1. Non-parametric methods in statistical testing
Douwe Postmus

2. Content Comparison of a continuous outcome variable between two groups
Independent samples
Student t-test (parametric)
Mann-Whitney-Wilcoxon test (non-parametric)
Dependent samples
Paired samples t-test (parametric)
Wilcoxon signed rank test (non-parametric)

3. Is a mother’s smoking status during pregnacy associated with the birth weight of an infant?

4. Parametric model
The birth weight of infants from mothers who smoked during the pregnacy is normally distributed with mean 𝜇 1 and variance 𝜎 2
The birth weight of infants from mothers who did not smoke during the pregnacy is normally distributed with mean 𝜇 2 and variance 𝜎 2
The study data are independent samples from these two probability distributions

5. Hypotheses
H0: µ1 = µ2 (or µ1 - µ2 = 0)
H1: µ1 ≠ µ2 (or µ1 - µ2 ≠ 0)

6. Results
If H0 is true, t follows a t-distribution with 𝑛 1 + 𝑛 2 −2 degrees of freedom

7. Are our assumptions reasonable?

8. What if the assumptions are not met?
Normality
Equal variances
Approach + -
Welch’s t-test
Intermediate or large samples (n>30):
Student t-test (central limit theorem)
Small samples:
Mann-Whitney-Wilcoxon test
Welch’s t-test performed on ranked data

9. Mann-Whitney-Wilcoxon test
The observations from group 1 are a random sample from some probability distribution F
The observations from group 2 are a random sample from some probability distribution G
The observations from group 1 are independent of the observations from group 2 (no pairing or clustering)

10. Hypotheses H0: F is equal to G
H1: F and G have the same shape but a different median (location-shift interpretation)

11. Test statistic (Wilcoxon W)
Step 1: group all observations together and rank them in order of increasing size
Step 2: calculate the sum of the ranks W of the observations that came from one of the groups
Group selection is arbitrary as the sum of the two rank sums is known
SPSS selects the group with the least amount of observations

12. Example - 10 randomly selected infants

13. Step 1: rank the observations

14. Step 2: calculate the sum of the ranks for infants from mothers who smoked during pregnacy

15. Distribution of W under H0
If H0 is true, each assignment of ranks to the 10 infants is equally likely
In total, there are 10*9*8*…*1 = 10! = 3,628,800 possible ways to assign ranks to the 10 infants
The distribution of W is obtained by enumerating the rank sums for all of these assignments

16. Results

17. Full dataset

18. Results
U = W – n2*(n2+1)/2
If H0 is true, U is approximately normally distributed with mean (n1*n2)/2 and variance n1*n2*(n1+n2+1)/12

19. Content Comparison of a continuous outcome variable between two groups
Independent samples
Student t-test (parametric)
Mann-Whitney-Wilcoxon test (non-parametric)
Dependent samples
Paired samples t-test (parametric)
Wilcoxon signed rank test (non-parametric)

20. How effective is a low-carb diet?
Twenty subjects submitted to a low-carb diet
Two measurements per subject
Body weight at the start of the diet (kg)
Body weight 16 weeks after the start of the diet (kg)

21. Study data

22. Paired samples t-test Assumptions: Hypotheses:
The difference in body weight between baseline and week 16 is normally distributed with mean 𝜇 and variance 𝜎
Hypotheses:
H0: µ = 0
H1: µ ≠ 0

23. Results
If H0 is true, t follows a t-distribution with n-1 degrees of freedom

24. Are our assumptions reasonable?

25. Wilcoxon signed-rank test
Assumptions:
The distribution of the difference in body weight between baseline and week 16 is symmetric around the median
Hypotheses:
H0: The distribution of the differences is symmetric around zero
H1: The distribution of the differences is symmetric around a value other than zero

26. Test statistic (W+)
Step 1: rank the absolute values of the differences
Exclude pairs for which the difference is zero
Step 2: obtain signed ranks by restoring the signs of the differences to the ranks
Step 3: calculate the sum of the ranks W+ that have a positive sign

27. Step 1: rank the absolute values of the differences

28. Step 2: obtain signed ranks by restoring the signs of the differences to the ranks

29. Step 3: calculate the sum of the ranks W+ that have a positive sign

30. Distribution of W+ under H0
If H0 is true, any particular assignment of signs to the 20 ranks is equally likely
In total, there are 220 = 1,048,576 possible ways to assign signs to the 20 ranks
The distribution of W+ is obtained by enumerating the sum of the ranks that have a positive sign for all of these assignments

31. Results

32. Summary
Non-parametric methods do not assume that the data follow any particular distributional form
More generally applicable than the corresponding parametric models at the cost of a (slight) reduction in power
Non-parametric methods are not assumption free!

33. Next lectures Date Location Speaker Topic 12 February TBD H. Burgerhof
Multiple testing: problems and some solutions
9 April
D. Postmus
Kaplan-Meier survival curves and the log-rank test
11 June

34. contact: d.postmus@umcg.nl