1. Non-Parametric Methods Professor of Epidemiology and Biostatistics
Statistics for Health Research
Non-Parametric Methods
Peter T. Donnan
Professor of Epidemiology and Biostatistics

2. Objectives of Presentation
Introduction
Ranks & Median
Wilcoxon Signed Rank Test
Paired Wilcoxon Signed Rank
Mann-Whitney test
Spearman’s Rank Correlation Coefficient
Others….

3. What are non-parametric tests?
‘Parametric’ tests involve estimating parameters such as the mean, and assume that distribution of sample means are ‘normally’ distributed
Often data does not follow a Normal distribution eg number of cigarettes smoked, cost to NHS etc.
Positively skewed distributions

4. A positively skewed distribution

5. What are non-parametric tests?
‘Non-parametric’ tests were developed for these situations where fewer assumptions have to be made
NP tests STILL have assumptions but are less stringent
NP tests can be applied to Normal data but parametric tests have greater power IF assumptions met

6. Ranks
Practical differences between parametric and NP are that NP methods use the ranks of values rather than the actual values
E.g.
1,2,3,4,5,7,13,22,38,45 - actual
1,2,3,4,5,6, 7, 8, 9,10 - rank

7. Median
The median is the value above and below which 50% of the data lie.
If the data is ranked in order, it is the middle value
In symmetric distributions the mean and median are the same
In skewed distributions, median more appropriate

8. Median
BPs:
135, 138, 140, 140, 141, 142, 143
Median=

9. Median
BPs:
135, 138, 140, 140, 141, 142, 143
Median=140
No. of cigarettes smoked:
0, 1, 2, 2, 2, 3, 5, 5, 8, 10
Median=

10. Median
BPs:
135, 138, 140, 140, 141, 142, 143
Median=140
No. of cigarettes smoked:
0, 1, 2, 2, 2, 3, 5, 5, 8, 10
Median=2.5

11. T-test
T-test used to test whether the mean of a sample is sig different from a hypothesised sample mean
T-test relies on the sample being drawn from a normally distributed population
If sample not Normal then use the Wilcoxon Signed Rank Test as an alternative

12. Wilcoxon Signed Rank Test
NP test relating to the median as measure of central tendency
The ranks of the absolute differences between the data and the hypothesised median calculated
The ranks for the negative and the positive differences are then summed separately (W- and W+ resp.)
The minimum of these is the test statistic, W

13. Wilcoxon Signed Rank Test: Example
The median heart rate for an 18 year old girl is supposed to be 82bpm. A student takes the pulse rates of 8 female students (all aged 18):
83, 90, 96, 82, 85, 80, 81, 87
Do these results suggest that the median might not be 82?

14. Wilcoxon Signed Rank Test: Example
H0:

15. Wilcoxon Signed Rank Test: Example
H0: median=82
H1:

16. Wilcoxon Signed Rank Test: Example
H0: median=82
H1: median≠82

17. Wilcoxon Signed Rank Test: Example
H0: median=82
H1: median≠82
Two-tailed test
Because one result equals 82 this cannot be used in the analysis

18. Wilcoxon Signed Rank Test: Example
Result
Above or below median
Absolute difference from median=82
Rank of difference 83 + 1
1.5 90 8 6 96 14 7 85 3 4 80 - 2 81 87 5
W+= =23.5 W-= 3+1.5=4.5 So, W=4.5
n=7, so the value of W > tabulated value of 2, so p>0.05

19. Wilcoxon Signed Rank Test: Example
Therefore, the student should conclude that these results could have come from a population which had a median of 82 as the result is not significantly different to the null hypothesis value.

20. Wilcoxon Signed Rank Test Normal Approximation
As the number of ranks (n) becomes larger, the distribution of W becomes approximately Normal
Generally, if n>20
Mean W=n(n+1)/4
Variance W=n(n+1)(2n+1)/24
Z=(W-mean W)/SD(W)

21. Wilcoxon Signed Rank Test Assumptions
Population should be approximately symmetrical but need not be Normal
Results must be classified as either being greater than or less than the median ie exclude results=median
Can be used for small or large samples

22. Paired samples t-test
Disadvantage: Assumes data are a random sample from a population which is Normally distributed
Advantage: Uses all detail of the available data, and if the data are normally distributed it is the most powerful test

23. The Wilcoxon Signed Rank Test for Paired Comparisons
Disadvantage: Only the sign (+ or -) of any change is analysed
Advantage: Easy to carry out and data can be analysed from any distribution or population

24. Paired And Not Paired Comparisons
If you have the same sample measured on two separate occasions then this is a paired comparison
Two independent samples is not a paired comparison
Different samples which are ‘matched’ by age and gender are paired

25. The Wilcoxon Signed Rank Test for Paired Comparisons
Similar calculation to the Wilcoxon Signed Rank test, only the differences in the paired results are ranked
Example using SPSS:
A group of 10 patients with chronic anxiety receive sessions of cognitive therapy. Quality of Life scores are measured before and after therapy.

26. Wilcoxon Signed Rank Test example
QoL Score
Before
After 6 9 5 12 3 4 2 1 8 10

27. Wilcoxon Signed Rank Test example

31. SPSS Output
p < 0.05

32. Mann-Whitney test
Used when we want to compare two unrelated or INDEPENDENT groups
For parametric data you would use the unpaired (independent) samples t-test
The assumptions of the t-test were:
The distribution of the measure in each group is approx Normally distributed
The variances are similar

33. Example (1) The following data shows the number
of alcohol units per week collected in a
survey:
Men (n=13): 0,0,1,5,10,30,45,5,5,1,0,0,0
Women (n=14): 0,0,0,0,1,5,4,1,0,0,3,20,0,0
Is the amount greater in men compared
to women?

34. Example (2) How would you test whether the
distributions in both groups are
approximately Normally distributed?

35. Example (2) How would you test whether the
distributions in both groups are
approximately Normally distributed?
Plot histograms
Stem and leaf plot
Box-plot
Q-Q or P-P plot

36. Boxplots of alcohol units per week by gender

37. Example (3)
Are those distributions symmetrical?

38. Example (3) Are those distributions symmetrical? Definitely not!
They are both highly skewed so not
Normal. If transformation is still not Normal
then use non-parametric test – Mann Whitney
Suggests perhaps that males tend to
have a higher intake than women.

39. Mann-Whitney on SPSS

44. Normal approx (NS)
Mann-Whitney (NS)

45. Spearman Rank Correlation
Method for investigating the relationship between 2 measured variables
Non-parametric equivalent to Pearson correlation
Variables are either non-Normal or measured on ordinal scale

46. Spearman Rank Correlation Example
A researcher wishes to assess whether
the distance to general practice
influences the time of diagnosis of
colorectal cancer.
The null hypothesis would be that
distance is not associated with time to
diagnosis. Data collected for 7 patients

47. Distance from GP and time to diagnosis
Distance (km)
Time to diagnosis (weeks) 5 6 2 4 3 8 20 45 10

48. Scatterplot

49. Distance from GP and time to diagnosis
(km)
Time
(weeks)
Rank for
distance
time
Difference
in Ranks D2 2 4 1 3 -2 5 6 7 -4 16 8 10 20
5.5
0.5
0.25 45
1.5
2.25
Total = 0
d2=28.5

50. Spearman Rank Correlation Example
The formula for Spearman’s rank
correlation is:
where n is the number of pairs

51. Spearman’s on SPSS

52. Spearman’s in SPSS

53. Spearman’s in SPSS

54. Spearman’s in SPSS

55. Spearman Rank Correlation Example
In our example, rs=0.468
In SPSS we can see that this value is not significant, ie.p=0.29
Therefore there is no significant
relationship between the distance to a
GP and the time to diagnosis but note that correlation is quite high!

56. Spearman Rank Correlation
Correlations lie between –1 to +1
A correlation coefficient close to zero indicates weak or no correlation
A significant rs value depends on sample size and tells you that its unlikely these results have arisen by chance
Correlation does NOT measure causality only association

57. Chi-squared test
Used when comparing 2 or more groups of categorical or nominal data (as opposed to measured data)
Already covered!
In SPSS Chi-squared test is test of observed vs. expected in single categorical variable

58. More than 2 groups So far we have been comparing 2 groups
If we have 3 or more independent groups and data is not Normal we need NP equivalent to ANOVA
If independent samples use Kruskal-Wallis
If related samples use Friedman
Same assumptions as before

59. More than 2 groups

60. Parametric related to Non-parametric test
Parametric Tests
Non-parametric Tests
Single sample t-test
Paired sample t-test
2 independent samples t-test
One-way Analysis of Variance
Pearson’s correlation

61. Parametric / Non-parametric
Parametric Tests
Non-parametric Tests
Single sample t-test
Wilcoxon-signed rank test
Paired sample t-test
2 independent samples t-test
One-way Analysis of Variance
Pearson’s correlation

62. Parametric / Non-parametric
Parametric Tests
Non-parametric Tests
Single sample t-test
Wilcoxon-signed rank test
Paired sample t-test
Paired Wilcoxon-signed rank
2 independent samples t-test
One-way Analysis of Variance
Pearson’s correlation

63. Parametric / Non-parametric
Parametric Tests
Non-parametric Tests
Single sample t-test
Wilcoxon-signed rank test
Paired sample t-test
Paired Wilcoxon-signed rank
2 independent samples t-test
Mann-Whitney test (Note: sometimes called Wilcoxon Rank Sums test!)
One-way Analysis of Variance
Pearson’s correlation

64. Parametric / Non-parametric
Parametric Tests
Non-parametric Tests
Single sample t-test
Wilcoxon-signed rank test
Paired sample t-test
Paired Wilcoxon-signed rank
2 independent samples t-test
Mann-Whitney test (Note: sometimes called Wilcoxon Rank Sums test!)
One-way Analysis of Variance
Kruskal-Wallis
Pearson’s correlation

65. Parametric / Non-parametric
Parametric Tests
Non-parametric Tests
Single sample t-test
Wilcoxon-signed rank test
Paired sample t-test
Paired Wilcoxon-signed rank
2 independent samples t-test
Mann-Whitney test(Note: sometimes called Wilcoxon Rank Sums test!)
One-way Analysis of Variance
Kruskal-Wallis
Pearson’s correlation
Spearman Rank

66. Summary Non-parametric
Non-parametric methods have fewer assumptions than parametric tests
So useful when these assumptions not met
Often used when sample size is small and difficult to tell if Normally distributed
Non-parametric methods are a ragbag of tests developed over time with no consistent framework
Read in datasets LDL, etc and carry out appropriate Non-Parametric tests

67. References
Corder GW, Foreman DI. Non-parametric Statistics for Non-Statisticians. Wiley, 2009.
Nonparametric statistics for the behavioural Sciences. Siegel S, Castellan NJ, Jr. McGraw-Hill, 1988 (first edition was 1956)