OMEGA HAS A NONZERO BLOCK
From:"Bonate, Peter"
Subject:[NMusers] OMEGA HAS A NONZERO BLOCK
Date:Fri, 4 Oct 2002 13:13:20 -0500
Dear All,
I have been following this discussion with great interest and thought I would share with the
group the results of some simple simulations that I have done. In the first simulation I
simulated data from 125 individuals with intense serial sampling after single dose administration.
Concentration-time data were simulated using a 1-compartment model with oral
administration. Clearance, volume of distribution, and Ka had typical values of 3.7 L/h,
227 L, and 0.7 per hour, respectively. All parameters were modeled as log-normal. Omega was
3 x 3 with values
CL V Ka
0.1
0.16 0.3
0.00 0.0 0.1
Residual variability was proportional having a variability of 0.003. Thus the correlation
between CL and V was 0.92. The model was then fit using FOCE having the true values as the
initial estimates. The model minimized with no errors and had an OFV of 13409. The final
parameter estimates (standard errors) was 3.65 (0.104) L/h, 220 (11.0) L, and 0.698
(0.0215) per hour for CL, V, and Ka, respectively. Omega was estimated at
CL V Ka
0.0893
0.142 0.264
0.00 0.000 0.101
with a residual variance of 0.00422. The model fitted correlation between CL and V was 0.92.
The largest and smallest eigenvalues of this model were 3.18 and 0.00417, respectively, with a
condition number of 763. Hence, the model was unstable, as expected.
I then refit the model using the trick we are all talking about. V was modeled as
Theta(2)*exp(eta(1) *theta(4)), where theta(4) is the ratio of the standard deviations. This model
was refit, had no errors, and had an OFV of 14005. Hence, the new model had an increase in OFV of 596!!
The new parameter estimates were 4.32 (0.186) L/h, 295 (22.8) L, and 0.636 (0.0223) per hour for CL, V,
and Ka, respectively. Omega was estimated at 0.109 for CL and 0.112 for Ka with a residual variance
of 0.0123. Theta(4) was 1.72 (0.146). The true ratio of the SDs was 1.73. So, reparameterization resulted in
a better estimate of the variance of both CL and V, with essentially no change in Ka. But, although
theta(4) was accurately estimated, CL, V, and Ka had greater bias and larger standard errors under the
new model. However, this model was more stable having a condition number of 3.33/0.0325 = 102.
The second simulation built on the first simulation where a PD marker was measured. The marker was
simulated having an Emax model with parameters Emax = 100% and EC50 = 25 ng/mL. Between-subject variability
was modeled as a normal distribution. Omega was 2 x 2 with values
Emax EC50
100
17 3
Hence, Emax and EC50 had correlation 0.98. Residual error was modeled as a normal distribution with
variance 10. The PD model was then fit using FOCE having the true values as the initial estimates.
The model minimized with no errors and had an OFV of 6494. The final parameter estimates (standard
errors) was 99.7 (1.06) for Emax and 25.2 (0.531) for EC50.
Omega was estimated at
Emax EC50
95.0
42.3 20.6
with residual variance estimated at 9.75. The condition number of the model was 2.89/0.00273 = 1059,
indicating the model was unstable.
The model was then parameterized as EC50 = THETA(2) + THETA(3)*ETA(1) and refit. Minimization was
successful with an OFV of 6516, an increase of 22. The final parameter estimates were 99.5 (1.10)
for Emax and 25.2 (0.538) for EC50. The variance of CL was estimated at 87.2 with a residual
variance of 10.2. Theta(3) was estimated at 0.366. The theoretical value was 0.173 (IS THIS
RIGHT, KEN?) This time the new model had no change in the estimates of Emax and EC50, but the
variance components had worse accuracy than the original model.
I then tried another reformulation of the model to EC50 = THETA(1)*THETA(2) + ETA(1) and refit.
This time the OFV was 7313.5, an increase of 820. The estimates of Emax were 98.9 (1.30), 0.252
(0.0435) for Theta(2), and hence 24.9 for EC50. The variances were totally off, however. The
variance of CL was estimated at 35.3 with a residual variance of 16.3.
As I interpret this, you may be better parameter estimates when two random effects are correlated
but the estimates are unstable and become data dependent. By using the shared covariance term
("the trick") there is no guarantee the OFV will be near the current, highly correlated model.
You also may get slightly more biased estimates with greater imprecision, but the model becomes
more stable and less data dependent. The trick seems to works for however the model is
parameterized, although if one random effect was log-normal and one was normal, I am not
sure what the new theta would represent.
Anyway just some more thoughts on the subject,
pete
Peter L. Bonate, PhD
Director, Pharmacokinetics
ILEX Oncology, Inc
4545 Horizon Hill Blvd
San Antonio, TX 78229
phone: 210-949-8662
fax: 210-949-8487
email: pbonate@ilexonc.com