NON- PARAMETRIC NON- PARAMETRIC TESTSTESTS

M.Prasad NaiduM.Prasad Naidu

MSc Medical Biochemistry, MSc Medical Biochemistry, Ph.D,.Ph.D,.

Non-parametric testsNon-parametric tests Note: When valid use parametricNote: When valid use parametric Commonly usedCommonly used

WilcoxonWilcoxon

Chi square etc.Chi square etc. Performance comparable to Performance comparable to

parametricparametric Useful for non-normal dataUseful for non-normal data If normalization not possibleIf normalization not possible Note: CI Note: CI

derivation-difficult/impossiblederivation-difficult/impossible

Wilcoxon signed rank testWilcoxon signed rank test

To test difference between To test difference between paired datapaired data

STEP 1STEP 1 Exclude any differences which are Exclude any differences which are

zerozero

Put the rest of differences in Put the rest of differences in ascending orderascending order

Ignore their signsIgnore their signs

Assign them ranksAssign them ranks

If any differences are equal, average If any differences are equal, average their rankstheir ranks

STEP 2STEP 2

Count up the ranks of +ives as TCount up the ranks of +ives as T++

Count up the ranks of –ives as TCount up the ranks of –ives as T--

STEP 3STEP 3 If there is no difference between drug If there is no difference between drug

(T(T++) and placebo (T) and placebo (T--), then T), then T+ + & T& T- - would would be similarbe similar

If there were a difference If there were a difference one sum would be much smaller and one sum would be much smaller and

the other much larger than the other much larger than

expectedexpected

The smaller sum is denoted as TThe smaller sum is denoted as T

T = smaller of TT = smaller of T+ + and Tand T- -

STEP 4STEP 4 Compare the value obtained with Compare the value obtained with

the critical values (5%, 2% and the critical values (5%, 2% and 1% ) in table1% ) in table

N is the number of differences N is the number of differences that were ranked (that were ranked (not the total not the total number of differencesnumber of differences))

So the zero differences are So the zero differences are excludedexcluded

PatientPatient

Hours of sleepHours of sleep

DifferenceDifferenceRankRank

Ignoring signIgnoring signDrugDrug PlaceboPlacebo

11 6.16.1 5.25.2 0.90.9 3.5*3.5*

22 7.07.0 7.97.9 -0.9-0.9 3.5*3.5*

33 8.28.2 3.93.9 4.34.3 1010

44 7.67.6 4.74.7 2.92.9 77

55 6.56.5 5.35.3 1.21.2 55

66 8.48.4 5.45.4 3.03.0 88

77 6.96.9 4.24.2 2.72.7 66

88 6.76.7 6.16.1 0.60.6 22

99 7.47.4 3.83.8 3.63.6 99

1010 5.85.8 6.36.3 -0.5-0.5 11

3rd & 4th ranks are tied hence averaged

T= smaller of T+ (50.5) and T- (4.5)

Here T=4.5 significant at 2% level indicating the drug (hypnotic) is more effective than placebo

Wilcoxon rank sum testWilcoxon rank sum test

To compare two groupsTo compare two groups

Consists of 3 basic stepsConsists of 3 basic steps

Non-parametric equivalent of Non-parametric equivalent of t testt test

Step 1Step 1

Rank the data of both the groups Rank the data of both the groups in ascending orderin ascending order

If any values are equal average If any values are equal average their rankstheir ranks

Step 2Step 2

Add up the ranks in group with Add up the ranks in group with smaller sample sizesmaller sample size

If the two groups are of the If the two groups are of the same size either one may be same size either one may be pickedpicked

T= sum of ranks in group with T= sum of ranks in group with smaller sample sizesmaller sample size

Step 3Step 3 Compare this sum with the critical Compare this sum with the critical

ranges given in tableranges given in table

Look up the rows corresponding to Look up the rows corresponding to the sample sizes of the two groupsthe sample sizes of the two groups

A range will be shown for the 5% A range will be shown for the 5% significance levelsignificance level

Non-smokers (n=15)Non-smokers (n=15) Heavy smokers (n=14)Heavy smokers (n=14)Birth wt (Kg)Birth wt (Kg) RankRank Birth wt (Kg)Birth wt (Kg) RankRank3.993.99 2727 3.183.18 773.793.79 2424 2.842.84 553.60*3.60* 1818 2.902.90 663.733.73 2222 3.273.27 11113.213.21 88 3.853.85 26263.60*3.60* 1818 3.523.52 14144.084.08 2828 3.233.23 993.613.61 2020 2.762.76 443.833.83 2525 3.60*3.60* 18183.313.31 1212 3.753.75 23234.134.13 2929 3.593.59 16163.263.26 1010 3.633.63 21213.543.54 1515 2.382.38 223.513.51 1313 2.342.34 112.712.71 33

Sum=272Sum=272 Sum=163Sum=163

* 17, 18 & 19are tied hence the ranks are averaged

THANK YOUTHANK YOU