Bulacan State University
City of Malolos
College of Science
Categorical Data Analysis (MAS 401)
Contingency Tables
12 October 2021
Prepared by: Carla M. Clemente
Schedule:
October 19, 2021: Asynchronous session
October 26, 2021: Online session, 2:30 PM to 3:30 PM
October 15, 2021: Submission of Activity 1
Objectives:
● understand the probability structure of contingency
tables
● describe the associations between categorical variables
and conduct inferences
Contingency Table
Table 1. Cross Classification of Belief in Afterlife by Gender
Definition
Suppose there are two categorical variables, denoted by X and Y . Let
I denote the number of categories of X and J the number of
categories of Y . A rectangular table having I rows for the categories
of X and J columns for the categories of Y has cells that display the I J
possible combinations of outcomes.
A table of this form that displays counts of outcomes in the cells is
called a contingency table.
Joint Distribution of X and Y
Let πij = P(X = i,Y = j) denote the probability that (X, Y ) falls in the cell in row i and
column j . The probabilities {πij } form the joint distribution of X and Y .
Marginal Distribution
The marginal distributions are the row and column totals of the joint
probabilities. We denote these by {πi+} for the row variable and {π+j} for the
column variable. For 2x2 tables,
Joint and marginal distributions
Notation for Cell Counts
Conditional distributions
A conditional distribution refers to probability distribution of Y at fixed
level of X.
Notation for Cell Counts
Independence
Two variables are said to be statistically independent if the population
conditional distributions of Y are identical at each level of X.
Statistical independence is the property that all joint probabilities equal the
product of their marginal probabilities,
COMPARING PROPORTIONS IN 2x2 TABLES
● Difference of Proportions
● Relative Risk
Difference of proportions
The difference of proportions π1 − π2 compares the success probabilities
in the two rows. This difference falls between −1 and +1.
Let p1 and p2 denote the sample proportions of successes.
The sample difference p1 − p2 estimates π1 −π2. When the counts in the two rows are
independent binomial samples, the estimated standard error of p1 − p2 is
Difference of proportions
The estimated standard error of p1 − p2 is
A large sample 100(1- alpha)% confidence interval for π1 − π2 is
Example
Table 2. Cross Classification of Aspirin Use and Myocardial Infarction
This was a five-year randomized study testing whether regular intake of
aspirin reduces mortality from cardiovascular disease. Every other day, the
male physicians participating in the study took either one aspirin tablet or a
placebo.
Example
Table 2. Cross Classification of Aspirin Use and Myocardial Infarction
Relative risk
For 2x2 tables, the relative risk is the ratio
Two groups with sample proportions p1 and p2 has a sample relative risk of
It can be any nonnegative real number and value of 1.0 corresponds to independence.
Example
Table 2. Cross Classification of Aspirin Use and Myocardial Infarction
Relative risk
Relative risk is the ratio of the probability of an event occurring with an exposure versus
the probability of the event occurring without the exposure.
One must know the exposure status of all individuals (either exposed or not exposed) to
calculate the relative risk.
This implies that relative risk is only appropriate for cases where the exposure status and
incidence of disease can be accurately determined, such as prospective cohort studies.
Tenny, S. & Hoffman, M. Relative risk.
https://www.ncbi.nlm.nih.gov/books/NBK430824/.
References
Agresti, A. (2007). An Introduction to Categorical Data Analysis, 2nd Edition. New
York: Wiley.
Analysis of Discrete Data. https://online.stat.psu.edu/stat504/
Bluman, A.G. (2014). Elementary Statistics: A Step By Step Approach (9th ed).
McGraw-Hill Education.
Gallistel,
CR.
Bayes
for
Beginners:
Probability
and
Likelihood.
https://ruccs.rutgers.edu/images/personal-charles-r-gallistel/publications/2015-A
PS-Bayes-for-Beginners-1-Probability-and-Likelihood---Association-for-Psycholo
gical-Science.pdf
Tenny, S. & Hoffma, M. Relative risk.
https://www.ncbi.nlm.nih.gov/books/NBK430824/.