11.1.1 Binary marginal models and marginal homogeneity
Denote
T
binary responses by (
Y
1
,
Y
2
, ...,
Y
T
). The marginal logit
model is
logit[
P
(
Y
t
=1)]=
α
+
β
t
,
t
=1
,
2
, ...,
T
,
(11
.
1)
with
β
T
=0
. Inpart
icu
lar
,
β
1
=
β
2
=
...
=
β
T
implies
marginal
homogeneity (MH)
.
For the outcome
i
=(
i
1
, ...,
i
T
)w
ith
i
t
= 1 or 0, let the joint mass
probability be
π
i
=
P
(
Y
1
=
i
1
,
Y
2
=
i
2
, ...,
Y
T
=
i
T
)
,
and the set of mass probabilities be
π
=
{
π
i
}
. Of course,
#
π
=2
T
.
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Let
n
i
denote the sample count of outcome
i
. The kernel of the log
likelihood is
L
(
π
)=
±
i
n
i
log
π
i
.
To test marginal homogeneity, the likelihood-ratio test uses
−
2[
L
(ˆ
π
MH
)
−
L
(
p
)] = 2
±
i
n
i
log(
p
i
/
ˆ
π
MH
i
)
∼
χ
2
df
,
where
df
=
T
−
1, ˆ
π
MH
and
p
are the ML estimators under and
without MH assumption, respectively.
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11.1.2 Crossover drug comparison example
Table 11.1 comes from a crossover study in which each subject
used each of three drugs for treatment of a chronic condition at
three times (±rst drug A, then B and C). Responses are binary
(favorable or unfavorable), 2
3
= 8 di²erent outcomes in total.
TABLE 11.1
Responses to Three Drugs in a Crossover Study
Drug A Favorable
Drug A Unfavorable
B Favorable
B Unfavorable
B Favorable
B Unfavorable
C Favorable
6
2
2
6
C Unfavorable
16
4
4
6
Ž.
Source:
Reprinted with permission from the Biometric Society Grizzle et al. 1969 .
The sample proportion favorable was (0
.
61
,
0
.
61
,
0
.
35) for drugs A,
B, C. The likelihood-ratio statistic for testing marginal
homogeneity is 5
.
95 (
df
=2)
.
For more analysis details, see the book.
7/32
11.1.3 Modeling margins of a multicategory response
Multinomial response
. With baseline-category logits for I outcome
categories, the saturated model is
log[
P
(
Y
t
=
j
)
/
P
(
Y
t
=
I
)] =
β
ij
,
t
, ...,
T
,
j
, ...,
I
−
1
.