From stuart Mon Feb 5 10:06:36 1996
Subject: How to fix elements of the OMEGA and SIGMA matrices

This is an exposition stimulated by an inquiry from user Vladimir Piotrovskij, who wants to fix the typical value and interindividual variability of Vd (because he only has steady-state trough levels, but has reasonable estimates of these parameters from another source), but not fix the typical value and interindividual variability of Cl. Moreover, he also has a reasonable estimate of what appears to be a significant covariability between CL and V, and wants to fix this covariability.

In the above, and in what follows, I may refer to the variability in PK parameters, or to the covariability between PK parameters, but what I really mean is the variability in, and covariability between, the ETA's corresponding to the PK parameters.

Most users figure out that the OMEGA (and SIGMA) matrix can be regarded as a block diagonal matrix, and that the elements of any block can be fixed *altogether* to their intial estimates during the Estimation Step. E.g. When a OMEGA matrix is specified to be (fully) diagonal, each block is actually a 1x1 matrix; these little matrices are arranged along the diagonal of OMEGA, and any one or more of these matrices (each of which are really just single numbers) can be fixed. An example might be

.04 .0 .0
.0 .06 .0
.0 .0 .01

where the values shown are the initial estimates, and the variance element .06 is to be fixed. We can specify this with the \$OMEGA record

\$OMEGA .04 .06 FIXED .01

Here is an example where OMEGA consists of two blocks, the first of which is a 2x2 matrix and (its elements are to be) fixed, and the second of which is a 1x1 matrix.

.04 .01 .0
.01 .06 .0
.0 .0 .01

We might specify this example as follows.

\$OMEGA BLOCK(2) .04 .01 .06 FIX
\$OMEGA BLOCK(1) .01

Note that only the elements of the lower triangular part of the first block, i.e.

.04
.01 .06

are listed in the first \$OMEGA record, since a diagonal block matrix is always taken to be itself a diagonal matrix. However, *all* 4 elements of the first block are being fixed.

The covariance elements of OMEGA occuring outside the blocks (i.e. "off the block diagonal") are fixed to 0 during the Estimation Step.

What if we want to fix the covariance elements of a given block, and one or more of its variance elements, but not fix all the elements of the block altogether? If we want to fix *all* the covariance elements to 0, all we need do is represent the block as a sequence of 1x1 diagonal blocks, and fix each individual 1x1 block as we please. The nontrivial question, and that represented by Vladimir's inquiry, is:

What if we want to fix the covariance elements of a given block to *some nonzero values*, and one or more of its variance elements, but not fix the elements of the block altogether?

OM11 OM12
OM21 OM22

where OM11 (corresponding to CL) is not fixed, but where OM22 (corresponding to Vd) and OM12 (=OM21) are fixed.

The answer to the question is that this cannot be done! BUT perhaps the spirit of what is being asked can be accomplished. We first need to recall that any variance-covariance block gives rise to a correlation matrix wherein the variance information is ignored, and each covariance element is replaced by a corresponding correlation element. E.g. the above 2x2 block has correlation form

1.0 R12
R21 1.0

where R12 (=R21) is OM12/SQRT(OM11*OM22). Every diagonal element of a correlation martix is always 1 (a random variable is always perfectly correlated with itself). Then we need to rephrase the question:

What if we want to fix the *correlation* elements of a given block to some nonzero values, and one or more of its variance elements, but not fix the elements of the block altogether?

In other words, and in terms of the above example, how can we fix the variance on Vd and the *correlation* between Vd and Cl? We would want to fix the correlation to that number given by the above formula for R12, *using an initial estimate of OM11*. Since, though, this is only an initial estimate, and we are actually somewhat unsure about the final estimate, the value for R12 may not be ideal. However, in this case, it is also unlikely that our estimate for OM12 itself is ideal.

To give the answer, first imagine the specification of the PK parameters in the \$PK block, using ETA variables, as we would normally do this. This could look like:

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

Then construct a modified control stream wherein the correlation matrix (from the 2x2 block) is specified with an \$OMEGA record, using the FIX option, and new theta elements are introduced to *represent* the standard deviations of the ETA's shown above (i.e. to represent the square roots of the variances of these ETA's), fixing these elements (on an individual-basis) as we please.

Take our example, where we wish to fix R12 to .6 and the variance of Vd to .04=(.2**2). The control stream will entail:

CL=THETA(1)*EXP(THETA(3)*ETA(1))
V =THETA(2)*EXP(THETA(4)*ETA(2))
\$OMEGA BLOCK(2) 1 .6 1
\$THETA x x ;numbers to be filled in
\$THETA .2 (.2 FIX)

Note that *all* the elements of the correlation matrix are being fixed. In particular, the diagonal elements are being fixed to 1 so that the correlation form of the matrix is preserved. But the correlation value .6 is also being fixed. In contrast, only the standard deviation of Vd is being fixed; the standard deviation of CL is not being fixed. Note too that the ETA's shown in the modified control stream are formally different from the ones shown in the original control stream; their variances are fixed to1, so that the variances of the original ETA's are given by THETA(3)**2 and THETA(4)**2.

One may or may not also constrain the standard deviations to be nonnegative (e.g. (0,2) (0,2 FIXED)); it doesn't matter. Not only will the OMEGA block of the new ETA's remain positive semidefinite during the Estimation Step (NONMEM always implements an implicit constraint keeping an OMEGA block positive semidefinite), but the OMEGA block that would correspond to that of the original ETA's will also always be positive semidefinite. In particular, the variance of the original ETA(1) will be THETA(1)**2, which must always be nonnegative. The full OMEGA block that would correspond to that of the original ETA's is given by

OM11 OM12
OM21 OM22

where

OM11=(THETA(3)**2)*om11, OM22=(THETA(4)**2)*om22,
OM12=THETA(3)*THETA(4)*om12, and where in these expressions
om11 om12
om21 om22

is the OMEGA block of the new ETA's.

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