Chapter 10. Models for Matched Pairs
Deyuan Li
School of Management, Fudan University
Feb. 28, 2011
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Outline
•
10.1: Comparing Dependent Proportions;
•
10.2: Conditional Logistic Regression for Binary Matched
Paries;
•
10.3: Marginal Models for Squared Contingency Tables;
•
10.4: Symmetry, Quasi-symmetry and Quasi-independence;
•
10.5: Measure Agreement Between Observers;
•
10.6: Bradley-Terry Models for Paired Preferences.
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In this chapter, we introduce methods for comparing categorical
responses for two samples when each observation in one sample
pairs with an observation in the other.
For easy understanding, we assume
n
independent
subjects and let
Y
i
=(
Y
i
1
,
Y
i
2
, ...,
Y
it
i
)
is the observation of subject
i
at di±erent time.
In statistics,
{
Y
1
,
Y
2
, ...,
Y
n
}
are called
longitudinal data
(
±
±
±
±
); in econometrics it is called
panel data
(
±
±
±
±
).
For ²xed
i
,
Y
i
is a time series; for ²xed time
j
,
{
Y
1
j
,
Y
2
j
, ...,
Y
nj
}
is
a sequence of independent random variables.
If
t
i
=2fora
l
l
i
,
{
Y
1
,
Y
2
, ...,
Y
n
}
is called
matched-pairs data
(
±
±
±
±
). Note that the two samples
{
Y
11
,
Y
21
, ...,
Y
n
1
}
and
{
Y
12
,
Y
22
, ...,
Y
n
2
}
are
not independent
.
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Table 10.1 illustrates matched-pairs data.
TABLE 10.1
Rating of Performance of Prime Minister
Second Survey
First
Survey
Approve
Disapprove
Total
Approve
794
150
944
Disapprove
86
570
656
Total
880
720
1600
For a poll of a random sample of 1600 voting-age British citizens,
944 indicated approval of the Prime Minister’s performance in
oﬃce. Six months later, of these same 1600 people, 880 indicated
approval.
For matched pairs with a categorical response, a two-way
contingency table with the same row and column categories
summarizes the data. The table is
square
.
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