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Nov 30, 2013 (3 years and 6 months ago)

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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.