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
NDR
23
34
BDR
PPDR
(State 2)
(State 3)
45
PDR
Blindness
(State 1)
(State 4)
(State 5)
Let ij represent the instantaneous progression rate from state i to state
j . As in traditional stochastic processes, the movement between states is
denoted by the following intensity matrix :
1
1  - 12

2 0
Previous 
Q
3 0

state
4 0
5  0
Current state
2
3
12
-  23
0
0
0
0
23
 34
0
0
4
5
0
0





45 
0 
34
- 45
0
0
0
0
Let d1, d2, d3, d4 and d5 (d1=0, d 2  12 , d 3   23 , d 4   34 , d 5  45 ) be the
eigenvalues of Q. The right eigenvector corresponding to d is denoted as A
with 55 matrix. Thus
Q  A D A1
(A -1)
where D=diag(d1, d2, d3, d4, d5).
It should be noted that due to the Markov property the inverse of λ23, λ34,
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 method1 are given by


Pt   A d i a egd1t , e d 2t , e d3t , e d 4t , e d5t A 1
(A - 2)
subject to P(0)=1. The matrix of transition probabilities is denoted as follows:
Current state
1
2
3
1  P11 t  P12 t  P13 t 

2 0
P22 t  P23 t 
Previous 
3 0
0
P33 t 

state
4 0
0
0

5 0
0
0
4
5
P14 t  P15 t  

P24 t  P25 t 
P34 t  P35 t 

P44 t  P45 t 

0
1 
For calculation of transition probabilities, see Chen et al2.
1
Kalbfleisch D, Lawless JF. The analysis 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 P15(5).
Given these transition probabilities, one can develop the likelihood function
based on the above retrospective cohort.
Let n1, n2, n3 and n4 denote the number of NDR, BDR, PPDR and PDR,
respectively. The likelihood function given this information is:
1
2
3
 P11 (ma ) 
 P (m ) 
 P (m ) 
 P (m ) 

   12 b    13 c    14 c 

a 1 b 1 c 1 d 1  1  P15 ( m a ) 
 1  P15 (mb ) 
 1  P15 (mc ) 
 1  P15 (mc ) 
n1
n2
n3
n4
4
(A -3)
where ma, mb, mc and md 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 have suffered from blindness. This
means that subjects with blindness at baseline are 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 evolution of DR may be delineated as follows:
λc21
λc32
NDR
BDR
PPDR
PDR
Blindness
λc12
λc23
λc34
λc45
Similarly, the transition model under clinical control or treatment is
denoted by the following matrix :
1
1  - C 12

2  C 21
Previous 
Q
3
 0
state
4 0

5 0
Current state
2
3
C 12
0
C
C
C
-  21   23 
 23
C 32
 C 32  C 34 
0
0
0
0
4
5
0
0





C 45 

0 
C 34
- C 45
0
0
0
0
The likelihood function of this model can be derived in the same way as
above.
Suppose successive transitions following initial state Yi(t) (Y(t)=1,2,3,4 or
5) at first examination is observed intermittently at times
ti ,0  ti ,1 ,    ti.mi 1  ti.mi , i  1,......, n individual
and the states observed at these times are
y i 0 , y i1 ,... y i mi
The likelihood function for this part is:
n
mi
i 1
d 1
  (P
y i d 1 . y i d
(t i.d  t i.d 1 ))
(A -4)
Note that a similar model was also developed while right-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.