Non-parametric Tests: When Things Aren’t ‘Normal’ Scientific Practice Non-parametric Tests: When Things Aren’t ‘Normal’

Where We Are/Where We Are Going Statistical tests rely on knowing how data distribute and then trying to determine if that distribution changes BP in response to a drug FVC as a function of height All the statistical tests so far have assumed that data distributes normally ie the normal distribution (defined by mean and SD) But what is data are not normally distributed? need tests which take this into account called non-parametric Often seen as less ‘preferable’ to tests such as t

Mann-Whitney U Test A non-parametric test for unpaired data Calculations usually done on rank order rather than actual values ranks are very resistant to extreme values; eg… 10, 20, 30, 40, 50 has the same rank order as 0.1, 20, 39, 40, 500 Data from both conditions are ranked; eg Group A values : 120 80 90 110 95 Group B values : 105 130 145 125 115 Group A ranks : 7 1 2 5 3 Group B ranks : 4 9 10 8 6

Mann-Whitney U Test Sum the ranks for each Group A : 7 1 2 5 3  Ta 18 Group B : 4 9 10 8 6  Tb 37 Work out the test statistic U for each… U = Ta – (na(na+1))/2 = 18-(5(5+1))/2 = 3 U = Tb – (nb (nb+1))/2 = 37-(5(5+1))/2 = 22 (na(na+1))/2 gives ‘average’ if all low ranks (5(5+1))/2 = 15 = 1+2+3+4+5 Take the smaller of the two U values (3) why?!?! slight diversion….

Mann-Whitney U Test Imagine two rankings that ‘fully’ overlap Group A : 1 3 6 8 10  Ta 28 Group B : 2 4 5 7 9  Tb 27 U = Ta – (na (na+1))/2 = 28 – 15 = 13 U = Tb – (nb (nb+1))/2 = 27 – 15 = 12 Imagine two rankings that don’t overlap Group A : 1 2 3 4 5  Ta 15 Group B : 6 7 8 9 10  Tb 40 U = Ta – (na (na+1))/2 = 15 – 15 = 0 U = Tb – (nb (nb+1))/2 = 40 – 15 = 25 one U becomes small when groups diverge

Mann-Whitney U Test Look up critical value for U needs to be smaller than critical to reject Null Hypo

Wilcoxon Signed Rank Sum Test A non-parametric test for paired data Calculating the test statistic… calculate the differences between the pairs, keeping track of the signs rank the absolute values of the differences ie ignore the sign ignore any zeroes add up the rankings of all the positive differences add up the rankings of all the negative differences take the smaller of the two sums (T) compare it to the critical value in the table

Wilcoxon Signed Rank Sum Test Example: 6 patients Lymphocyte beta-adrenergic receptor counts.. Control 1162 1095 1164 1261 1103 1235 Drug 892 903 1327 1002 961 875 diffs 270 192 -163 259 142 360 abs diffs 270 192 163 259 142 360 abs ranks 5 3 2 4 1 6 pos ranks 19 neg ranks 2 T = 2 the more ‘different’ are the data, the smaller T is

Wilcoxon Signed Rank Sum Test Critical value for T, alpha = 0.05, n = 6 (pairs) is 0 Our T is 2 it is not <= to crit value cannot reject Null Hypo no diff between Cont and Drug data

Spearman Rank Correlation Use it when you want to look at correlation of x and y data, but data are not normal Calculation… separately rank the X and the Y values calculate the diffs between each XY pair of ranks square them, then sum them (sumdsq) Rank Coeff = 1 – ( 6sumdsq / n(n2-1) ) Use a table to look up the critical value depends on alpha (0.05) and df (number of pairs - 2) Our Rank Coeff needs to be > critical to reject Null Hypo

Spearman Rank Correlation Example…

Spearman Rank Correlation Coeff = 0.67, alpha = 0.05, n = 10 (or df = 8)

Summary For most tests where normality is assumed there is an equivalent one if data not normal called non-parametric tests Non-parametric tests… tend to work on the rankings of data eliminates ‘extreme’ effects wrongly regarded as less useful Mann-Whitney U replaces Unpaired t-test Wilcoxon Signed Rank replaces Paired t-test Spearman Rank Correlation replaces Correlation