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Non-Parametric Methods

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Title: 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
  • Paired Wilcoxon Signed Rank
  • Mann-Whitney test (or Wilcoxon Rank Sum 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
  • Sometimes called Distribution-free tests
  • 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 tests
  • Frank Wilcoxon was Chemist
  • In USA who wanted to develop
  • test similar to t-test but without requirement of
    Normal distribution
  • Presented paper in 1945
  • Wilcoxon Signed Rank ? paired t-test
  • Wilcoxon Rank Sum ? independent t-test

13
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

14
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)

15
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

16
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

17
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

18
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

19
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.

20
Wilcoxon Signed Rank Test example
QoL Score QoL Score
Before After Diff Rank -/
6 9 3 5.5
5 12 7 10
3 9 6 9
4 9 5 8
2 3 1 4
1 1 0 3 tied
3 2 -1 2 -
8 12 4 7
6 9 3 5.5
12 10 -2 1 -
W- 2 W 7 1 tied
21
Wilcoxon Signed Rank Test example
22
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25
SPSS Output
p lt 0.05
26
Wilcoxon tests
  • Frank Wilcoxon was Chemist
  • In USA who wanted to develop
  • test similar to t-test but without requirement of
    Normal distribution
  • Presented paper in 1945
  • Wilcoxon Signed Rank ? paired t-test
  • Wilcoxon Rank Sum ? independent t-test

27
Mann-Whitney test ? Wilcoxon Rank Sum
HB Mann
  • 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

28
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?

29
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

30
Boxplots of alcohol units per week by gender
31
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.

32
Mann-Whitney on SPSS
33
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34
Normal approx (NS)
Mann-Whitney (NS)
35
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

36
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

37
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
38
Scatterplot
39
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
40
Spearman Rank Correlation Example
  • The formula for Spearmans rank
  • correlation is
  • where n is the number of pairs

41
Spearmans in SPSS
42
Spearmans in SPSS
43
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!

44
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

45
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

46
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

47
More than 2 groups
48
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
49
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
50
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
51
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 Sum test!)
One-way Analysis of Variance
Pearsons correlation
52
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 Sum test!)
One-way Analysis of Variance Kruskal-Wallis
Pearsons correlation
53
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
Repeated Measures Friedman
54
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

55
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)
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