1
Psych 100A
Winter 2010
Lecture 14: Statistical Inference
Categorical data
•
Multinomial experiment
•
Chisquare test
•
Contingency table
Categorical data
•
Previous inference methods applicable to quantitative
data.
•
Inference on qualitative data.
•
Number of observations at each level of a
qualitative variable = count or enumeration data.
•
Population: individuals can be placed into various
categories according to some characteristic.
•
Sample: count of number of individuals who fall into
each category.
•
Data is characteristic of multinomial experiment.
1. Multinomial experiment.
•
A natural extension of a binomial experiment
•
Multinomial experiments are defined by the
following conditions:
1. Experiment consists of n iid trials
2
Polychotomous outcome on each trial: each
2. Polychotomous outcome on each trial: each
trial results in one of k outcomes
3. Pr(outcome i) =
i
, i = 1,…,k; constant
from trial to trial;
I
= 1.
4. Variables of interest are n
i
= the number
of trials with outcome i observed during
the n trials.
Formula: Pr(outcomes) in a multinomial
experiment
•
Probability for the number of observations
resulting in each of the k outcomes is given by
k
n
n
n
k
k
n
n
n
n
n
n
n
3
2
1
2
1
2
1
2
1
!
!
!
!
,
,
,
Pr
•
where
•
n = number of trials
•
i
= probability of i
th
outcome
•
n
i
= number of i
th
outcomes
•
x! = x(x1)(x2)…(2)(1)
Multinomial experiment (cont.)
•
One use is to test specified probabilities (
iO
) for
each outcome in a categorical study.
•
In an study with k outcomes, the expected number
of outcomes of type I in n trials equals:
E
i
= n
iO
•
In 1900, Pearson devised a test statistic to test
specified categorical probabilities called the
2
(chisquare) goodnessoffit statistic:
•
where n
i
’s = observed cell counts and E
i
’s =
expected cell counts.
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 Winter '10
 FIRSTENBERG,I.
 Chisquare distribution, Pearson's chisquare test, Multinomial Experiment, eij

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