E F G R (2) 3 n( k 1) k 2 2 R Ri nk( k 1) i 12 7(3)(4) (7)(4)
38 χ 2 Distribution Upper Tail Probability df
39 Friedman Test Example Sommelier Wine 1 Wine 2 Wine 3 A B C D E F G R (2) 3 n( k 1) k 2 2 R Ri nk( k 1) i 12 7(3)(4) (7)(4) crit 5.99, retain H 0 In this case, we would conclude that there is no significant difference in quality between the 3 wines or that the sommeliers are unable to discern any differences.
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