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 • 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 20 Frequency 15 10 5 Mean = 8.03 Std. Dev. = 12.952 N = 30 0 0 10 20 30 Units of alcohol per week 40 50 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 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 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: 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? Wilcoxon Signed Rank Test: Example H0: Wilcoxon Signed Rank Test: Example H0: median=82 H1: Wilcoxon Signed Rank Test: Example H0: median=82 H1: median≠82 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 Wilcoxon Signed Rank Test: Example Result Above or below median Absolute difference from median=82 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+= 1.5+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 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. 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 6 5 3 4 2 1 3 8 6 12 After 9 12 9 9 3 1 2 12 9 10 Wilcoxon Signed Rank Test example SPSS Output p < 0.05 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: 1. The distribution of the measure in each group is approx Normally distributed 2. 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? 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 50 7 Units of alcohol per week 40 6 30 25 20 10 0 Male Female Gender Example (3) Are those distributions symmetrical? 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 4 3 8 4 20 5 45 5 10 4 Scatterplot Distance from GP and time to diagnosis D2 Distance (km) Time (weeks) Rank for distance Rank for time Difference in Ranks 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 d2=28.5 Spearman Rank Correlation Example The formula for Spearman’s rank correlation is: rs 1 6 d 2 n n 1 2 where n is the number of pairs Spearman’s on SPSS Spearman’s in SPSS 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 KruskalWallis If related samples use Friedman Same assumptions as before More than 2 groups Parametric related to Nonparametric test Parametric Tests Single sample t-test Paired sample t-test 2 independent samples t-test One-way Analysis of Variance Pearson’s correlation Non-parametric Tests Parametric / Non-parametric Parametric Tests Single sample t-test Paired sample t-test 2 independent samples t-test One-way Analysis of Variance Pearson’s correlation Non-parametric Tests Wilcoxon-signed rank test 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 Sums 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 Sums 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 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 NonStatisticians. Wiley, 2009. Nonparametric statistics for the behavioural Sciences. Siegel S, Castellan NJ, Jr. McGraw-Hill, 1988 (first edition was 1956)