This document discusses parametric and non-parametric statistical tests. It begins by defining different types of data and the standard normal distribution curve. It then covers hypothesis testing, including the different types of errors. Both parametric and non-parametric tests are examined. Parametric tests discussed include z-tests, t-tests, and ANOVA, while non-parametric tests include chi-square, sign tests, McNemar's test, and Fischer's exact test. Examples are provided to illustrate several of the tests.
Parametric vs Nonparametric Tests: When to use whichGönenç Dalgıç
There are several statistical tests which can be categorized as parametric and nonparametric. This presentation will help the readers to identify which type of tests can be appropriate regarding particular data features.
Statistical tests of significance and Student`s T-TestVasundhraKakkar
Statistical tests of significance is explained along with steps involve in Statistical tests of significance and types of significance test are also mentioned. Student`s T-Test is explained
Parametric vs Nonparametric Tests: When to use whichGönenç Dalgıç
There are several statistical tests which can be categorized as parametric and nonparametric. This presentation will help the readers to identify which type of tests can be appropriate regarding particular data features.
Statistical tests of significance and Student`s T-TestVasundhraKakkar
Statistical tests of significance is explained along with steps involve in Statistical tests of significance and types of significance test are also mentioned. Student`s T-Test is explained
Today’s overwhelming number of techniques applicable to data analysis makes it extremely difficult to define the most beneficial approach while considering all the significant variables.
The analysis of variance has been studied from several approaches, the most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear across factor levels. Interpretation is easy when data is balanced across factors but much deeper understanding is needed for unbalanced data.
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.Analysis of variance (ANOVA) is an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The systematic factors have a statistical influence on the given data set, while the random factors do not. Analysts use the ANOVA test to determine the influence that independent variables have on the dependent variable in a regression study.
Sir Ronald Fisher pioneered the development of ANOVA for analyzing results of agricultural experiments.1 Today, ANOVA is included in almost every statistical package, which makes it accessible to investigators in all experimental sciences. It is easy to input a data set and run a simple ANOVA, but it is challenging to choose the appropriate ANOVA for different experimental designs, to examine whether data adhere to the modeling assumptions, and to interpret the results correctly. The purpose of this report, together with the next 2 articles in the Statistical Primer for Cardiovascular Research series, is to enhance understanding of ANVOA and to promote its successful use in experimental cardiovascular research. My colleagues and I attempt to accomplish those goals through examples and explanation, while keeping within reason the burden of notation, technical jargon, and mathematical equations.
OBJECTIVES:
Run the test of hypothesis for mean difference using paired samples. Construct a confidence interval for the difference in population means using paired samples.
Observation of interest will be the difference in the readings
before and after intervention called paired difference observation.
Paired t test:
A paired t-test is used to compare two means where you have two samples in which observations in one sample can be paired with observations in the other sample.
Examples of where this might occur are:
Before-and-after observations on the same subjects (e.g. students’ test
results before and after a particular module or course).
A comparison of two different methods of measurement or two different treatments where the measurements/treatments are applied to the same subjects (e.g. blood pressure measurements using a sphygmomanometer and a dynamap).
When there is a relationship between the groups, such as identical twins.
This test is concerned with the pair-wise differences
between sets of data.
This means that each data point in one group has a related data point in the other group (groups always have equal numbers).
ASSUMPTIONS:
The sample or samples are randomly selected
The sample data are dependent
The distribution of differences is approximately normally
distributed.
Note: The under root is onto the entire numerator and denominator, so you should take the root after solving it entirely
where “t” has (n-1) degrees of freedom and “n” is
the total number of pairs.
Today’s overwhelming number of techniques applicable to data analysis makes it extremely difficult to define the most beneficial approach while considering all the significant variables.
The analysis of variance has been studied from several approaches, the most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear across factor levels. Interpretation is easy when data is balanced across factors but much deeper understanding is needed for unbalanced data.
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.Analysis of variance (ANOVA) is an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The systematic factors have a statistical influence on the given data set, while the random factors do not. Analysts use the ANOVA test to determine the influence that independent variables have on the dependent variable in a regression study.
Sir Ronald Fisher pioneered the development of ANOVA for analyzing results of agricultural experiments.1 Today, ANOVA is included in almost every statistical package, which makes it accessible to investigators in all experimental sciences. It is easy to input a data set and run a simple ANOVA, but it is challenging to choose the appropriate ANOVA for different experimental designs, to examine whether data adhere to the modeling assumptions, and to interpret the results correctly. The purpose of this report, together with the next 2 articles in the Statistical Primer for Cardiovascular Research series, is to enhance understanding of ANVOA and to promote its successful use in experimental cardiovascular research. My colleagues and I attempt to accomplish those goals through examples and explanation, while keeping within reason the burden of notation, technical jargon, and mathematical equations.
OBJECTIVES:
Run the test of hypothesis for mean difference using paired samples. Construct a confidence interval for the difference in population means using paired samples.
Observation of interest will be the difference in the readings
before and after intervention called paired difference observation.
Paired t test:
A paired t-test is used to compare two means where you have two samples in which observations in one sample can be paired with observations in the other sample.
Examples of where this might occur are:
Before-and-after observations on the same subjects (e.g. students’ test
results before and after a particular module or course).
A comparison of two different methods of measurement or two different treatments where the measurements/treatments are applied to the same subjects (e.g. blood pressure measurements using a sphygmomanometer and a dynamap).
When there is a relationship between the groups, such as identical twins.
This test is concerned with the pair-wise differences
between sets of data.
This means that each data point in one group has a related data point in the other group (groups always have equal numbers).
ASSUMPTIONS:
The sample or samples are randomly selected
The sample data are dependent
The distribution of differences is approximately normally
distributed.
Note: The under root is onto the entire numerator and denominator, so you should take the root after solving it entirely
where “t” has (n-1) degrees of freedom and “n” is
the total number of pairs.
2.0.statistical methods and determination of sample sizesalummkata1
statistical methods and determination of sample size
These guidelines focus on the validation of the bioanalytical methods generating quantitative concentration data used for pharmacokinetic and toxicokinetic parameter determinations.
• Non parametric tests are distribution free methods, which do not rely on assumptions that the data are drawn from a given probability distribution. As such it is the opposite of parametric statistics
• In non- parametric tests we do not assume that a particular distribution is applicable or that a certain value is attached to a parameter of the population.
When to use non parametric test???
1) Sample distribution is unknown.
2) When the population distribution is abnormal
Non-parametric tests focus on order or ranking
1) Data is changed from scores to ranks or signs
2) A parametric test focuses on the mean difference, and equivalent non-parametric test focuses on the difference between medians.
1) Chi – square test
• First formulated by Helmert and then it was developed by Karl Pearson
• It is both parametric and non-parametric test but more of non - parametric test.
• The test involves calculation of a quantity called Chi square.
• Follows specific distribution known as Chi square distribution
• It is used to test the significance of difference between 2 proportions and can be used when there are more than 2 groups to be compared.
Applications
1) Test of proportion
2) Test of association
3) Test of goodness of fit
Criteria for applying Chi- square test
• Groups: More than 2 independent
• Data: Qualitative
• Sample size: Small or Large, random sample
• Distribution: Non-Normal (Distribution free)
• Lowest expected frequency in any cell should be greater than 5
• No group should contain less than 10 items
Example: If there are two groups, one of which has received oral hygiene instructions and the other has not received any instructions and if it is desired to test if the occurrence of new cavities is associated with the instructions.
2) Fischer Exact Test
• Used when one or more of the expected counts in a 2×2 table is small.
• Used to calculate the exact probability of finding the observed numbers by using the fischer exact probability test.
3) Mc Nemar Test
• Used to compare before and after findings in the same individual or to compare findings in a matched analysis (for dichotomous variables).
Example: comparing the attitudes of medical students toward confidence in statistics analysis before and after the intensive statistics course.
4) Sign Test
• Sign test is used to find out the statistical significance of differences in matched pair comparisons.
• Its based on + or – signs of observations in a sample and not on their numerical magnitudes.
• For each subject, subtract the 2nd score from the 1st, and write down the sign of the difference.
It can be used
a. in place of a one-sample t-test
b. in place of a paired t-test or
c. for ordered categorial data where a numerical scale is inappropriate but where it is possible to rank the observations.
5) Wilcoxon signed rank test
• Analogous to paired ‘t’ test
6) Mann Whitney Test
• similar to the student’s t test
7) Spearman’s rank correlation - similar to pearson's correlation.
Chapter 7
Hypothesis Testing Procedures
Learning Objectives
• Define null and research hypothesis, test
statistic, level of significance and decision rule
• Distinguish between Type I and Type II errors
and discuss the implications of each
• Explain the difference between one- and two-
sided tests of hypothesis
Learning Objectives
• Estimate and interpret p-values
• Explain the relationship between confidence interval
estimates and p-values in drawing inferences
• Perform analysis of variance by hand
• Appropriately interpret the results of analysis of
variance tests
• Distinguish between one and two factor analysis of
variance tests
Learning Objectives
• Perform chi-square tests by hand
• Appropriately interpret the results of chi-square tests
• Identify the appropriate hypothesis testing procedures
based on type of outcome variable and number of
samples
Hypothesis Testing
• Research hypothesis is generated about
unknown population parameter
• Sample data are analyzed and determined to
support or refute the research hypothesis
Hypothesis Testing Procedures
Step 1
Null hypothesis (H0):
No difference, no change
Research hypothesis (H1):
What investigator
believes to be true
Hypothesis Testing Procedures
Step 2
Collect sample data and determine whether sample
data support research hypothesis or not.
For example, in test for m, evaluate .
X
Hypothesis Testing Procedures
Step 3
• Set up decision rule to decide when to believe null
versus research hypothesis
• Depends on level of significance, a = P(Reject H0|H0
is true)
Hypothesis Testing Procedures
Steps 4 and 5
• Summarize sample information in test statistic (e.g.,
Z value)
• Draw conclusion by comparing test statistic to
decision rule. Provide final assessment as to whether
H1 is likely true given the observed data.
P-values
• P-values represent the exact significance of the
data
• Estimate p-values when rejecting H0 to
summarize significance of the data (can
approximate with statistical tables, can get
exact value with statistical computing
package)
• P-value is the smallest a where we still reject
H0
Hypothesis Testing Procedures
1. Set up null and research hypotheses, select a
2. Select test statistic
2. Set up decision rule
3. Compute test statistic
4. Draw conclusion & summarize significance
Errors in Hypothesis Tests
Hypothesis Testing for m
• Continuous outcome
• 1 Sample
H0: m=m0
H1: m>m0, m<m0, m≠m0
Test Statistic
n>30 (Find critical
value in Table 1C,
n<30 Table 2, df=n-1)
ns/
μ-X
Z
0
=
ns/
μ-X
t
0
=
Example 7.2.
Hypothesis Testing for m
The National Center for Health Statistics (NCHS)
reports the mean total cholesterol for adults is 203. Is
the mean total cholesterol in Framingham Heart
Study participants significantly different?
In 3310 participants the mean is 200.3 with a standard
deviation of 36.8.
Example 7.2.
Hypothesis Test ...
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Chapter 3 - Islamic Banking Products and Services.pptx
Statistical test
1. Parametric and
Non Parametric Tests
Presenter: Dr. Mrigesh
Facilitator: Dr. Swati
Moderator: Dr. Ranjan Das
Dr. Manish Goel
12/6/2017 1
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.
12/6/2017 2
3. • Non Parametric test
Pre-requisites of non Parametric test
Types of non Parametric test
Limitations.
• Quick recap.
• References.
12/6/2017 3
4. Data: A collection of items of information.
A dictionary of Epidemiology, Miquel Porta
DATA
Qualitative Quantitative
Ordinal Nominal Continuous Discrete
Interval Ratio
12/6/2017 4
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.
12/6/2017 5
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.
12/6/2017 7
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
12/6/2017 9
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
12/6/2017 10
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
12/6/2017 11
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)
12/6/2017 12
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.
12/6/2017 13
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
12/6/2017 14
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
12/6/2017 15
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
12/6/2017 16
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
12/6/2017 17
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
12/6/2017 18
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
12/6/2017 19
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.
12/6/2017 20
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.
12/6/2017 22
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
12/6/2017 23
24. Test with single mean (One sample t-test):
t = ẋ - μ
SEẋ - μ
where, SE ẋ - μ = SD/√n
DF = n-1
12/6/2017 24
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
12/6/2017 25
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
12/6/2017 26
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
12/6/2017 28
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
12/6/2017 30
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
12/6/2017 31
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
12/6/2017 33
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
12/6/2017 34
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)
12/6/2017 36
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
12/6/2017 37
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
12/6/2017 38
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
12/6/2017 41
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)
12/6/2017 49
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
12/6/2017 50
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
12/6/2017 51
50. • Test statistic, χ2 = Σ(O – E)2
E
where, E = Row Total x Column Total
Grand Total
• DF = (r-1)(c-1)
12/6/2017 52
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
12/6/2017 53
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
12/6/2017 54
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
12/6/2017 55
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
12/6/2017 56
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
12/6/2017 58
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
12/6/2017 59
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
12/6/2017 60
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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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
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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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