Appendix I
Derivation of transition matrix, transition probabilities,
and the likelihood function for the disease natural history model
The five

state model is
depicted
as follows:
12
23
34
4
5
N
DR
BDR PPDR PDR
Blindness
(State 1) (State 2) (State 3) (State 4) (State 5)
Let
ij
represent the instantaneous progression rate from state
i
to s
tate
j
. As in traditional stochastic processes, the movement between states
i
s
denoted by the following intensity matrix
:
0
0
0
0
0

0
0
0
0
0
0
0
0

0
0
0
0

5
4
3
2
1
state
Previous
Q
5
4
3
2
1
state
Current
45
45
34
34
23
23
12
12
Let d
1
, d
2
, d
3
, d
4
and d
5
(d
1
=0,
12
2
d
,
23
3
d
,
34
4
d
,
45
5
d
) be the
eigenvalues of Q. The right eigenvector corresponding to d is denoted as A
with 5
5 matrix. Thus
1
ADA
Q
(A

1)
where D=diag(d
1
, d
2
, d
3
, d
4
, d
5
).
It shou
ld be noted that due to the Markov property the inverse of
λ
23
,
λ
34
,
λ
45
gives
average dwelling times
staying at BDR, PPDR, and PDR,
respectively.
Assuming Q follows a time

homogeneous Markov model, the transition
probabilities following
Kalbfleisch
and Lawless method
1
are given by
1
5
4
3
2
1
,
,
,
,
A
e
e
e
e
e
Ad i a g
t
P
t
d
t
d
t
d
t
d
t
d
(A

2)
subject to P(0)=1. The matrix of transition probabilities is denoted as follows:
1
0
0
0
0
0
0
0
0
0
0
5
4
3
2
1
state
Previous
5
4
3
2
1
state
Current
45
44
35
34
33
25
24
23
22
15
14
13
12
11
t
P
t
P
t
P
t
P
t
P
t
P
t
P
t
P
t
P
t
P
t
P
t
P
t
P
t
P
For calculation of transition probabilities, see Chen
et al
2
.
1
Kalbfleisch D, Lawless JF. The an
alysis of panel data under a Markov assumption.
J Am Stat
Assoc
1985;
80
: 863

871.
2
Chen THH, Kuo HS, Yen MF, et al. Estimation of sojourn time in chronic disease screening
without data on interval cases.
Biometrics
2000;
56
: 167

172
.
Transition probabilities in the above represent the probability of
progressing from one state to another state. For example, the risk of transition
from DM without DR to blindness during a five

year period is denoted by P
15
(5).
Given these tra
nsition probabilities, one can develop the likelihood function
based on the above retrospective cohort.
Let n
1
, n
2
, n
3
and n
4
denote the number of NDR, BDR, PPDR and PDR,
respectively. The likelihood function given this information
i
s:
4
3
1
2
1
2
3
4
)
(
1
)
(
)
(
1
)
(
)
(
1
)
(
)
(
1
)
(
15
14
1
15
13
15
12
1
1
1
15
11
c
c
n
a
c
c
b
b
n
b
n
c
n
d
a
a
m
P
m
P
m
P
m
P
m
P
m
P
m
P
m
P
(A

3)
where m
a
, m
b
, m
c
and m
d
represents age at first examination, and
i
is index
variable for individual in state i.
It should be noted that the probabilities in
the parenthesis of the above
expression (A

3)
are
conditional probabilities because there
is
no possibility of
receiving fundus examination if subjects ha
ve
suffered from blindness. This
means that subjects with blindness at baseline
a
re truncated from our
retrospective cohort.
Appendix II
Derivation of transition matrix, transition probabilities, and
the likelihood function for the intervention model
In clinical reality, patients may regress to DM without DR after clinical control or
treatment. The e
volution of DR may be delineated as follows:
λ
c
21
λ
c
32
N
DR
BDR
PPDR
PDR
Blindness
λ
c
12
λ
c
23
λ
c
34
λ
c
45
Similarly, the tra
nsition model under clinical control or treatment
i
s
denoted by the following matrix :
0
0
0
0
0

0
0
0
0
0
0
0

0
0
0

5
4
3
2
1
state
Previous
Q
5
4
3
2
1
state
Current
45
45
34
34
32
32
23
23
21
21
12
12
C
C
C
C
C
C
C
C
C
C
C
C
The likelihood function of this model can be derived in the same way as
above.
Suppose successive transitions following initial state Y
i
(t) (Y(
t)=1,2,3,4 or
5) at first examination
is
observed intermittently at times
i
i
i
m
m
i
m
i
i
i
y
y
y
n
i
t
t
t
t
i
1
i
0
i
.
.
1
,
0
,
,...
,
are
times
at these
observed
states
the
and
individual
,......,
1
,
,
1
The likelihood function for this part
is
:
i
d
i
d
i
m
d
d
i
d
i
y
y
n
i
t
t
P
1
1
.
.
.
1
))
(
(
1
(A

4)
Note that
a
similar model w
as
also developed while
r
ight

censoring due to
death was
taken into account.
Also note that as seen in the equation (A

4), our
model can
accommodate
irregular intervals because individual intervals were
modeled
to estimate transition rates.
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