Introduction to Categorical Data Analysis

4 INTRODUCTION
models for continuous data, the normal distribution plays a central role. This section
presents the key distributions for categorical data: the binomial and multinomial
distributions.
1.2.1 Binomial Distribution
Often, categorical data result from n independent and identical trials with two possible
outcomes for each, referred to as “success” and “failure.” These are generic labels,
and the “success” outcome need not be a preferred result. Identical trials means that
the probability of success is the same for each trial. Independent trials means the
response outcomes are independent random variables. In particular, the outcome of
one trial does not affect the outcome of another. These are often called Bernoulli
trials. Let π denote the probability of success for a given trial. Let Y denote the
number of successes out of the n trials.
Under the assumption of n independent, identical trials, Y has the binomial distribution
with index n and parameter π. You are probably familiar with this distribution,
but we review it briefly here. The probability of outcome y for Y equals
P(y) =
n!
y!(n − y)! π y (1 − π) n−y , y = 0, 1, 2,...,n (1.1)
To illustrate, suppose a quiz has 10 multiple-choice questions, with five possible
answers for each. A student who is completely unprepared randomly guesses the
answer for each question. Let Y denote the number of correct responses. The probability
of a correct response is 0.20 for a given question, so n = 10 and π = 0.20. The
probability of y = 0 correct responses, and hence n − y = 10 incorrect ones, equals
P(0) =[10!/(0!10!)](0.20) 0 (0.80) 10 = (0.80) 10 = 0.107.
The probability of 1 correct response equals
P(1) =[10!/(1!9!)](0.20) 1 (0.80) 9 = 10(0.20)(0.80) 9 = 0.268.
Table 1.1 shows the entire distribution. For contrast, it also shows the distributions
when π = 0.50 and when π = 0.80.
The binomial distribution for n trials with parameter π has mean and standard
deviation
E(Y) = μ = nπ, σ = � nπ(1 − π)
The binomial distribution in Table 1.1 has μ = 10(0.20) = 2.0 and σ =
√ [10(0.20)(0.80)] =1.26.
The binomial distribution is always symmetric when π = 0.50. For fixed n, it
becomes more skewed as π moves toward 0 or 1. For fixed π, it becomes more