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

Objectives of Presentation Introduction Ranks & Median Paired Wilcoxon Signed Rank Mann-Whitney test (or Wilcoxon Rank Sum test) Spearman’s Rank Correlation Coefficient Others….

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

A positively skewed distribution

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

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

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

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

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=

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

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

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 Ξ paired t-test Wilcoxon Rank Sum Ξ independent t- test

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

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)

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

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

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

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

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.

Wilcoxon Signed Rank Test example QoL Score Before After Diff Rank -/+ 6 9 3 5.5 + 5 12 7 10 4 8 2 1 tied -1 - -2 W- = 2 W+ = 7 1 tied

Wilcoxon Signed Rank Test example

SPSS Output p < 0.05

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 Ξ paired t-test Wilcoxon Rank Sum Ξ independent t- test

Mann-Whitney test Ξ 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

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?

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

Boxplots of alcohol units per week by gender

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.

Mann-Whitney on SPSS

Normal approx (NS) Mann-Whitney (NS)

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

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

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

Scatterplot

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

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

Spearman’s in SPSS

Spearman’s in SPSS

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!

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

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

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

More than 2 groups

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

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

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

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 Pearson’s correlation

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 Pearson’s correlation

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 Repeated Measures Friedman

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

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)