Page 442 - Contributed Paper Session (CPS) - Volume 2

CPS1917 Trijya S.
3. Result
Result In order to compare the performance of the proposed method to
the one proposed by Shah (1973) we have used the data reported in his paper.
The model being considered is as follows. After administering a dose of the
(10)
drug, its concentrations ( ), ( )..., ( ) are measured by taking a sample

1
0
from the central compartment at times , , . . . , . For this sample, the model
1

0
could be written as
( ) = − + − + ,

where = 0, 1, . . . , . We assume that ’s are independently distributed

with ( ) = 0 and ( ) = for all . The data set in Shah (1973) can
2

be described as follows. A panel of 8 subjects was administered 250 milligrams
(mg) of the drug Tetracycline Hyrdochloride after a specified breakfast. The
concentration of the drug in blood stream was measured at different points
of time. The time and the lowest concentration for all time points are given in
the first two columns of Table 1. Using the last six points of the elimination
phase, we obtain the decay function as
() = 0.2793 −0.1331(−15) , ≥ 15.
Therefore, = 0.2793, = 0.1331 and = = 14.1349. Using the
̂
̂
̂

methodology given in the previous section, the estimates of the first four raw
moments are given by = 8.1648, = 123.4072, = 2782.1156 and
2
1
3
= 83634.2720. The quadratic equation (2) for this data set turns out to be
4
59.5243 − 481.3506 + 257.2676 = 0 . The roots of this equation are
2 ̂
̂
= 7.5112 and = 0.5754 since the smaller root is taken to be .
̂
̂
̂
1
2
1
The first approach (Method 1) used to estimate A, B, and in (10) utilizes
only the first three moments and can be described as follows. Given the
1
̂
̂
estimates and , the estimates of and are obtained as, ̂ = ̂ and
2
1
1
̂
= 1 . We know that is the coefficient of the first term of (9). Therefore,
̂
2
−1

is the estimate of ( + ) . But + = so given the estimate we can
̂

find the estimate of as and the estimate of A as = ̂ . Using the
̂
̂
̂

estimate of , we can obtain the estimate of as (1 − ), and hence the
̂

estimate of B is given by = (1 − ).
̂
̂
̂
The second approach (Method 2) uses a two stage procedure of estimation
which also utilizes the first four moments. In the first stage we estimate
̂
1
and using the cubic equation (8), which, in this case is given by
̂
2
− 60.7012 + 31.3279 = 0 . Since 4 + 27 = −867125.8 < 0 , all
3
2
3
1
roots are real. Two roots are given by estimate and are obtained as
̂
̂
2
1
0.4835 and 7.5190 which are then used to obtain estimates ̂ and . In the
̂
second stage, given the estimates ̂ and we transform the data as follows.
̂
We define () = (−̂) and () = (− ). Using
̂
1
2
{( ), ( ), ( )}, = 0, 1, . . . , , we regress ( ) on ( ) and

1

2
1
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