Chapter 11. Analyzing Repeated Categorical
Response Data
Deyuan Li
School of Management, Fudan University
Feb. 28, 2011
1/32
Outline
•
11.1: Comparing marginal distributions: multiple responses;
•
11.2: Marginal modeling: maximum likelihood approach;
•
11.5: Markov chains: transitional modeling.
2/32
Repeated categorical response data:
(1) response variable for each subject is measured repeatedly (at
several times or under various conditions); dependence among the
repeated responses.
For example, blood measures of one person at several times.
(2) responses of subjects in a set or cluster; dependence in the set
or cluster.
For example, survival response for each fetus in a litter for a
sample of pregnant mice.
3/32
11.1 Comparing marginal distributions: multiple responses
Usually, the multivariate dependence among repeated responses is
of less interest than their marginal distributions. For instance, in
treating a chronic condition with some treatment, the primary goal
might be to study whether the probability of success increases over
the
T
weeks of a treatment period.
In Sections 10.2.1 and 10.3 we compared marginal distributions for
matched pairs (
T
= 2) using models that apply directly to the
marginal distributions. In this section we extend this approach to
T
>
2.
4/32
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document11.1.1 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
.
5/32
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 likelihoodratio 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.
6/32
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 likelihoodratio 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 baselinecategory 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
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 DeyuanLi
 Markov Chains, Markov chain, Logit, Yt, marginal distributions

Click to edit the document details