Subject: One ETA...for which parameter?
Date: Tue, 8 May 2001 01:59:52 -0400

Sometimes, having more than one ETA in the model makes the model over parameterized, given the data. Placing this ETA with different parameters produces a set of different parameter estimates. How can we interpret those parameters? Which set is true? What is the meaning of this "lump sum" interindividual variability parameter (ETA)?

Regards,
University of Georgia

*****

From: "Gibiansky, Leonid" <gibianskyl@globomax.com>
Subject: RE: One ETA...for which parameter?
Date: Tue, 8 May 2001 08:06:02 -0400

To fix the question, consider three situations:

CL=THETA(1)*EXP(ETA(1)) (1)
V=THETA(2)*EXP(ETA(2))

CL=THETA(1)*ETA(1)
V=THETA(2)*EXP(THETA(3)*ETA(1)) (2)

CL=THETA(1)*ETA(1) (3)
V=THETA(2)*EXP(ETA(1))

The first corresponds to independent variations of CL and V, with K=CL/V= [THETA(1)/THETA(2)] *EXP(ETA(1)-THETA(3)*ETA(2)) (1a)
Here individual clearance and volume are independent of each other.

If one observes (using ETA1 vs. ETA2 plot or block structure of the OMEGA matrix) that ETA1 and ETA2 are actually strongly correlated, one may try the second variant in which case

K=CL/V=[THETA(1)/THETA(2)] *EXP( [1-THETA(3)]*ETA(1)) (2a)
Here patients with higher CL should have higher V, according to the model, K varies together with CL and V, all with different variances.

Finally, if variances of CL and V are equal, and they are strongly correlated, K may be constant (variant 3):

K=CL/V=THETA(1)/THETA(2) (3a)
Here the patients with higher CL should have higher V, moreover, the CL/V ratio is constant.

These variants correspond to different models (actually (2) may degenerate into (3) when THETA(3)=1)

Hope this helps
Leonid