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Non-Parametric Methods

Statistics for Health Research. Non-Parametric Methods. Peter T. Donnan Professor of Epidemiology and Biostatistics. Objectives of Presentation. Introduction Ranks & Median Wilcoxon Signed Rank Test Paired Wilcoxon Signed Rank Mann-Whitney test Spearman’s Rank Correlation Coefficient

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Non-Parametric Methods

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  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 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 W+= 1.5+6+7+4+5=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

  27. Wilcoxon Signed Rank Test example

  28. SPSS Output p < 0.05

  29. 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

  30. 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?

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

  32. 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

  33. Boxplots of alcohol units per week by gender

  34. Example (3) Are those distributions symmetrical?

  35. 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.

  36. Mann-Whitney on SPSS

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

  38. 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

  39. 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

  40. Distance from GP and time to diagnosis

  41. Scatterplot

  42. Distance from GP and time to diagnosis

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

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