Non-Parametric Tests

Transcription

1 Non-Parametric Tests

2 Non Parametric Tests Do not make as many assumptions about the distribution of the data as the t test. Do not require data to be Normal Good for data with outliers Non-parametric tests based on ranks of the data Work well for ordinal data (data that have a defined order, but for which averages may not make sense).

3 We ll cover three non-parametric tests: Sign Test Wilcoxon Signed-rank Test Wilcoxon Rank-sum Test (Mann-Whitney U Test) Kruskal-Wallis Test * paired data 2 independent samples > 2 independent samples * The Kruskal-Wallis Test won t be discussed further, but explanation can be found in Rosner 12.7

4 Paired data example: body image Children in an orthodontia study were asked to rate how they felt about their teeth on a 5 point scale. Survey administered before and after treatment. How do you feel about your teeth? 1. Wish I could change them 2. Don t like, but can put up with them 3. No particular feelings one way or the other 4. I am satisfied with them 5. Consider myself fortunate in this area

5 Paired data example: body image These data are Ordinal They have a definite order, but averages may not have a clear interpretation. Paired Two observations (before and after treatment) are made on each child. How do you feel about your teeth? 1. Wish I could change them 2. Don t like, but can put up with them 3. No particular feelings one way or the other 4. I am satisfied with them 5. Consider myself fortunate in this area

6 Sign Test Used for paired data Can be ordinal or continuous Very simple and easy to interpret Makes no assumptions about distribution of the data Not very powerful

7 Sign Test: null hypothesis The null hypothesis for the sign test is H 0 : the median difference is zero To evaluate H 0 we only need to know the signs of the differences If half the differences are positive and half are negative, then the median = 0 (H 0 is true). If the signs are more unbalanced, then that is evidence against H 0.

8 child Rating before Rating after Example: Body image data Use the sign test to evaluate whether these data provide evidence that ortho tx improves children s image of their teeth.

9 child Rating before Rating after difference Example: Body image data Use the sign test to evaluate whether these data provide evidence that ortho tx improves children s image of their teeth. First, for each child, compute the diffference between the two ratings

10 differences The sign test looks only at the signs of the differences One negative difference Four zero differences (ties) Fifteen positive differences When calculating the sign test we ignore the zero differences

11 P-value for sign test The p-value is the probability of an outcome as or more extreme (under H 0 ) than that observed. We observed 15 positives and 1 negative. If H 0 were true we d expect an equal number of positive and negative differences. Outcomes that would be more extreme: more that 15 positives or less than 1 positives

12 P-value for sign test P-value = P(X > 15) + P(X <1) X is the number of positive differences Under H 0, X is Binomial(n = 16, p = 0.5) n = 16 because the sign test disregards the zero differences Compute P-value using Binomial tables P-value = P(X=15) + P(X=16) + P(X=1) + P(X=0) = =

13 Wilcoxon Signed-rank test Wilcoxon Signed-rank test is another nonparametric test used for paired data. It uses the magnitudes of the differences the sign test does not More powerful than the sign test More difficult to interpret than the sign test

14 differences The Wilcoxon signed-rank test also looks at the (nonzero) differences It focuses on magnitudes of differences Keep track of which differences are positive (red) and negative (black)

15 Ranks Avg. Ranks magnitudes of differences Assign ranks to the observations based on the magnitudes of the differences When there are ties in the data assign average ranks for the tied observations e.g. average of ranks 2,3,4,5 is 3.5

16 Ranks Avg. Ranks magnitudes of differences The statistic for the signed-rank test is the sum of the ranks of the positive differences R 1 = 1 + 3* * *15 = 132.5

17 R 1 : What does it mean? With 16 observations R 1 could range from 0 (all differences are negative) to 136 (all differences are positive). If H 0 were true we d expect R 1 to be near the middle of this range, in this case, 68. R 1 = appears to be evidence against H 0 Need a p-value

18 Signed-rank test p-value For n > 15, can use a normal approximation ( +1) µ = nn 4 ( 3 ) 2 n ( n + 1)( 2n + 1) t = i t σ i where n is the number of non-zero differences and t i are the numbers of ties in each group of ties (note that if t i = 1 then the term is 0) The two-sided p-value is given by R1 p value = 2 P N(0,1) > µ 0.5 σ

19 p-value for body image example 16( ) µ = = 4 68 There are 4 people tied with difference 2, 8 with difference 3, and 3 tied with difference 4. So ( 3 ) ( 3 ) ( 3 ) ( 3 t ) i t = + + = 588 i And so, σ 2 = =

20 p-value for body image example p value = 2 P N(0,1) > ( (0,1) > 3 26) = 2 P N. = =

21 p-value for signed-rank test If n < 15 then should not use Normal approximation, but instead use an exact p- value. See 13.2 in text for example of calculating an exact p-value. In body image example, exact p-value is

22 Wilcoxon Rank Sum Test Used to compare two independent samples Equivalent to Mann-Whitney U test. Like the Signed-rank test, the Rank-Sum test is based on the ranks of the data.

23 Example: Shear Strengths Strength (MPa) Shear Strengths of Ceramics press layer Application Type Wish to compare two methods of preparing ceramics in terms of product strength. Two methods of preparation Press (n=10) Layer (n=10) Note outliers in press group T test not appropriate Large outliers in press group Data likely not Normal

24 Shear Strengths of Ceramics Computing the Wilcoxon Rank-Sum Test Statistic Rank of Strengths Assign ranks to the combined sample press layer Application Type

25 Rank of Strengths Shear Strengths of Ceramics press layer Application Type Computing the Wilcoxon Rank-Sum Test Statistic Assign ranks to the combined sample Choose a group, and sum the ranks in that group The Rank Sum is R 1 = = 77

26 Interpretation of rank sum Like the signed-rank statistic, the rank sum does not have an obvious interpretation. It will depend on the numbers of observations in the entire sample and in the chosen group. In this case (total = 20, number in group = 10), R 1 could range from 55 to 155. If the groups are equal we d expect R 1 to be in the middle, around 105. R 1 = 77 seems rather on the low end

27 Rank-sum p-value distribution of rank sum (N=20, n1=10) rank sum The null hypothesis is that the two distributions are equivalent The distribution of R 1 under H 0 is all possible rank sums that could occur when 10 ranks are randomly chosen from 20.

28 Rank-sum p-value distribution of rank sum (N=20, n1=10) p =.0178 p =.0178 The p-value is the percentage of possible combinations that result in a result as extreme as R 1 = 77 2-sided (exact) p-value is p = = rank sum

29 Normal approximation p-value Exact p-value is difficult to compute. Can use Normal approximation when both groups have at least 10 observations See text 13.3 (p. 126) for computation details In this example, the Normal approximation p- value is p=

30 Non Parametric Tests wrap up Non-parametric tests: Do not require data to be Normal Good for data with outliers Useful for ordinal data Three tests presented Sign test Wilcoxon signed-rank test Wilcoxon rank-sum test (Mann-Whitney U test)