NON-PARAMETRIC TESTS by Prajakta Sawant

Presented By:
Ms. Prajakta Sawant
Second Year M.Pharm (Roll No. 5)
Class: PH- Sem III
(Dept. of Pharmaceutics)
Sub: Research Methodology and Biostatistics
Alard College Of Pharmacy, Pune.
Under the Guidance of:
Dr. Nalanda Borkar
Head of Department
(Dept. of Pharmaceutics)
Alard College Of Pharmacy, Pune.
NON-PARAMETRIC TESTS
(Wilcoxan rank tests, Analysis of variance,
Correlation, Chi square test)
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INTRODUCTION
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Parametric & nonparametric concept
 Most of the statistical methods referred to as
parametric require the use of interval- or ratio-
scaled data.
 Nonparametric methods are often the only way to
analyze nominal or ordinal data and draw
statistical conclusions.
 Nonparametric methods require no assumptions
about the population probability distributions.
 Nonparametric methods are often called
distribution-free methods.

What are Nonparametric Tests?
 Nonparametric tests are methods of statistical
analysis that do not require a distribution to meet the
required assumptions to be analyzed (especially if
the data is not normally distributed).
 Due to this reason, they are sometimes referred to
as distribution-free tests.
 Nonparametric tests serve as an alternative to
parametric tests such as T-test or ANOVA that can
be employed only if the underlying data satisfies
certain criteria and assumptions.
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 Note that nonparametric tests are used as an alternative
method to parametric tests, not as their substitutes. In other
words, if the data meets the required assumptions for
performing the parametric tests, the relevant parametric test
must be applied.
 In addition, in some cases, even if the data do not meet the
necessary assumptions but the sample size of the data is
large enough, we can still apply the parametric tests instead
of the nonparametric tests.
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Reasons to Use Nonparametric Tests:
 In order to achieve the correct results from
the statistical analysis, we should know the situations
in which the application of nonparametric tests is
appropriate. The main reasons to apply the
nonparametric test include the following:
1. The underlying data do not meet the
assumptions about the population sample
 Generally, the application of parametric tests requires
various assumptions to be satisfied. For example, the
data follows a normal distribution and the population
variance is homogeneous. However, some data
samples may show skewed distributions.

 The skewness makes the parametric tests less
powerful because the mean is no longer the best
measure of central tendency because it is strongly
affected by the extreme values. At the same time,
nonparametric tests work well with skewed
distributions and distributions that are better
represented by the median.
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2. The population sample size is too small
 The sample size is an important assumption in
selecting the appropriate statistical method. If a
sample size is reasonably large, the applicable
parametric test can be used. However, if a sample
size is too small, it is possible that you may not be
able to validate the distribution of the data. Thus, the
application of nonparametric tests is the only suitable
option.
3. The analyzed data is ordinal or nominal
 Unlike parametric tests that can work only with
continuous data, nonparametric tests can be applied
to other data types such as ordinal or nominal data.
For such types of variables, the nonparametric tests
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 IMPORTANT TERMS
1) PARAMETRIC TEST: The test in which, the population
constants like mean, std deviation, std error, correlation
coefficient, proportion etc. and data tend to follow one
assumed or established distribution such as normal, binomial,
poisson, etc.
2) NON PARAMETRIC TEST: the test in which no constant of a
population is used. Data do not follow any specific distribution
and no assumption are made in these tests. E.g. to classify
good, better and best we just allocate arbitrary numbers or
marks to each category.
3) HYPOTHESIS: It is a definite statement about the population
parameters.
4) NULL HYPOTHESIS: (H0) states that no association exists
between the two cross-tabulated variables in the population,
and therefore the variables are statistically independent. E.g. if
we want to compare 2 methods method A and method B for its
superiority, and if the assumption is that both methods are
equally good, then this assumption is called as NULL
HYPOTHESIS.

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6) DEGREE OF FREEDOM: It denotes the extent of
independence (freedom) enjoyed by a given set of
observed frequencies Suppose we are given a set of n
observed frequencies which are subjected to k
independent constraints(restrictions) then,
d.f. = (number of frequencies) – (number of
independent
constraints
on them)
In other terms, df = (r – 1)(c – 1)
where, r = the number of rows
c = the number of columns
7) CONTINGENCY TABLE: When the table is prepared by
enumeration of qualitative data by entering the actual
frequencies, and if that table represents occurance of two
sets of events, that table is called the contingency table.

 Variable: A characteristic that is observed or
manipulated.
• Dependent
• Independent
 Data: Measurements or observations of a variable
1. Nominal or Classificatory Scale: Gender, ethnic
background, eye colour, blood group
2. Ordinal or Ranking Scale: School performance, social
economic class
3. Interval Scale: Celsius or Fahrenheit scale
4. Ratio Scale: Kelvin scale, weight, pulse rate,
respiratory rate
 Parameter: is any numerical quantity that characterizes
a given population or some aspect of it. Most common
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 Assumptions
 The general assumptions of parametric tests
are
− The populations are normally distributed (follow
normal distribution curve).
− The selected population is representative of
general population
− The data is in interval or ratio scale
 Non-parametric tests can be applied when:
– Data don’t follow any specific distribution and no
assumptions about the population are made
– Data measured on any scale
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 Testing normality
 Normality: This assumption is only broken if there are
large and obvious departures from normality .
 This can be checked by ;
 Inspecting a histogram
 Skewness and kurtosis ( Kurtosis describes the peakof
the curve Skewness describes the symmetry of the
curve)
 Kolmogorov-Smirnov (K-S) test (sample size is ≥50 )
 Shapiro-Wilk test (if sample size is <50)
(Sig. value >0.05 indicates normality of the distribution)
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 Commonly used Non Parametric Tests
are:
1. Wilcoxan rank tests
2. Analysis of variance
3. Correlation
4. Chi Square test
5. Microbial Interdependent Test

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1. Wilcoxan rank tests
A. Wilcoxan signed-rank test
 Non-parametric tests for comparing two
related or paired groups or conditions.
 Used when you have two conditions, both
performed by the same subjects.
 Each subject produces two scores, one for
each condition.
 Tests whether there is a statistically
significant difference between the two
conditions.

 Assumptions
1. Dependent variable should be measured at the
ordinal or continuous level (i.e., interval or ratio).
2. Independent variable should consist of two
categorical, related or matched groups.
3. Not normally distributed and distributions in each
group have the same variability
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 EXAMPLE
 The 14 difference scores in BP among
hypertensive patients after giving drug A were:
-20, -8, -14, -12, -26, +6, -18, -10, -12, -10, -8, +4,
+2, -18
The statistic T is found by calculating the sum of
the positive ranks, and the sum of the negative
ranks.
 The smaller of the two values is considered.
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SCORE RANK
+2 1
+4 2
+6 3
-8 4.5
-8 4.5
-10 6.5
-10 6.5
-12 8
-14 9
-16 10
-18 11.5
-18 11.5
-20 13
-26 14
Sum of positive ranks =
6
Sum of negative ranks =
99
For N = 14, and α = .05, the
critical value of T = 21.
If T is equal to or less than
T critical, then null
hypothesis is rejected i.e.,
drug A decreases the BP
among hypertensive
patients.
T=6

B. Wilcoxan rank sum test / Mann-Whitney U test
• Mann-Whitney U – similar to Wilcoxon signed-ranks test except
that the samples are independent and not paired.
• Null hypothesis: the population means are the same for the two
groups.
• Rank the combined data values for the two groups. Then find
the average rank in each group.
• Then the U value is calculated using formula;
U= N1*N2+ N x (Nx+1) - Rx (where, Rx is larger rank 2 total)
• To be statistically significant, obtained U has to be equal to or
LESS than this critical value.
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 Find the sum of ranks assigned to the values of the first sample (R1) and also
the sum of the ranks assigned to the values of the second samples(R2).
 Then use test statistics U, which is a measurement of the difference between
the ranked observations of the two samples
 n1& n2 are the sample sizes and R1 &R2 are the sum of ranks assigned to the
values of the first & second samples respectively.
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 Example:
 10 dieters following Atkin’s diet vs. 10 dieters following Jenny Craig diet
Hypothetical RESULTS:
 Atkin’s group loses an average of 34.5 lbs.
 J. Craig group loses an average of 18.5 lbs.
Conclusion: Atkin’s is better?
 When individual data is seen
• Atkin’s, change in weight (lbs): +4, +3, 0, -3, -4, -5, -11, -14, -15, -300
• J. Craig, change in weight (lbs): -8, -10, -12, -16, -18, -20, -21, -24, -26, -30
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 RANK the values combinedly, 1 being the least
weight loss and 20 being the most weight loss.
• Atkin’s
+4, +3, 0, -3, -4, -5, -11, -14, -15, -300
1, 2, 3, 4, 5, 6, 9, 11, 12, 20
• J. Craig
-8, -10, -12, -16, -18, -20, -21, -24, -26, -30
7, 8, 10, 13, 14, 15, 16, 17, 18, 19
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 Sum of Atkin’s ranks:
1+ 2 + 3 + 4 + 5 + 6 + 9 + 11+ 12 + 20=73
 Sum of Jenny Craig’s ranks:
7 + 8 +10+ 13+ 14+ 15+16+ 17+ 18+19=137
 Jenny Craig clearly ranked higher.
 Calculated U value (18) < table value (27), Null hypothesis
is rejected.
 Hypothesis:
 Null hypothesis (Ho) : the two population are identical
 Alternative Hypothesis H1: the two population are not
identicald.
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Critical values of wilcoxon signed rank test:

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2. Analysis of Variance (One Way ANOVA)
 Kruskal Wallis one way analysis of variance by rank is a non-
parametric method for testing whether samples originate from
the same distribution .
 Used for comparing two or more samples that are independent,.
 And that may have different sample sizes, and extended the
Mann - Whitney U test to more than two groups.
 Just like one way ANOVA it is applied to populations from which
the samples drawn are not normally distributed with equal
variances or when the data for analysis consists of only ranks.
 It is computed exactly like the Mann-Whitney test, except that
there are more groups (>2 groups).
 Applied on independent samples with the same shape (but not
necessarily normal).
 Analysis of variance (ANOVA) is a collection of statistical
models used to analyze the differences between group means
and their associated procedures (such as "variation" among and
between groups).
 Compares multiple groups at one time.
 Developed by R.A.

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• Like the One-way ANOVA, the Kruskal – Wallis Test is
used to determined whether the probability distributions
involved are all same or not.
• The Kruskal – Wallis Test is the Non-Parametric
counterpart of One Way ANOVA Test.
• H Statistics is computed in this test.
Ho : all probability distribution involved are identical
Ha : atleast two of the probability distribution involved
are different.

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Example
 In study of cerebrovascular disease, the patients from 16
socioeconomic background were thoroughly investigated.
 One characteristic measured was diastolic blood pressure
in mm/hg.
 Is there any reason to believe that three groups differ with
respect to this characteristic?
 Study of cerebrovascular disease in 3 socioeconomic
backgrounds.
GROUP
A
GROUP
B
GROUP
C
100 92 81
1031 97 102
89 88 86
78 84 83
105 90 99
95
ni=5 ni=6 ni=5
Total (n) = 5+6+5=16

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 Null hypothesis (Ho): there is no difference in the
diastolic pressure of the three groups.
 Alternative hypothesis (Ho): there is difference in the
diastolic pressures of the three groups.
16

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 Friedman ANOVA
 Friedman ANOVA: When either a matched-
subjects or repeated-measure design is used and
the hypothesis of a difference among three or
more (k) treatments is to be tested, the Friedman
ANOVA by ranks test can be used.
 Friedmann's test compares the medians of three
or more dependent groups and in the
nonparametric equivalent of the two - way
ANOVA.

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3. Correlation
Correlation analysis measures the degree of
association of two
variables.
Methods of studying of correlation:
1. Scatter diagram( Graphical method of
representation of relationship)
2. Karl Pearson’s correlation coefficient (for
quantitative data)
3. Spearman’s rank correlation coefficient (ordinal
data)

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1. Scatter diagram (scatter plot) method
• Simples method of studying relationship between
two variables by graphically.
• Fist step of showing the relationship between
variables.
• Give the direction correlation but fail to give the
degree of relationship.
Fig. Independent variable ( X) variable is plot along
with the X-axis (horizontal) and dependent variable
(Y)

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 Merits
• Simple and non mathematical method for studying
correlation.
• Easy to understand and easy to interpret.
• First step to study the relation.
 Demerits
• It gives just an idea about the direction correlation. It
does not establish the exact degree of correlation.
• Just qualitative method of showing the relationship
between two variables.

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2. Karl Pearson’s coefficient of correlation
• Mathematical method to measure the degree of
relationship between two quantitative variable.
• Denoted by r .
• Is a parametric method of finding the relationship
between variables
• k/n bivariate analysis
• The value of correlation coefficient lies in between -1
to +1.
 Interpretation of correlation coefficient :
• If r= -1, there is perfect negative correlation between X
& Y
• If r=+1, there is perfect positive correlation between X
& Y

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 Merits
• It gives the exact measure of degree of correlation
between two variables.
• It gives whether the correlation is positive or
negative.
 Demerits
• Affected by extreme values.
• Gives only linear relationship.
• Tedious calculation.
• Uses only in quantitative measurement.

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3. Spearman rank correlation
• The data obtained from bi-variate population which is
not in normal then the previous Karl Pearson
coefficient correlation is not applied.
• Instead, we give the ranks for each variable.
• Used to find the relationship.
• We use this method when the variables are taken
from qualitative nature such as intelligence, honesty,
ability, beauty, color etc..
• The spearman’s rank correlation is also called non-
parametric test or distribution free test.
• Denoted by rs.
• Lies in between -1 to +1.

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 Limitations:
 Does not deal individual data .
 Technique deals with the quantitative data only. It
ignores qualitative aspects like beauty, goodness,
intelligence, gender, pain, knowledge etc..
 Laws are not exact like mathematical like
mathematical laws .
 They are based on the average.
 Sometimes it gives absurd result .
 The greatest limitation of biostatistics is that only
who has a sound knowledge of statistical methods
can efficiently handle statistical data, Person with
poor expertise knowingly or unknowingly can draw
faulty conclusion

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4. Chi Square Test
 The chi-square test is an important test amongst the
several tests of significance developed by
statisticians.
 It was developed by Karl Pearson in1900.
 CHI SQUARE TEST is a non parametric test not
based on any assumption or distribution of any
variable.
 This statistical test follows a specific distribution
known as chi square distribution.
 In general The test we use to measure the
differences between what is observed and what is
expected according to an assumed hypothesis is
called the chi-square test.

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 Application of chi-square test:
1. TEST OF GOODNESS OF FIT OF DISTRIBUTIONS:
This test enables us to see how well does the
assumed theoretical distribution (such as Binomial
distribution, Poisson distribution or Normal
distribution) fit to the observed data.
2. TEST OF INDEPENDENCE OF ATTRIBUTES: Test
enables us to explain whether or not two attributes are
associated. For instance, we may be interested in
knowing whether a new medicine is effective in
controlling fever or not, x2 test is useful.
3. TEST OF HOMOGENITY: This test can also be used
to test whether the occurance of events follow
uniformity or not e.g. the admission of patients in
government hospital in all days of week is uniform or

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 STEPS INVOLVED IN CALCULATING x2:
1) Calculate the expected frequencies and the
observed frequencies:
 Expected frequencies (e) : the cell frequencies that
would be expected in a contingency table if the two
variables were statistically independent.
 Observed frequencies (o): the cell frequencies
actually observed in a contingency table.
 e = (column total)(row total) / N
 To obtain the expected frequencies for any cell in
any cross- tabulation in which the two variables are
assumed independent, multiply the row and column
totals for that cell and divide the product by the total
number of cases in the table.

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2) CALCULATION OF CHI SQUARE

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 EXAMPLE:

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Yates’ correction: applies when we have two categories (one
degree of freedom).
 Used when sample size is ≥ 40, and expected frequency of <5 in
one cell.
 Subtracting 0.5 from the difference between each observed value
and its expected value in a 2 × 2 contingency table.
 χ2 = ∑ [O- E-0.5]2 / E

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 4. Microbial Interdependent Test
 A Non-parametric Microbial Interdependence Test
(NMIT) is a non-parametric distance based test for
group comparison on microbial temporal
interdependence reaction in longitudinal study design.
 Component 1: We capture the individual microbial
dependencies over time by performing pair-wise
correlation analysis within each subject using the
longitudinal microbial measurements.
 Component 2: We test whether the correlation
structure is different between groups or associated
with an interested outcome or not using permutation
MANOVA.

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 NMIT step 1: Taxon screening

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 NMIT step 2: Temporal correlation analysis

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 NMIT step 3: Distance between taxa temporal
correlations

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 NMIT step 4: Permutation MANOVA

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• Summary of Non-parametric Microbial
interdependence test (NMIT)
 NMIT is designed for longitudinal microbial study.
 NMIT tests overall interdependence structure.
 NMIT is a non-parametric distance based test.
 NMIT allows both balanced and unbalanced design.
 NMIT can adjust confounders.
• Assumptions:
 NMIT assumes subjects are exchangeable
(independent).
 NMIT assumes taxa correlation do not change over
time.
REF: Zhang Y, Han SW, Cox LM., and Li H. (2017) A Multivariate Distance
Based Test on Microbial Temporal Interaction Group Comparison. Genetic
Epidemiology.Dr. Yilong Zhang

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 Advantages of non-parametric tests
 These tests are distribution free.
 Easier to calculate & less time consuming than
parametric tests when sample size is small.
 Can be used with any type of data.
 Many non-parametric methods make it possible to
work with very small samples, particularly helpful in
collecting pilot study data or medical researcher
working with a rare disease.

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 Limitations of non-parametric
methods
 Statistical methods which require no
assumptions about populations are usually less
efficient.
 As the sample size get larger, data manipulations
required for non-parametric tests becomes
laborious.
 A collection of tabulated critical values for a
variety of nonparametric tests under situations
dealing with various sample sizes is not readily
available.

References:
 https://www.slideshare.net/raghurh/non-parametric-test Assessed Date: 12/10/2021
 https://www.slideshare.net/kapilgautam8/parametric-and-non-parametric-test-in-
biostatistics Assessed Date: 12/10/2021
 https://www.slideshare.net/cutechellam/nonparametric-tests-29634320 Assessed
Date: 12/10/2021
 https://onlinelibrary.wiley.com/doi/10.1002/9780470905173.ch24 Assessed Date:
12/10/2021
 https://www.slideshare.net/RizwanSa/a-introduction-to-nonparametric-tests
Assessed Date:12/10/2021
 https://corporatefinanceinstitute.com/resources/knowledge/other/nonparametric-
tests/ Assessed Date: 12/10/2021
 https://www.slideshare.net/parth241989/chi-square-test-16093013 Assessed Date:
12/10/2021
 https://slideplayer.com/slide/14833646/ Assessed Date:12/10/2021
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