1. Chapter 14: Nonparametric Statistics

2. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
Where We’ve Been
Presented methods for making inferences about means and correlation
Methods required the data or the sampling distributions to be normally distributed
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

3. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
Where We’re Going
Inferential techniques requiring fewer or less stringent assumptions
Nonparametric tests based on ranks
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

4. 14.1: Distribution-Free Tests
Testing non-normal data with test based on normality may lead to
P(Type I error) > 
less than maximum power of the test (1 - ).
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

5. 14.1: Distribution-Free Tests
Parametric tests
(z, t, F)
Data or sampling distribution are normal
Non-parametric tests
(Rank-ordered, no assumed distribution)
Data or sampling distribution are skewed, or data is ordinal
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

6. 14.1: Distribution-Free Tests
Parametric tests
(z, t, F)
Data or sampling distribution are normal
Nonparametric statistics (or tests) based on the ranks of measurements are called rank statistics (or rank tests.)
Non-parametric tests
(Rank-ordered, no assumed distribution)
Data or sampling distribution are skewed, or data is ordinal
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

7. 14.2: Single-Population Inferences
The sign test provides inferences about population medians, or central tendencies, when skewed data or an outlier would invalidate tests based on normal distributions.
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

8. 14.2: Single-Population Inferences
One-tailed test for a Population Median, 
Test Statistic:
S = number of sample measurements greater than (less than) 0
Two-tailed test for a Population Median, 
Test Statistic:
S = larger of S1 and S2 where S1 is the number of measurements less than 0 and S2 the number greater than 0
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

9. 14.2: Single-Population Inferences
One-tailed test for a Population Median, 
Observed significance level:
p-value = P(xS)
Two-tailed test for a Population Median, 
Observed significance level:
p-value = 2P(xS)
where x has a binomial distribution with parameters
n and p = .5.
Reject H0 if p-value .
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

10. 14.2: Single-Population Inferences
Median time to failure for a band of compact disc players is 5,250 hours. Twenty players from a competitor are tested, with failure times from 5 hours to 6,575 hours. Fourteen of the players exceed 5,250 hours.
Do the competitor’s machines perform differently?
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

11. 14.2: Single-Population Inferences
Median time to failure for a band of compact disc players is 5,250 hours. Twenty players from a competitor are tested, with failure times from 5 hours to 6,575 hours. Fourteen of the players exceed 5,250 hours.
Do the competitor’s machines perform differently?
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

12. 14.2: Single-Population Inferences
Median time to failure for a band of compact disc players is 5,250 hours. Twenty players from a competitor are tested, with failure times from 5 hours to 6,575 hours. Fourteen of the players exceed 5,250 hours.
Do the competitor’s machines perform differently?
Do not reject H0
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

13. 14.3: Comparing Two Populations: Independent Samples
Wilcoxon Rank Sum Test
Used to test whether two independent samples have the same probability distribution
Samples must be random and independent.
Probability distributions must be continuous.
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

14. 14.3: Comparing Two Populations: Independent Samples
Wilcoxon Rank Sum Test
One-tailed test
H0: D1 and D2 are identical
Ha: D1 is shifted right of D2 or Ha: D1 is shifted left of D2
Test statistic:
T1, if n1 < n2
T2, if n1 > n2
Either if n1 = n2
Rejection region:
T1: T1 TU or T1  TL
T2: T2  TL or T2  TU
Two-tailed test
H0: D1 and D2 are identical
Ha: D1 is shifted either right or left of D2
Test statistic:
T1, if n1 < n2
T2, if n1 > n2
Either if n1 = n2
Rejection region:
T TL or T  TU
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

15. 14.3: Comparing Two Populations: Independent Samples
Reaction Times of Subjects Under the Influence of Drug A or B
Rank:
Value: 1 2 3 4 5 6 7
1.62
1.71
1.93
1.96
2.07
2.11
2.24 8 9 10 11 12 13
2.41
2.43
2.50
2.71
2.84
2.88
H0 : DA and DB are identical
Ha: DA is shifted right of DB or Ha: DA is shifted left of DB
 =.05
TA = = 25
TB = = 66
Test Statistic is TA, since nA < nB
TL (=.05, nA= 6, nB= 7) = 28 > TA = 25
Reject H0
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

16. 14.3: Comparing Two Populations: Independent Samples
Wilcoxon Rank Sum Test for Large Samples
One-tailed test
H0: D1 and D2 are identical
Ha: D1 is shifted right of D2 or Ha: D1 is shifted left of D2
Rejection region:
| z | > za
Two-tailed test
H0: D1 and D2 are identical
Ha: D1 is shifted either right or left of D2
Rejection region:
| z | > za/2
Test Statistic:
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

17. 14.4: Comparing Two Populations: Paired Difference Experiment
Judge
Product A
Product B
A - B
|A – B|
Rank of 1 6 4 2 5 8 3
7.5 -1 9 7 -2 10
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

18. 14.4: Comparing Two Populations: Paired Difference Experiment
Judge
Product A
Product B
A - B
|A – B|
Rank of 1 6 4 2 5 8 3
7.5 -1 9 7 -2 10
T- = Sum of negative ranks = 9
T+ = Sum of positive ranks = 46
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

19. 14.4: Comparing Two Populations: Paired Difference Experiment
Judge
Product A
Product B
A - B
|A – B|
Rank of 1 6 4 2 5 8 3
7.5 -1 9 7 -2 10
H0: The probability distributions of the ratings for products A and B are identical
Ha: The probability distributions of the ratings differ
= .05, two-tailed test
Test statistic: T = Smaller of T+ and T-
Rejection region: T 8 (see Table XIII in Appendix A)
T- = 9 > 8
Do not reject H0
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

20. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.5: Comparing Three or More Populations: Completely Randomized Design
Kruskal-Wallis H – test
Compares probability distributions for k populations or treatments
No assumption about the distributions
H0 : The k probability distributions are identical
Ha: At least two of the k probability distributions differ
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

21. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.5: Comparing Three or More Populations: Completely Randomized Design
Kruskal-Wallis H – test
k samples are random and independent.
For each sample nj  5.
The k probability distributions are continuous.
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

22. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.5: Comparing Three or More Populations: Completely Randomized Design
Kruskal-Wallis H – test
Test statistic:
n = total sample size
nj = measurements in sample j
Rj = rank sum of sample j
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

23. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.5: Comparing Three or More Populations: Completely Randomized Design
Kruskal-Wallis H – test
H = 0 All samples have the same mean rank
Large H Larger differences between sample mean ranks
If H0 is true, H ~ 2, with df = (k-1)
Reject H0 if H > 2
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

24. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.5: Comparing Three or More Populations: Completely Randomized Design
A study of three populations yielded the following:
Population A B C nj 15 Rj
235
439
361
Rj2
55,225
192,721
130,321
H0 : The k probability distributions are identical
Ha: At least two of the k probability distributions differ
= .05
df = 3-1= 2
2 .05=
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

25. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.5: Comparing Three or More Populations: Completely Randomized Design
A study of three populations yielded the following:
Population A B C nj 15 Rj
235
439
361
Rj2
55,225
192,721
130,321
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

26. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.5: Comparing Three or More Populations: Completely Randomized Design
A study of three populations yielded the following:
Population A B C nj 15 Rj
235
439
361
Rj2
55,225
192,721
130,321
H0 : The k probability distributions are identical
Ha: At least two of the k probability distributions differ
= .05
df = 3-1= 2
2.05 =
Since H = 8.19 > 2 .05= ,
reject H0
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

27. 14.6: Comparing Three or More Populations: Randomized Block Design
Friedman Fr-statistic
H0 : The p probability distributions are identical
Ha: At least two of the p probability distributions differ in location
Test statistic:
b = number of blocks (>5)
k = number of treatments (>5)
Rj = rank sum of treatment j
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

28. 14.6: Comparing Three or More Populations: Randomized Block Design
Friedman Fr-statistic
Treatments are randomly assigned to experimental units within the blocks.
Measurements can be ranked within blocks.
The p probability distributions from which the samples within each block are drawn are continuous.
Fr ~ 2 with k – 1 degrees of freedom.
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

29. 14.6: Comparing Three or More Populations: Randomized Block Design
A study of four treatments and six blocks yielded the following:
Population A B C D Rj 11 21 7
Rj2
121
441 49
H0 : The probability distributions for the p treatments are identical
Ha: At least two of the p probability distributions differ in location
= .05
df = 4-1= 3
2 .05 =
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

30. 14.6: Comparing Three or More Populations: Randomized Block Design
A study of four treatments and six blocks yielded the following:
Population A B C D Rj 11 21 7
Rj2
121
441 49
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

31. 14.6: Comparing Three or More Populations: Randomized Block Design
A study of four treatments and six blocks yielded the following:
Population A B C D Rj 11 21 7
Rj2
121
441 49
H0 : The probability distributions for the p treatments are identical
Ha: At least two of the p probability distributions differ in location
= .05
df = 4-1= 3
2 .05 =
Since H = 15.2 > 2 .05= ,
reject H0
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

32. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.7: Rank Correlation
Spearman’s Rank Correlation Coefficient
where
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

33. 14.7: Rank Correlation Spearman’s Rank Correlation Coefficient
where (cont.) ui = Rank of the ith observation in sample 1
vi = Rank of the ith observation in sample 2
n = Number of pairs of observations
Shortcut Formula for rs*
where
* A good approximation when there are few ties relative to n
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

34. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.7: Rank Correlation
Spearman’s Rank Correlation Coefficient
Perfect negative correlation
Perfect positive correlation
No correlation
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

35. 14.7: Rank Correlation
Spearman’s Nonparametric Test for Rank Correlation
One-Tailed Test Rejection region: |rs | > rs,
Two-Tailed test Rejection region: |rs | > rs,/2
Test Statistics: rs
Conditions
1. The sample of experimental units on which the two variables are measured must be randomly selected, and
2. The probability distributions of the two variables must be continuous.
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

36. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.7: Rank Correlation
Preseason Predictions for 2007 ACC Atlantic Division Football
Team
Predictor A
Predictor B
Boston College 1 5
Florida State 2
Wake Forest 3
Clemson 4
Maryland 6
N.C. State
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

37. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.7: Rank Correlation
Preseason Predictions for 2007 ACC Atlantic Division Football
Team
Predictor A
Predictor B
Boston College 1 5
Florida State 2
Wake Forest 3
Clemson 4
Maryland 6
N.C. State
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

38. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.7: Rank Correlation
Preseason Predictions for 2007 ACC Atlantic Division Football
Team
Predictor A
Predictor B
Boston College 1 5
Florida State 2
Wake Forest 3
Clemson 4
Maryland 6
N.C. State
From Table XIV,
with n = 6,
rs =.05 =.829, so
H0 is not rejected.
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics