Parametric and non parametric test in biostatistics

PARAMETRIC AND NON-
PARAMETRIC TEST IN
BIOSTATISTICS
KAPIL GAUTAM
INSTITUTE OF MEDICINE, MAHARAGJUNJ, KTM NEPAL
TEACHING HOSPITAL

Presentation layout
• Biostatistics –introduction/history
- Statistics & differences
- Uses and scope
• Parametric test
• Non-parametric test
• Limitations
• Summary
• References

History
• Father of biostatistics – Francis Galton
• Concept- correlation
• He used questionnaires

Statistics – Statistics is a branch of mathematics that deals
with the methods of
• Collection
• compilation
• Analysis
• Presentation, and interpretation of Data
Biostatistics
• It is defined as application of statistical methods to medical,
Biological and Public health related problems

Bio-statistics
• Is the application of statistical techniques to scientific
research in health-related fields, including medicine,
public health , pharmacy, nursing and biology.
• Biostatistics has became an indispensable tool in
improving health and reducing illness

Biostatics can be applied to
• Determine of major risk factors for heart disease, lung
cancer and related disease
• Test of new drugs combat e.g. HIV/AIDS
• Evaluate the potential environmental factors harmful to
human health, such as tobacco smoke, vehicle smoke etc.

Uses of biostatistics
• Biostatistics helps to define what is normal and it also gives
limits of normality. E.g.: Hemoglobin level.
• It allows us to compare information from one city or region
to that of other .
• Find out correlation between two variables
• Provides useful information for Planning and
Implementation of various health program, Monitoring and
Evaluation of such program.

Uses in preventive medicine
• To provide the magnitude of any health problem in the
community.
• To find out the basic factors underlying the ill-
health.
• To evaluate the health programs which was
introduced in the community (success/failure).

Parametric vs Non-parametric tests
Parametric test - Assume the data is of sufficient “quality”
• The results can be misleading if assumptions are wrong
• Quality is defined in terms of certain properties of
data
Non-Parametric tests – Can be used when the data is not of
sufficient quality to satisfy the assumptions of
parametric test
• Parametric tests are preferred when the assumptions
are met because they are more sensitive.

Parametric test
Assumption
• Random independent samples
• Interval or ratio level of measurement
• Normally distributed
• No-outliers
• Homogeneity of variance
• Sample size larger than minimum for many non-parametric test
Tests
• More power- more like to detect a difference than truly exists
• Less likely than non-parametric tests to make a type-II error

Types of parametric tests
1. Large sample tests
• Z-test
2. Small sample tests
• T-test
• Independent/unpaired t-test
• Paired t-test
• ANOVA (Analysis of variance)
• One way ANOVA
• Two way ANOVA
3. F-test

Z-test
• A Z-test is used for testing the mean of a
population versus a standard.
• OR
• Comparing the means of two populations.
(With large (n≥ 30) samples whether we know the
population standard deviation or not)

It is also used for testing the proportion of some
characteristic versus a standard proportion, or comparing
the proportions of two populations.
E.g. Comparing the average engineering salaries of men
versus women.
E.g. Comparing the fraction defectives from two production
lines.

T- test
• Derived by W S Gosset in 1908.
Properties of t distribution:
i. It has mean 0
ii. It has variance greater than one
iii. It is bell shaped symmetrical distribution about mean
Assumption for t test:
i. Sample must be random, observations independent
ii. Standard deviation is not known
iii. Normal distribution of population

Uses of t test
i. The mean of the sample
ii. The difference between means or to compare two
samples
iii. Correlation coefficient
• Types of t test:
a. Paired t test
b. Unpaired t test

Paired t test
Consists of a sample of matched pairs of similar units, or
one group of units that has been tested twice (a
"repeated measures" t-test )
• Ex. where subjects are tested prior to a treatment, say
for high blood pressure, and the same subjects are
tested again after treatment with a blood-pressure
lowering medication.

Unpaired t test:
• When two separate sets of independent and identically
distributed samples are obtained, one from each of the
two populations being compared.
• Ex: 1. compare the height of girls and boys.
2. compare 2 stress reduction interventions
• when one group practiced mindfulness meditation while
the other learned progressive muscle relaxation

Analysis of variance (ANOVA)
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. Fisher
• Two types: i. One way ANOVA
ii. Two way ANOVA

One way ANOVA
It compares three or more different groups when data
are categorized in one way E.g.
1. Compare control group with three different doses of
aspirin in rats
2. Effect of supplementation of vit C in each subject
before, during and after the treatment

Two way ANOVA:
• Used to determine the effect of two nominal predictor
variables on a continuous outcome variable.
• A two-way ANOVA test analyzes the effect of the
independent variables on the expected outcome along
with their relationship to the outcome itself

Difference between one & two way
ANOVA
• E.g. In one-way ANOVA - if we want to determine if
there is a difference in the mean height of stalks of three
different types of seeds.
Since there is more than one mean, we can use a one-way
ANOVA since there is only one factor that could be making
the heights different.

Now, if we take these three different types of seeds, and
then add the possibility that three different types of
fertilizer is used, then we would want to use a two-way
ANOVA.
• The mean height of the stalks could be different for a
combination of several reasons

• The types of seed could cause the change, the types of
fertilizer could cause the change, and/or there is an
interaction between the type of seed and the type of
fertilizer.
• There are two factors here (type of seed and type of
fertilizer), so, if the assumptions hold, then we can use a two-
way ANOVA.

F’ test
• A statistical test is used to determine whether two
populations having normal distribution have the same
variances or standard deviation
• Same as ANOVA test or just another name
Estimate of σ2 from means
F=
Estimate of σ2 from individuals

• Summary of parametric tests applied for different type
of data Sly no Type of Group Parametric test 1.
Comparison of two paired groups Paired ‘t’ test 2.
Comparison of two unpaired groups Unpaired ‘t’ test 3.
Comparison of three or more matched groups Two way
ANOVA 4. Comparison of three or more matched
groups One way ANOVA 5. Correlation between two
variables Pearson correlation
Unmatched

Non-parametric test
• Test are based on certain probability distribution such as
normality of the population distribution and randomness of
the sample.
• Dot not use the parameter of the distribution.
• Test without a model i.e. Called distribution free test or
non-parametric tests
• Can be use for ordinal data, Skewed data etc.
• When the continuous data are not normally distributed, we
select the nonparametric test.

• The distribution can be checked whether it is normal or
abnormal by using
• Histogram plot
• Normal plot
• Kolmogorov- Smironve test and Shaprio
Wilk test

Situation for the non-parametric test
• When quick data analysis is required to have a rough idea
about how the things are working with data.
• When only comparative rather than absolute magnitudes are
available as in the case of clinical data.
E.g. The patients can be categorized as better, unchanged and
worse
• When data is available in non parametric nature
• When data does not satisfy the assumption of a parametric
procedure.

Advantages
• Can be applied for small sample size
• Most suitable for ranked data
• Can be used for observations drawn from
different populations
• Much easier to learn than parametric tests
• Can be calculated in very quick time

Disadvantages
• Don’t use the actual measurements
• Effects of variable may not find out in non-
parametric test
• Less powerful than the parametric test

Types of Non-parametric test
1. Run test
2. Sign test
3. Ranks & Median test
4. Paired Wilcoxon Signed Rank
5. Mann-Whitney test (or Wilconxon Rank Sum Test)
6. Kruskal Wallis Test
7. Fisher’s exact test

Run test
• Most frequently to test the randomness ( or lack of
randomness) of data.
• A sequence of data that possess a common property.
• The test statistic in this test is V, the number of runs
observed
• Used to decide if a data set is from a random process.
• Run is defined as a series of increasing values or a
series of decreasing values

• The no. of increasing or decreasing, values is the length of
the run
• In a random data set, the probability that the (L+1)th value
is larger or smaller than the Lth value follows a binomial
distribution which forms the basis of the runs
• A run is a sequence of similar events
• E.g. In flipping coins, the number of “heads” in a row
• In a series of patients, the number of “female Patients” in a
row

Median test
• Is the value above and below which 50% of the data lie.
• If the data is ranked in order, it is the middle value
• In symmetric distribution the mean and median are the
same,
• In skewed distributions, median more appropriate

E.g.
• Let us suppose the blood pressure of 7 patients are
given below
• BP: 135, 138, 140, 141, 142, 143, 144
Median - 141
In parametric test, mean is compared to test the null
hypothesis whereas in non - Parametric test median is
used

Q- No. of cigarettes smoked: 0, 1, 2, 2, 2, 3, 5, 5, 8, 10
Median = ??
Answer = (2.5)
Question on floor
Bonus slide

Sign test
• One of oldest & easiest non-parametric test in statistics
• Based on the direction of plus or minus sings of
observation rather than numerical values in a samples.
• Frequently used method instead of t-test

• The IQ SCORES OF 10 mentally retarded women are given
• Let us assume the population median = 5
• Null Hypothesis (Ho): the population median is 5= {p(+) = p(-)}
=0.5
Test statistic: the test statistics for the sign test is either the
observed no. of plus signs or the observed no. of minus signs
Women Scores Women Scores
1 4 6 9
2 5 7 10
3 8 8 7
4 8 9 6
5 9 10 6

• Distribution of test statistic: the plus sign are assigned if
observations are greater than population median and
minus sign are assigned if observations are less than the
population median
• Scores above + and below – hypothesized median based
on the data
• No of positive sign = 8 No. of negative sign =1 and no. of zero = 1
Women 1 2 3 4 5 6 7 8 9 1
0
Score relative to hypothesized median - 0 + ++ + + + + +

Use binomial probability distribution to know the probability
of negative or positive sign
For probability of observing fewer minus sign is = 0.0195
Which is less than 0.5 we reject the null hypothesis
We conclude that the median score is not 5

Two Independed sample: The Mann-Whitney test
• Mann Whitney test is also known as Wilcoxon rank sum
test.
• Popular test amongst the rank sum tests
• Used to determine whether two independent samples
have been drawn from the same populations having
same median or not

Assumptions
• Two samples are drawn randomly and independently
• Two population have same median
• The measurement scale of dependent variable is at
least ordinal

Steps
• First arrange the both group data in ordered from
(ascending or descending)
• Give the rank from low to high according to the magnitude.
• For tied observation, give the mean rank to all tied
observations.

• 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.

For larger sample
The test statistic is
This is similar to the procedure of Z test

• A researcher wishes to find the assess the effects of prolonged
inhalation of cadmium oxide. He had taken 11 animals as experimental
subjects and 8 similar animals served as controls. The variable of
interest was hemoglobin level.
• How conclude that prolonged inhalation of cadmium oxide reduces
hemoglobin level ?
• Experimental Group: 14.0 15.3 16.7 13.7 15.3 15.7 15.6 14.1 15.6
15.1 15.9 16.6 14.1
• Control group: 16.2 17.1 17.4 17.5 15.0 16.0 16.9 15.0

• Ho- prolonged inhalation of cadmium oxide doesn’t reduce
hemoglobin level
• H1: prolonged inhalation of cadmium oxide reduces
hemoglobin level
• To find the test statistics, we combine the two samples and
rank all observations from smallest to largest. Tied
observations are assigned a mean rank to all observations

Observation in
ordered from
Rank Rank of X Rank of Y
13.7
14.0
14.1
14.1
15.0
15.O
15.3
15.3
15.6
15.7
15.9
16.0
16.2
16.6
16.7
16.9
17.1
17.4
17.5
1
2
3.5
3.5
5.5
5.5
7.5
7.5
9
10
11
12
13
14
15
16
17
18
19
1
2
3.5
3.5
7.5
7.5
9
10
11
14
15
5.5
5.5
12
13
16
17
18
19

Here, n1= 11, and n=8
R1= 84 and R2= 106, so take the larger sum ran, in this care
R2 is taken
Hence, test statistic
• Tabulated value of U is as U0.05(18),11,8 = 19
• Since calculation value (18) is less than the tabulated value
(19), Ho is rejected
Conclusion
• We can conclude that prolonged inhalation cadmium oxide
does not reduce hemoglobin

Wilcoxon Matched pairs test ( signed rank
test): for dependent samples
• It is equivalent to paired t-test.
• Alternative method to test the paired data when the
observations do not follow the criteria of normality
• Used to test paired data such as: before and after studies,
studies of twins or other relatives, two different medicines
used on the same group at different time period etc.

• Each subject produces two scores, one for each condition
• Test is done to show whether there is a statistically
significant difference between the two conditions
Step
• First find the differences(d) of the scores of the two
matched samples
• Assign the rank to the differences ( ignoring the sign)
• Positive ranks are summed
• Negative ranks are summed
• T is the smaller sum of ranks
• n is the number of matched pairs

Conditions of test
• If n>15, T is approximately normally distribution,
and a Z test is used
• If n<15, a special “small sample” procedure is
followed
• The paired data are randomly selected
• The underlying distributions are symmetrical

Hypothesis
Null hypothesis (Ho) : the two population are
identical
Alternative Hypothesis H1: the two population are
not identical

• The data in the table below the duration of tolerance of
pain by 11 subject before and after the administration of a
drug (0.04mg/20g) does the data provides sufficient
evidence in support that drug increases the duration of
endurance of pain
• Calculation of Wilcoxon’s Signed Rank Test (small sample)

SN Before Drug After Drug Difference Rank
differen
ce
Ranked with
signs
1 154.5 21.2 -133.3 8 8
2 12.7 20.1 7.4 11 +11
3 14.8 17.2 2.4 7 +7
4 16.7 22.7 6 9 +9
5 20.1 20 -0.1 1 -1
6 22 19.8 -2.2 6 -6
7 20.2 19.8 -0.4 3 -3
8 18.1 18.8 0.7 5 +5
9 17.4 17.9 0.3 2 +2
10 17.6 24.3 6.9 10 +10
11 19.1 18.6 -0.5 4 -4

• Sum of negative ranks = -1-6-3-4 = -14
• Sum of positive ranks = 8+11+7+9+5+2+10= 52
• The null hypothesis is tested using the smaller value of
the sums of negative ranks (T), In this case, the sum of
negative ranks (T) = 14
• Tabulated value (critical value) T at 5% level of
significance at 11 pairs = 7

Note: if the difference of the pair is zero, that pair should
be subtracted from the total pairs
Conclusion
• Experimental value of T is 14, while tabled value of T is
7, it means that null hypothesis can be rejected. So, two
population are not identical

Kruskal Wallis one way analysis of
variance by rank
• One-way analysis of variance by ranks 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

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
n=5 n=6 n=5
Total (n) = 5+6+5 =16

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.

Calculation
• Arranged all the data in ordered from
obs. 78 81 83 84 86 88 89 90 92 95 97 99 100 102 103 105
Ran
k
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 15
Group A Group B Group c
13
15
7
1
16
9
11
6
4
8
10
2
14
5
3
12
R1= 52 R2=48 R3= 36

Observed H = 1.2
• Since H is distributed with chi square with (3-1) = 2d.f.
and tabulated value of chi square at 2 d.f. = 5.99.
• Therefore, the null hypothesis is not rejected
Conclusion: It means that there is no difference in the
diastolic pressures of the three groups.

Chi - Square test =
• The most commonly used nonparametric test in the biological
experiments
• It is computed on the basis of frequencies in a sample and is
applied only for qualitative data (nominal and ordinal scale data)
E.g.. Health, response to drug etc.
• Test is used as a test of significance when data is expressed in
frequencies or in terms of percentages.
• Enable us to determine the degree of deviation between
observed frequencies and expected frequencies and to conclude
whether the deviation between observed frequencies and
expected frequencies is due to error sampling or due to chance.

• Formula
Where O= observed frequency in a class and E = expected
frequency in a class.

• Application of tests
• To test the goodness of fit
• To test independence of attribute
• To test the homogeneity of the attribute in
respect of a particular characteristic or it
may be used to test the population variance

• Criterion for using test
• Data should be categorical or qualitative
• One or more categories
• Observations must be independent
• Adequate sampling size (50 or more than 50)
• Sample should be taken randomly
• Data are in frequency form
• The expected frequency of any item of cell should not
be less than 5.
if it is less than 5, then frequencies taking from the
preceding or succeeding frequency be pooled together
in order to make it 5 or more than 5

Step involved in this case are as
1. formulate the hypothesis
2. Test statistic
3. Level of significant
4. Decision
5. Conclusion

For example, to test the hypothesis that a random sample
of 1000 people has been drawn from a population in
which men and women are equal in frequency, the
observed number of men and women would be compared
to the theoretical frequencies of 50 men and 50 women. If
there were 44 men in the sample and 56 women, then

Hypothesis
• Ho: sex ration is equal
• H1: sex ration is significantly different
Test statistic
Expected value is 50:50
O = 44(M) or 56(F) E= 50
By solving we got 1.44

• Degree of freedom = k-1 = 2-1 = 1
• Level of significance = 5% = 0.05
• Tabulated value of chi square at 1 d.f. and 5% level of
significance = 3.84
Comparison = calculated chi square (1.44) is less than
tabulated chi square (3.84). So we accept the null
hypothesis.
Conclusion: since null hypothesis is accepted, we conclude
that the sex ratio is equal

McNemar’s test
• Test is a statistical test used on paired Nominal data
• A non parametric chi-square procedure that compares
proportions obtained from a 2*2 contingency table.
• Used on paired nominal data.
• Applied to 2*2 contingency tables with a dichotomous trail,
with matched pairs of subjects, to determine whether the
row and column marginal frequencies are equal.

1- Pair-matched data can come from
• Case-control studies where each case has a matching
control (matched on age, gender, race etc.)
Twins studies – the matched pairs are twins
• Before & after data
The outcome is present (+) or absence (-) of some
characteristic measured on the same individual at two
time points

Example
• Brest cancer patients receiving mastectomy followed by
chemotherapy were matched to each other on age and
cancer stage.
Pair-matched Data for case-control Study: outcome is
exposure to some risk factor

• The counts in the table for a before-after study are numbers
of pairs and no. of individuals.
• It is used to test the two types of diagnosis test or two types
of medicine whether they give the same result or not
• The Null hypothesis is Ho: Pb= Pc
• The alternative hypothesis is H1 : Pb ≠ Pc

2- Null hypotheses for paired data
3- Matched case-control study

Fisher’s exact test
• Comparing binary outputs produced by two methods
• The significance of the deviation can be calculated
exactly
• Null hypothesis : Output difference between two
methods is zero

Methods of studying of correlation
• Correlation analysis measures the degree of association
of two variables
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)

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)

• 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

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
• If r=0, there is no correlation between X & Y

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

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

• Spearman’s Rank correlation (R) =
r= spearman’s rank correlation
D= difference between two ranks
n = Number of pairs of observations

limitation
• 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

Parametric test Non-parametric test
1. Large sample tests
Z-test
2. Small sample tests
T-test
Independent/unpaired
t-test
Paired t-test
ANOVA (Analysis of
variance)
One way ANOVA
Two way ANOVA
3. F test
1. Run test
2. Sign test
3. Ranks & Median test
4. Paired Wilcoxon Signed
Rank
5. Mann-Whitney test (or
Wilconxon Rank Sum Test)
6. Kruskal Wallis Test
7. Fisher’s exact test
Summary

• Simplifies complexity
• Collects the information scientific methods
• Analyzes the data
• Helps in formulation of suitable polices
• Facilitates comparison
• Helps in forecasting
Cond..

Parametric and non parametric test in biostatistics

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