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

2. 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

3. 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 :
Group B values :
Group A ranks :
Group B ranks :

4. Mann-Whitney U Test Sum the ranks for each
Group A :  Ta 18
Group B :  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 =
Take the smaller of the two U values (3)
why?!?!
slight diversion….

5. Mann-Whitney U Test Imagine two rankings that ‘fully’ overlap
Group A :  Ta 28
Group B :  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 :  Ta 15
Group B :  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

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

7. 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

8. Wilcoxon Signed Rank Sum Test
Example: 6 patients
Lymphocyte beta-adrenergic receptor counts..
Control
Drug
diffs
abs diffs
abs ranks
pos ranks 19
neg ranks 2
T = 2
the more ‘different’ are the data, the smaller T is

9. 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

10. 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

11. Spearman Rank Correlation
Example…

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

13. 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