RE: Help: Non-positive semi-definite message
From: "KOWALSKI, KENNETH G. [R&D/1825]" <kenneth.g.kowalski@pharmacia.com>
Subject: RE: Help: Non-positive semi-definite message
Date: Mon, 20 Aug 2001 18:01:05 -0500
Alan,
The trick I suggested when a correlation goes to unity does reduce the
dimensionality by a parameter. In the unrestricted case you are estimating
a variance component for both etas (e.g., V and CL) as well as the
covariance between the etas (the off-diagonal element in OMEGA). Thus, in
the case of two etas you are estimating 3 parameters. In the example I gave
where an eta is shared, then you only have 2 parameters being estimated as
the correlation is restricted to unity.
If you are not using a full OMEGA to begin with you should look at Lew
Sheiner's recent message on this topic regarding diagonal OMEGAs with
variance components being estimated as zero. He comments on the problem of
having too many etas than is supported by the data and that you should look
at a full OMEGA as a diagnostic and see if any of the off-diagonal elements
have correlations approaching unity. I do this as a matter of good practice
regardless of whether a diagonal OMEGA gives me problems or not. I'm pretty
sure I went through an example illustrating this when I visited Cognigen a
few months ago.
Blocking of OMEGA is also a useful way to reduce the dimensionality by
restricting certain covariances to zero. Basically, all etas within a block
are assumed to be correlated and etas between blocks are uncorrelated. If
for example you have a one compartment model with tlag, ka, V, and CL, a
full OMEGA would be specified as BLOCK(4) on the $OMEGA statement. If this
leads to an over-parameterized OMEGA that suggests that tlag and ka are
uncorrelated with V and CL it might be reasonable to reduce the
dimensionality by assuming tlag and ka are in one block and V and CL are in
a second block with code something like
$OMEGA BLOCK(2)
0.04; Var(tlag)
0.01; Cov(tlag, ka)
0.04; Var(ka)
$OMEGA BLOCK(2)
0.04; Var(V)
0.01; Cov(V, CL)
0.04; Var(CL)
Note in the full BLOCK(4) OMEGA there are 10 parameters to be estimated (4
diagonal elements and 6 off-diagnonal elements). In the above example where
the covariances between the random effects for tlag and ka are uncorrelated
with those for V and CL there are only 6 parameters to be estimated.
Furthermore, if the BLOCK(4) OMEGA yields off-diagonal correlations near
unity then you can use the trick I suggest with the shared etas to further
reduce the dimensionality. Another way to reduce the dimensionality in
OMEGA is the use of band diagonal matrices. Diane Mould discussed this on
the NONMEM network a couple of months ago. I think blocking, shared etas
and banding give us quite a bit of flexibility in reducing the
dimensionality of OMEGA to combat the problems with over-parameterized
OMEGAs which can lead to those nasty non-positive semi-definite messages.
Regards,
Ken