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Non – Parametric Test
Dr.L.Jeyaseelan
Dept. of Biostatistics
Christian Medical College
Vellore, India
Introduction

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 NonParametric Test (NP) Test

Applied to a wide variety of situations
continuous ordinal scores
When are non-parametric tests used?
 Assumptions of parametric test are violated
 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
Compare the distribution of scores on a quantitative variable
obtained from two independent groups.
Group A : 2
Group B : 8
4 2 6 4 8
8 4 10 12 11
Ho: There is no significant difference between the medians
of the two samples
Ha: There is a significant difference between the medians
of the two samples
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
U  E (U )
Z
var(U )
n1n2
U
2
Z
(n1  n2  1)
n1n2
12
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
Wilcoxon Signed-Rank Test
Below is the measurement of anxiety before and after
two weeks of treatment.
SN
O
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?
Result???
Exercise: Wilcoxon signed rank Test
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
subjects before and after receiving a standard
9
149
143
10
treatment
142
136
11
149
145
12
143
140
13
145
143
14
146
142
15
143
140
16
146
141
Examine whether there is any significant
difference in the systolic blood pressure of 16
Result???
Advantages of Nonparametric Methods

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.
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
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
Pearson Correlation
Spearman Rank
Correlation
Type of Data
Quantify relation between
two variables

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