Non-Parametric Methods

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

• Introduction
• Ranks Median
• Paired Wilcoxon Signed Rank
• Mann-Whitney test (or Wilcoxon Rank Sum test)
• Spearmans Rank Correlation Coefficient
• Others.

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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

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A positively skewed distribution
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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

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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

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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

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Median

• BPs
• 135, 138, 140, 140, 141, 142, 143
• Median

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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

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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

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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

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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

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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

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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)

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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

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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

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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

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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

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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.

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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
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Wilcoxon Signed Rank Test example
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SPSS Output
p lt 0.05
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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

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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

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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?

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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

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Boxplots of alcohol units per week by gender
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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.

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Mann-Whitney on SPSS
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Normal approx (NS)
Mann-Whitney (NS)
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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

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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

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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
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Scatterplot
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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
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Spearman Rank Correlation Example

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

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Spearmans in SPSS
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Spearmans in SPSS
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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!

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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

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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

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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

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More than 2 groups
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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
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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
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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
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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
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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
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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
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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

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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)