Non – Parametric Test

Dr.L.JeyaseelanDept. of Biostatistics

Christian Medical CollegeVellore, India

IntroductionIntroduction

No rigid assumptions about the distribution of the populations - “Distribution-free tests”

Answers the same sort of questions as the parametric test – for each Parametric tests (PT) there is an alternative Non-Parametric Test (NP) Test

Applied to a wide variety of situations continuous ordinal scores

Assumptions of parametric test are violated

When are non-parametric tests used?When are non-parametric tests used?

Non-normal or skewed

Variance very high in relative to mean

Data is on an ordinal scale

Very few observations

Mann-Whitney U Test

Most powerful of the NP test – Wilcoxon Rank Sum test

Alternative to the parametric independent t-test

To test whether two independent groups have been drawn from the same population

Assumptions:

Two sample are selected independently and at random from their respective population

Variable of interest is continuous

Measurement scale is at least ordinal.

Mann-Whitney U Test

HHoo:: There is no significant difference between the medians There is no significant difference between the medians of the two samplesof the two samples

HHaa:: There is a significant difference between the medians There is a significant difference between the medians of the two samplesof the two samples

Compare the distribution of scores on a quantitative variable obtained from two independent groups.

Group A : 2 4 2 6 4 8

Group B : 8 8 4 10 12 11

Man Whitney U test output

Mann-Whitney U Test (contd.)

A researcher designed an experiment to assess the effects of prolonged inhalation of cadmium oxide. 15 lab animals served as experimental subjects while 10 similar animals served as controls. Variable of interest was Hb level following the experiment. Conclude that there is no difference between exposed and unexposed animals in their Hb level

sno

Exposed Unexposed

1 13.1 17.4

2 15.6 16.2

3 17.3 17.1

4 16.4 17.5

5 14.3 15

6 15.5 16

7 14.9 16.9

8 15.6 15.1

9 14.1 16.1

10 15.3 17.2

11 15.7

12 16.7

13 13.7

14 15.6

15 14

Deciding Normality

Conclusion???

Large Sample: When n1,n2 > 10 (normal approximation)

The test statistic is given as

12)1(

2

2121

21

nnnn

nnU

Z

)var(

)(

U

UEUZ

where ‘U’ is the smallest sum of ranks between U1 and U2

Wilcoxon Signed-Rank Test

An alternative to the parametric paired t-test Used to compare 2 samples from populations are not

independent eg., measure a variable in each subject before and after an intervention

Assumptions Samples must be paired Pairs are randomly selected from the larger population Probability distribution from which the sample of paired

differences drawn is continuous

Below is the measurement of anxiety before and after two weeks of treatment.

SNO

BEFORE Rx

AFTER Rx

1 130 120

2 170 163

3 125 120

4 170 135

5 130 143

6 130 136

7 145 144

8 160 120

Does the treatment have any beneficial effects?

Wilcoxon Signed-Rank Test

Result???

Exercise: Wilcoxon signed rank Test

Examine whether there is any significant

difference in the systolic blood pressure of 16

subjects before and after receiving a standard

treatment

Sno Before

After

1 142 143

2 148 146

3 144 147

4 142 138

5 140 136

6 144 139

7 146 141

8 150 145

9 149 143

10 142 136

11 149 145

12 143 140

13 145 143

14 146 142

15 143 140

16 146 141

Result???

Nonnumeric data such as Taste of food: bad, good, great and excellent, Smoking habit: light, moderate and heavy etc.

Involves simpler computations than the corresponding parametric methods and are therefore easier to understand.

Since the inference is based on ranks, Nonparametric methods are quick for small samples and less subject to measurement error than parametric methods.

Advantages of Nonparametric Methods

Disadvantages of NP Methods

No parameters to describe and it becomes more difficult to make quantitative statements about the actual difference between populations

Tend to waste information because it deals with ranks and discard the actual values

Less powerful where parametric test is applicable.

Selecting a Statistical Test

Type of Data

Measurement from Normal Population

Rank, Score or Measurement from

Non-normal

Describe one group Mean, SD Median, Interquartile Range

Compare two independent groups

Independent sample t-test (unpaired t test)

Mann-Whitney test

Compare two paired groups

Paired t test Wilcoxon Signed rank test

Quantify relation between two variables

Pearson Correlation Spearman Rank Correlation