Non-Parametric Methods

1
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
• Spearmans 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
• Median140
• 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
• Median140
• No. of cigarettes smoked
• 0, 1, 2, 2, 2, 3, 5, 5, 8, 10
• Median2.5

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 TestExample

• 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 TestExample

• H0

15
Wilcoxon Signed Rank TestExample

• H0 median82
• H1

16
Wilcoxon Signed Rank TestExample

• H0 median82
• H1 median?82

17
Wilcoxon Signed Rank TestExample

• H0 median82
• H1 median?82
• Two-tailed test
• Because one result equals 82 this cannot be used
  in the analysis

18
Wilcoxon Signed Rank TestExample
Result Above or below median Absolute difference from median82 Rank of difference
83 1 1.5
90 8 6
96 14 7
85 3 4
80 - 2 3
81 - 1 1.5
87 5 5
W 1.5674523.5 W- 31.54.5 So,
W4.5 n7, so the value of W gt tabulated value of
2, so pgt0.05
19
Wilcoxon Signed Rank TestExample

• 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 ngt20
• Mean Wn(n1)/4
• Variance Wn(n1)(2n1)/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
  resultsmedian
• 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 QoL Score
Before After
6 9
5 12
3 9
4 9
2 3
1 1
3 2
8 12
6 9
12 10
27
Wilcoxon Signed Rank Test example
28
(No Transcript)
29
(No Transcript)
30
(No Transcript)
31
SPSS Output
p lt 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 (n13) 0,0,1,5,10,30,45,5,5,1,0,0,0
• Women (n14) 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
40
(No Transcript)
41
(No Transcript)
42
(No Transcript)
43
(No Transcript)
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
4 3
8 4
20 5
45 5
10 4
48
Scatterplot
49
Distance from GP and time to diagnosis
Distance (km) Time (weeks) Rank for distance Rank for time Difference in Ranks D2
2 4 1 3 -2 4
4 3 2 1 1 1
5 6 3 7 -4 16
8 4 4 3 1 1
10 4 5 3 2 4
20 5 6 5.5 0.5 0.25
45 5 7 5.5 1.5 2.25
Total 0 ?d228.5
50
Spearman Rank Correlation Example

• The formula for Spearmans rank
• correlation is
• where n is the number of pairs

51
Spearmans on SPSS
52
Spearmans in SPSS
53
Spearmans in SPSS
54
Spearmans in SPSS
55
Spearman Rank Correlation Example

• In our example, rs0.468
• In SPSS we can see that this value is not
  significant, ie.p0.29
• 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
Pearsons 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
Pearsons 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
Pearsons 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
Pearsons 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
Pearsons 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
Pearsons 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)