1. Some Nonparametric Methods
stat120C
UCI

2. Introduction: Why use nonparametric methods?
A statistical model usually has several assumptions.
Parametric models make more assumptions than nonparametric models
As an example, consider one-sample t-test. What would happen if
The normality assumption does not hold?
There are several outliers?

3. Parametric vs Nonparametric
Nonparametric methods are usually more robust to violations of assumptions and outliers
When assumptions hold, parametric methods are usually more powerful
Nonparametric methods do not assume that the data follow any particular distribution
In many tests, data values are replaced by ranks. The results are invariant under any monotonic transformation

4. A nonparametric test for comparing paired samples – Wilcoxon signed rank test
An example: measurements before and after a treatment Before: After: Question: is this treatment efficient? If normality can be assumed, we can consider paired t-test. What if the distributional assumptions underlying t-test cannot be satisfied?

5. The Wilcoxon signed rank test
An example: measurements before and after a treatment
Before:
After:
Step 1: calculate Di
Step 2: Rank |Di|
Step 3: Calculate W+, which is defined as
Step 4: make statistical inference
Based on exact distributions
Based on asymptotic distributions (for large n)

6. The example W+=2+4=6 Under the null, the sign is random and W+
W+=2+4=6
Under the null, the sign is random and W+
should not be too large or too small.

7. Based on exact distributions
Probabilities under the null hypothesis:
signed ranks w+ prob
(observed)
Evidence of positive change
P-value
= 0.875
Evidence of positive change

8. What is the null hypothesis
H0: the random sample is drawn from a distribution with zero median
H1: the random sample is drawn from a distribution whose median is not zero

9. Inference based on exact distribution
We can use the R function “wilcox.test” to obtain p-values
> wilcox.test(c(2,-4,-1,10))
Wilcoxon signed rank test
data: c(2, -4, -1, 10)
V = 6, p-value = 0.875
alternative hypothesis: true location is not equal to 0

10. Inference based on asymptotic distributions
The asymptotic distribution of W+ under the null hypothesis of zero median

11. Proof

12. Asymptotic Wilcoxon signed rank test

13. A nonparametric test for comparing two samples: the Mann-Whitney-Wilcoxon test
In a study of four subjects, two are randomly assigned to the treatment group and two to the control group. The following values are observed
When sample size is small (such as the above example) or the normality assumption does not hold, the two-sample t-test is not accurate

14. An example

15. An example
Two-sided p-value=
1/6+1/6+1/6+1/6=2/3

16. The null hypothesis of the Mann-Whitney-Wilcoxon test
The null hypothesis for the two-sample t-test says that the two populations means are the same
The null hypothesis for the Mann-Whitney-Wilcoxon test says that the two random samples were drawn from the same distribution

17. A nonparametric test for comparing two samples: the Mann-Whitney-Wilcoxon test

18. Asymptotic Mann-Whitney-Wilcoxon test

19. Asymptotic Mann-Whitney-Wilcoxon test