STA 4504 - 5503: Outline of Lecture Notes, c Alan Agresti

187 Pages·2011·0.44 MB·English

STA 4504 - 5503: Outline of Lecture Notes, c Alan Agresti
(cid:13)
Categorical Data Analysis
Introduction
1.
Methods for response (dependent) variable Y having
•
scale that is a set of categories
Explanatory variables may be categorical or contin-
•
uous or both
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Example
Y = vote in election (Democrat, Republican, Indepen-
dent)
x’s - income, education, gender, race
Two types of categorical variables
Nominal - unordered categories
Ordinal - ordered categories
Examples
Ordinal
patient condition (excellent, good, fair, poor)
government spending (too high, about right, too low)
2
Nominal
transport to work (car, bus, bicycle, walk, . . . )
favorite music (rock, classical, jazz, country, folk, pop)
We pay special attention to
binary variables (success - fail)
for which nominal - ordinal distinction unimportant.
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Probability Distributions for Categorical Data
The binomial distribution (and its multinomial dis-
tribution generalization) plays the role that the normal
distribution does for continuous response.
Binomial Distribution
n Bernoulli trials - two possible outcomes for each
•
(success, failure)
π = P(success), 1 π = P(failure) for each trial
• −
Y = number of successes out of n trials
•
Trials are independent
•
Y has binomial distribution
4
n!
y n y
P(y) = π (1 π) , y = 0, 1, 2, . . . , n
−
y!(n y)! −
−
y! = y(y 1)(y 2) (1) with 0! = 1 (factorial)
− − · · ·
Example Vote (Democrat, Republican)
Suppose π = prob(Democrat) = 0.50.
For random sample size n = 3, let y = number of Demo-
cratic votes
3!
y 3 y
p(y) = .5 .5
−
y!(3 y)!
−
3!
0 3 3
p(0) = .5 .5 = .5 = 0.125
0!3!
3!
1 2 3
p(1) = .5 .5 = 3(.5 ) = 0.375
1!2!
y P(y)
0 0.125
1 0.375
2 0.375
3 0.152
1.0
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Note
E(Y ) = nπ
•
V ar(Y ) = nπ(1 π), σ = nπ(1 π)
− −
p
Y
p = = proportion of success (also denoted πˆ)
• n
Y
E(p) = E = π
n
(cid:18) (cid:19)
Y π(1 π)
σ = −
n n
r
(cid:18) (cid:19)
When each trial has > 2 possible outcomes, num-
•
bers of outcomes in various categories have multinomial
distribution
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Inference for a Proportion
We conduct inferences about parameters using
maximum likelihood
Definition: The likelihood function is the probability of
the observed data, expressed as a function of the param-
eter value.
Example: Binomial, n = 2, observe y = 1
p(1) = 2! π1(1 π)1 = 2π(1 π)
1!1! − −
= ℓ(π)
the likelihood function defined for π between 0 and 1
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If π = 0, probability is ℓ(0) = 0 of getting y = 1
If π = 0.5, probability is ℓ(0.5) = 0.5 of getting y = 1
Definition The maximum likelihood (ML) estimate
is the parameter value at which the likelihood function
takes its maximum.
Example ℓ(π) = 2π(1 π) maximized at πˆ = 0.5
−
i.e., y = 1 in n = 2 trials is most likely if π = 0.5.
ML estimate of π is πˆ = 0.50.
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Note
y
For binomial, πˆ = = proportion of successes.
• n
If y , y , . . . , y are independent from normal (or many
1 2 n
•
other distributions, such as Poisson), ML estimate µˆ = y¯.
In ordinary regression (Y normal) “least squares”
• ∼
estimates are ML.
For large n for any distribution, ML estimates are
•
optimal (no other estimator has smaller standard error)
For large n, ML estimators have approximate normal
•
sampling distributions (under weak conditions)
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ML Inference about Binomial Parameter
y
πˆ = p =
n
π(1 π)
Recall E(p) = π, σ(p) = .
−
n
q
Note σ(p) as n , so
• ↓ ↑
p π (law of large numbers, true in general for ML)
→
p is a sample mean for (0,1) data, so by Central
•
Limit Theorem, sampling distribution of p is approxi-
mately normal for large n (True in general for ML)
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