1. Categorical
Data Analysis
KRISHNAKUMAR D
AVMC&H

2. Categorical Data Analysis
Text Book: “ An
Introduction to
Categorical Data
Analysis”
By “ Alan Agresti”

4. Scales Of Measurement
 Four Scales Of Measurement:
 Nominal : No Order (e.g)- gender
 Ordinal: Order (e.g)- Income Status
 Ratio: Equal intervals with no True
0 (e.g): Height
 Interval: Equal intervals with True 0
(e.g): Temperature
Categorical Data Analysis

5. Categorical Data: Analysis
Strategies
 Hypothesis Testing: Is there any association?
 Chi Square Test, Fishers Exact test, etc
 Chapter- 1, 2 , 3.
 Modeling: What is the nature of Association?
 Logistic Regression, Log linear Models
 Chapter- 4, 5, 6,7

6. What is categorical data?
The measurement scale for the response
consists of a number of categories
Variable Measurement Scale
Farm system Organic & non organic
Education Good , average, poor
Food texture
Very soft, Soft, Hard,
Very hard
Nutrition
status
Grade 1, 2, 3
KAP- public
health
“yes” or “No”

7. Data Analysis considered:
 Response variable(s) –( Dependent Variable or Y variable)
is categorical
 Explanatory variable(s) –(Independent or X variable)
may be categorical or continuous or both
Example: Diabetes (categorical response) depend on the
explanatory variables?
Sex (categorical)
Age (continuous)
Example:
Y = Diabetes( Present, absent/ Normal, mild , moderate,
severe- Independent)
X’s = Income, Education, gender, age, Sedentary life style,
Hereditary etc.

8. Important Note
 Methods designed for nominal variables give the same results no
matter how the categories are listed
 Methods for ordinal variables utilize the category ordering. Whether we
list the categories from low to high or from high to low is irrelevant in
terms of substantive conclusions, but results would change if the
categories were reordered in any other way.
 Methods designed for ordinal variables cannot be used with nominal
variables
 However, Methods designed for nominal variables can be used with
nominal or ordinal variables
 If used, it results in serious loss of power.
•nominal < ordinal < interval

9. Probability Distributions
 For continuous response variable – Normal distribution
 For Categorical response variable – Binomial
distribution or multinomial distribution

10. Binomial Distribution
 n Bernoulli trials - two possible outcomes for each
(success, failure)
 ∏ = P(success), 1 − ∏ = P(failure) for each trial
 Y = number of successes out of n trials
 Trials are independent
Y has binomial distribution
, y= 0,1, 2,…, n

11. Example: Binomial
Distribution
 Vote (Democrat, Republican)
 Suppose = prob(Democrat) = 0.50.
For n = 3 persons, let y = number of
Democratic votes
then, p(0) = 0.125
p(1) = 0.375
p(2)= 0.375
p(3) = 0.125

12. Multinomial distribution
 When each trial has >2 possible outcomes, no of
outcomes in various categories have multinomial
distribution.
 Let c denote the number of outcome categories
 The binomial distribution is the special case with c = 2
categories.

13. Properties of the
Multinomial Experiment
1. The experiment consists of n identical trials.
2. There are k possible outcomes to each trial. These
outcomes are called classes, categories, or cells.
3. The probabilities of the k outcomes, denoted by p1,
p2,…, pk, remain the same from trial to trial,where
p1 + p2 + … + pk = 1.
4. The trials are independent.
5. The random variables of interest are the cell
counts, n1, n2, …, nk, of the number of
observations that fall in each of the k classes.

14. Statistical Inference for a
proportion
 The parameters of a Binomial and Multinomial
distribution are estimated using the sample data.
 Methods of estimation is “Maximum Likelihood
Estimation” (ML Estimation)
 The likelihood function(denoted by l) is the probability
of the observed data, expressed as a function of the
parameter value.

15. Contd…
Example:
Consider a Binomial case, n = 2, observe y = 1
 The likelihood function defined for between 0 and 1
 If = 0, probability is l (0) = 0 of getting y = 1
 If = 0.5, probability is l(0.5) = 0.5 of getting y = 1

16. Maximum Likelihood
 The maximum likelihood (ML) estimate is the
parameter value at which the likelihood function takes
its maximum.
 Example
l( ) = 2(1 − ) maximized at ˆ = 0.5
 i.e., y = 1 in n = 2 trials is most likely if = 0.5.
ML estimate of is ˆ = 0.50.
 In general, ML estimate of is p= y/n.

17. Binomial Likelihood functions for y=0
successes and y=6 successes in n
=10 trials
The result y = 6 in n = 10 trials is more likely to
occur when π = 0.60 than when π equals any other value.

18. Significance Test for
binomial parameter
 A significance test merely indicates whether a
particular value for a parameter is plausible.
 The ML estimator for the Binomial Distribution is the
sample proportion , p.

19. Confidence interval and
significance tests
 Three different test methods to find CI and test
statistic:
 Wald Method
 Likelihood-ratio method
 Score method

20. Wald Test
 Let be the ML estimator. Then the Wald Test
statistic to test is given by
Where SE is the Standard Error of the ML estimate
and this follows standard normal distribution and Z2
follows Chisquare distribution with d.f = 1.
 The z or chi-squared test using this test statistic is
called a Wald test.

21. Likelihood Ratio Test
This alternative test uses the likelihood function
through the ratio of two maximizations of it:
1. the maximum over the possible parameter values
that assume the null hypothesis,
2. the maximum over the larger set of possible
parameter values, permitting the null or the
alternative hypothesis to be true.

22. Contd..
Let l0 denote the maximized value of the likelihood
function under the null hypothesis, and let l1 denote
the maximized value more generally.
For instance, when there is a single parameter β, l0 is
the likelihood function calculated at β0, and 1 is the
likelihood function calculated at the ML estimate ˆ β.
Then l1 is always at least as large as l0, because l1
refers to maximizing over a larger set of possible
parameter values.

23. Remarks
 For ordinary regression models assuming a normal
distribution for Y , the three tests provide identical results.
 In other cases, for large samples they have similar
behaviour when H0 is true.
 Wald CI often has poor performance in categorical data
analysis unless n quite large.
 For inference about proportions, score method tends to
perform better than Wald method, in terms of having
actual error rates closer to the advertised levels.
 In practice, Wald inference is popular because of
simplicity, ease of forming it using software output

24. Thank you