Parametric and non parametric test

1. Parametric and
Non Parametric Tests
Presenter: Dr. Mrigesh
Facilitator: Dr. Swati
Moderator: Dr. Ranjan Das
Dr. Manish Goel
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2. Plan of presentation
• Data and its type.
• Standard normal distribution curve and its
characteristics.
• Hypothesis and types of error.
• Steps involved in testing of hypothesis.
• Parametric test
Pre-requisites of parametric test
Parametric test with examples
Limitations.
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3. • Non Parametric test
Pre-requisites of non Parametric test
Types of non Parametric test
Limitations.
• Quick recap.
• References.
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4. Data: A collection of items of information.
A dictionary of Epidemiology, Miquel Porta
DATA
Qualitative Quantitative
Ordinal Nominal Continuous Discrete
Interval Ratio
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5. Standard Normal Distribution Curve
Characteristics
• Has a bell shaped curve, symmetric.
• Mean = Median = Mode
• The total area under the curve is 1 (or 100%)
• The tails of the curve are infinite.
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6. 12/6/2017 6

7. Hypothesis : A supposition, arrived at from observation
or reflection that leads to refutable prediction.
• Two types:
A) Null hypothesis: No relationship between the
variables being studied.
B) Alternate hypothesis : There is a relationship
between the variables being studied.
– Eg. A new drug A came in market and the
manufacturer claims that it is more effective than
drug B for treating angina.
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8. Drug A=B Drug A ≠ B
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9. Steps in Testing of Hypothesis
1. Determine the appropriate test
2. Establish the level of significance (α)
3. Formulate the statistical hypothesis
4. Calculate the test statistic
5. Determine the degree of freedom
6. Compare computed test statistic against a
tabled/critical value
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10. Parametric Statistics
• A branch of statistics that assumes data comes
from a type of probability distribution and
makes inferences about the parameters of the
distribution
• Also called as Distribution dependent
statistics, Classical or Standard tests
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11. Assumptions
• Variable in question has a known underlying
mathematical distribution (Gaussian)
• They show the same variance
• Usually used for continuous data, ratio/interval data
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12. Types of parametric test
Test of Significance based on - Number of groups
- Sample size
 2 groups
 Large samples : Z-test
 Small samples : t-test
 To compare mean of more than 2 groups: F-test -
Analysis of Variance (ANOVA)
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13. Z-test
• Also called as Standard normal variate /standard
score / Z score/z-values, normal scores.
• Pre-requisites:
 Samples must be randomly selected
 Large sample size N ≥30
 Data can be quantitative or qualitative
 Variable assumed to follow normal distribution.
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14. • Test statistic, Z = difference observed
standard error
• If, Z > 1.96, p < 0.05 Then, reject NH
• If, Z < 1.96, p > 0.05 Then, fail to reject NH
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15. Qualitative data:
i) Test for single proportion:
Let, p = proportion of sample
P = proportion of population
Then, Z = p-P where, SEp = √PQ/n
SEp
Q = 1-P
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16. i) Test for two proportions:
Let, p1 = proportion of sample-1
p2 = proportion of sample-2
Then, Z = p1-p2 where, SEp1-p2 = p1q1 + p2q2
SEp1-p2 n1 n2
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17. Quantitative data:
i) Test for single mean:
Let, ẋ = mean of sample
μ = mean of population
Then, Z = ẋ - μ where, SEẋ = S/√n
SEẋ
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18. i) Test for two means:
Let, ẋ1 = mean of sample-1
ẋ2 = mean of sample-2
Then, Z = ẋ1 - ẋ2 where, SEẋ1-ẋ2 = s1
2 + s2
2
SEẋ1-ẋ2 n1 n2
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19. Ex: Complication rate of a drug which was tried on 100 pts was
15%. The new drug which was administered to another
group of 100 pts had complication rate of 7%. Test whether
the new drug is superior in terms of reducing complications.
Soln: NH: P1 = P2 AH: P1 ≠ P2
n1 = 100; p1 = 15% n2 = 100; p2 = 7%
So, Z = p1-p2
SEp1-p2
= 15-7
15 x 85 + 7 x 93
100 100
Z = 8/6.21 = 1.82
As, Z < 1.96, p > 0.05
Fail to reject NH
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20. t-Test
• Pre-requisites:
 Samples must be randomly selected
 Small sample size N < 30
 Data must be quantitative
 Variable assumed to follow normal distribution.
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21. 12/6/2017 21

22. Degrees of freedom (DF):
• Freedom to choose variables
• Hence, DF = n – k
where, n = no. of observations (sample size)
k = no. of population parameters
from this sample.
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23. • Critical value calculated according to degrees of freedom and
‘p’ value as per ‘t-table’
• Test statistic, td,p = difference observed
standard error
where, t = t value
d = DF
p = p value
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24. Test with single mean (One sample t-test):
t = ẋ - μ
SEẋ - μ
where, SE ẋ - μ = SD/√n
DF = n-1
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25. Test with two means:
i) Unpaired t-test:
t = ẋ1 - ẋ2
SEẋ1-ẋ2
where, SEẋ1-ẋ2 = S2 + S2 and, S2 = (n1-1)s1
2+(n2-1)s2
2
n1 n2 n1+n2-2
DF = n1+n2 -1
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26. Ex: In a clinical trial of comparing efficacy of two
antihypertensive drugs A & B, Drug A was given to
10 randomly selected pt’s & at the end of trial mean
DBP was 88mmHg with SD of 5mmHg. Drug B was
given to 8 randomly selected pt’s & at end of trial
mean DBP 94mmHg with SD of 6mmHg. Test
whether drug A differs from drug B in the treatment
of hypertension?
Soln: NH: Drug A= B AH: Drug A ≠ B
n1=10 ẋ1 = 88 SD1 =5
n2 = 8 ẋ2 = 94 SD2 =6
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27. Soln: NH: Drug A = B AH: Drug A ≠ B
n1=10 ẋ1 = 88 SD1 =5
n2 = 8 ẋ2 = 94 SD2 =6
t = ẋ1 –ẋ2 = 88-94/2.589 = 2.317
SE
DF = 10+8-2=16
t for 16 DF at P0.05= 1.746
Calculated t > t16,0.05
Hence, Reject NH and thus Drug A differs from drug B12/6/2017 27

28. ii) Paired t-test (pre-post comparison):
t = d
SEd where, d = Σd / n
SE d = SD d /√n
SDd = Σ (d-d)2
n-1
DF = n-1
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29. Pts Before
intervention
(x1)
After
intervention
(x2)
d = x1 – x2 (d-d)2
Total
d = Σd / n
SD = Σ (d-d)2
n-1
SE = SD/√n
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30. Ex: 5 persons were chosen randomly & their PR were recorded
before & after administration of drug. Results after 5min of
administration are as follows. Test whether drug changes PR?
Individual Before DA After DA
1 92 88
2 90 88
3 96 90
4 98 88
5 92 89
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31. Individual Before DA (x1) After DA
(x2)
d= x1-x2
1 92 88 -4
2 90 88 -2
3 96 90 -6
4 98 88 -10
5 92 89 -8
Total -30
d = Σd / n
= 30 / 5
= 6
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32. Individual Before DA
(x1)
After DA
(x2)
d= x1-x2 (d-d)2
1 92 88 -4 4
2 90 88 -2 16
3 96 90 -6 0
4 98 88 -10 16
5 92 89 -8 4
Total -30 40
SD = √40/(5-1)
= √10
= 3.16
SE = SD / √n
= 3.16 / 2.23
= 1.4112/6/2017 32

33. Soln: NH: PR d = 0 AH: PR d ≠ 0
• Paired t = d / SEd = 6 / 1.41 = 4.25
• DF = n-1=5-1=4
• t for 4 DF at P0.05 = 2.13
• Calculated Paired t = 4.25 > t4,0.05
• Hence, Reject NH and thus, Drug causes change in PR
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34. Analysis of Variance (ANOVA)
• Used to compare the means of two or more samples to see
whether they come from the same population.
• Compares variances within groups to variances between
groups (F-value)
• It compare groups (independent variable) based on single
continuous response variable(dependent variable) .
• Eg. Comparing test score by level of education
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35. 12/6/2017 35

36. Pre-requisites:
• Variables assumed to follow normal distribution
• Individuals in various groups must be randomly selected
• Samples comprising the groups should be independent
• All groups have same standard deviation (variance)
• Variables must be quantitative (means)
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37. • Here, variance is calculated - within group variation and
between group variation
• Between groups variation (E1) = sum of squares
k – 1
where, k=no. of groups
• Within group variation (E2) = sum of squares
n – k
where, n=total no. of observations
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38. • Test statistic, F = E1 (variance between groups)
E2 variance within groups)
• Degrees of Freedom (DF) = k-1, n-k
• F table: DF between groups (k-1) – column
DF within groups (n-k) – rows
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39. • Exercise:
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40. 12/6/2017 40

41. ANCOVA: Analysis of co-variance
• ANCOVA has single continuous response variable.
• It compares a response variable by both a factor and a
continuous independent variable
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42. 12/6/2017 42

43. MANOVA: Multivariate analysis of variance
• MANOVA is an ANOVA with two or more
continuous response variable.
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44. 12/6/2017 44

45. MANCOVA: Multivariate analysis of co-variance
• MANCOVA is ANCOVA with two or more
continuous response variable.
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46. 12/6/2017 46

47. NON-PARAMETRIC TEST
Assumptions
• Distribution of variables need not follow Gaussian
distribution.
• Sample observations are independent
• Variable is continuous or ordinal
• Small samples (n<30)
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48. Categorical Outcomes
 Chi square test
 Sign test
 Mc Nemar test
 Fischer exact test
Numerical Outcomes
 Wilcoxon Signed Rank
test
 Wilcoxon Rank Sum
test
 Kruskal Wallis test
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49. Chi square test
• Most commonly used non parametric test
• Karl Pearson invented in 1900.
Pre-requisites
• Random sample data
• To be applied on actual data and not percentages
• Adequate cell size – 5 or more in all cells of 2x2 table
and 5 or more in 80% of cells in larger tables but no
cells with zero count.
• Observations must be independent
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50. • Test statistic, χ2 = Σ(O – E)2
E
where, E = Row Total x Column Total
Grand Total
• DF = (r-1)(c-1)
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51. Characteristics of Chi Square
i) Can be applied to more than 2 groups (Test of
Homogeneity)
ii) Association can be found (Test of Association)
iii) Test for Goodness of fit – to test whether a given
distribution is a good fit to the given data
iv) Yates’ correction – Arbitrary , conservative adjustment to
chi square applied to a 2x2 table when, one or more cells
have expected value <5
χ2 = Σ [(O – E) – 0.5]2
E
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52. Ex: A anti hypertensive drug trial was conducted in
Belgaum and it found that of 60 patients who
received drug A, 45 had some complication and of 60
patients who received drug B, 33 had complications.
Is the drug safe?
Soln: NH: No relation between drug and complications
AH: Drug and complications are related
Drug Complications Total
No Yes
A 15 45 60
B 27 33 60
Total 42 78 120
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53. Χ2 = Σ (O-E)2
E
= (15-21)2 + (45-39)2 + (27-21)2 + (33-39)2
21 39 21 39
= 5.274
DF = (r-1)(c-1) = (2-1)(2-1) = 1
Χ2
1,0.05 = 3.84
Drug Complications Total
No Yes
A 15 (21) 45 (39) 60
B 27 (21) 33 (39) 60
Total 42 78 120
As, Χ2 > critical value
Reject NH
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54. Sign test
• Test to analyze the sign of difference between paired
observations either same individuals or related
individuals
• Alternative to ‘paired t-test’
• Probability is calculated
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55. Mc Nemar test
• Similar to 2x2 chi square test
• For comparison of variables from matched pairs
• Can also be used for pre and post samples
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56. Mc Nemar χ2 (χ2
c) = [(b-c)-1]2
(b+c)
Intervention Outcome Total
Yes No
Yes a b a+b
No c d c+d
Total a+c b+d a+b+c+d = N
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57. Fischer exact test
• Used when the expected values are <5 in more than
20% cells or one of them is zero
• Can be used for r x c tables
• It gives exact probability
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58. P = (a+b)!.(c+d)!.(a+c)!.(b+d)!
N!.a!.b!.c!.d!
Intervention Outcome Total
Yes No
Yes a b a+b
No c d c+d
Total a+c b+d a+b+c+d = N
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59. Wilcoxon Signed Rank test
• Comparison in a single sample
• For pre and post intervention comparison
• Medians compared
• Observations are ranked and then compared
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61. 12/6/2017 64

62. 12/6/2017 65

63. Wilcoxon Rank Sum test
• Comparing two independent samples
• Means are compared
• Observations are ranked and then compared
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64. 12/6/2017 67
• Example: To compare the scores on quantitative
variables obtained from two independent group
Group A: 2 4 2 6 4 8
Group B: 8 8 4 10 12 11

65. 12/6/2017 68

66. Kruskal Wallis test
• For comparison of more than 2 groups
• Observations are ranked altogether and then
compared
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68. 12/6/2017 72

69. Quick recap
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70. Quick recap contd..
Two sample
problem
All Expected values
<5?
Chi square test
Fischer’s Exact
test
Mc Nemar
test
Paired t test
Samples
independent?
Distribution
Normal?Yes
Yes No
No
YesNo
Large Small
Unpaired t
test
Sample Size
Samples
independent
Z test
Yes No
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71. • Quick recap contd…
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72. References
• Park K. Park’s textbook of preventive and social
medicine. 23rd ed. Jabalpur: M/s Banarsidas Bhanot;
2015.
• Bhalwar R. Textbook of public health and community
medicine. Pune: Department of community
medicine,AFMC;2009.
• Sunder L, Adarsh, Pankaj. Textbook of community
medicine preventive and social medicine. 4th ed. New
Delhi: CBS Publishers &Distributors Pvt Ltd;2014.
• Indrayan A., Holt P. Concise Encyclopedia of
Biostatistics for medical professionals. CRC press
Taylor and Francis group; 201712/6/2017 76

73. • Beaglehole R., Bonita R. Basic Epidemiology 2nd ed.
WHO library cataloguing-in-publication data; 2002
• Armitage P., Berry G. Statistical Methods in Medical
Research 4th ed.Blackwell Science;2002.
• Indrayan A., Sarmukaddam B.Medical Biostatistics
1st ed.. Marcel Dekker, Inc. New York. Basel; 2001.
• Das R., Das P. Biomedical Research Methodology
including Biostatistical Applications. Jaypee Brothers
Medical Publishers (P) Ltd.; 2011.
• Negi K. Methods in Biostatistics 1st ed. AITBS
Publisher India; 2012.
• Porta M. A Dictionary of Epidemiology 6th
ed.BEA;201412/6/2017 77

74. Thank you!
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