This gives the initial estimates of the diagonal elements of a diagonal block of OMEGA, e.g.,

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$OMEGA [DIAGONAL(n)] [ v11 v22 v33 ... vnn ]

;# Example:

;# Initial estimate of variance of eta(1) = .04
;# Initial estimate of variance of eta(2) = .12
  $OMEGA .04 .12

Each initial estimate may optionally be coded with one of the forms:

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  init options      (init options)        (options init)

NONMEM v7.3+ supports (value,value...)xn for repeated entries of $OMEGA may be entered easily. Any initial value or group of initial values may be enclosed in parentheses and followed by xn to indicate n replications of the values.

The following options apply only to a single initial estimate (i.e., a single 1x1 block) and must follow the initial estimate unless within parentheses.

• FIXED: the variance is fixed thus not a parameter to be estimated. When FIXED appears anywhere, then the block is described by NONMEM as consisting of separate blocks, each of dimension one.
• UNINT (NM75): used during the Optimal Design Step to identify an ETA as uninteresting. UNINT may be used anywhere that FIXED may be used.
• VARIANCE (default): the initial estimate is a variance of the ETA.
• STANDARD (with alias SD): the initial estimate is the standard deviation of the ETA. An initial estimate may be 0 only with VARIANCE + FIXED and SD + FIXED format.

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$OMEGA BLOCK(n) [ v11 v21 v22 v31 v32 v33 ... vn1 vn2 ... vnn ]

;# example
$OMEGA BLOCK(2) .04 .002 .12

Specifies a full block of \(\Omega\). The example gives initial estimates \(\text{var}(\eta_1)=0.04\), \(\text{cov}(\eta_1,\eta_2) = .002\), and \(\text{var}(\eta_2)=0.12\).

Any initial value or group of initial values may be enclosed in parentheses and followed by "xn", which means to replicate the values within parentheses n times ("repeated value").

The following options apply to the entire block and may appear anywhere among the list of initial estimates:

• FIXED: indicates that the entire block is constrained to be fixed to its initial estimate.
• UNINT (NM75): used during the Optimal Design Step to identify an \(\eta\) as uninteresting. UNINT may be used anywhere that FIXED may be used.
• VARIANCE (default): same as in DIAGONAL.
• STANDARD: same as in DIAGONAL.
• COVARIANCE (default): all initial estmates given for off-diagonal elements are for covariances of ETAs.
• CORRELATON: all initial estmates given for off-diagonal elements are for correlations of ETAs.
• CHOLESKY: the block is the lower triangular half of the Cholesky decomposition of \(\Omega\).

Internally NONMEM converts all initial estimates to variance and covariances, as shown in the NONMEM report and in the raw and additional output files.

Examples: The following describe the same block (within rounding errors):

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  $OMEGA BLOCK(2) ;# or $OMEGA VARIANCE COVARIANCE BLOCK(2)
  0.64
  -0.24 0.58

  $OMEGA STANDARD BLOCK(2)
  0.8
  -0.24 0.762

  $OMEGA STANDARD CORRELATION BLOCK(2)
  0.8
  -0.394 0.762

  $OMEGA VARIANCE CORRELATION BLOCK(2)
  0.64
  -0.394 0.58

  $OMEGA CHOLESKY BLOCK(2)
  0.8
  -0.3 0.7

The (entire) initial estimate of the block must be positive definite. The only exception is when the entire initial estimate of the block is 0, in which case it must be fixed to this estimate. Initial estimates of some of the elements of the block may be 0, while initial estimates of some other elements may be nonzero, but only in the case where the block is constrained to be of band symmetric form. That is, given the diagonal and a group of contiguous subdiagonals symmetrically ocurring across the diagonal, the elements off both the diagonal and the subdiagonals are constrained to be zero. To specify the initial estimates of such a block, the initial estimates of those elements that are to be constrained to 0 should be given as 0, while all other initial estimates should be given as nonzero. E.g., with these structures for $OMEGA BLOCK(3), the 0's are preserved:

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  x
  0x
  00x

  x
  xx
  0xx

With NONMEM 7.3 if the initial estimate of a block is not positive definite because of rounding errors, a value will be added to the diagonal elements to make it positive definite. A message in the NONMEM report file will indicate that this was done. E.g.,

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 DIAGONAL SHIFT OF 1.1000E-03 WAS IMPOSED TO ENSURE POSITIVE DEFINITENESS

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  $OMEGA BLOCK(n) SAME(m)

This describes a block whose initial estimates, as well as final estimates, are constrained to be equal to those of the preceding block. Values may not be given. (n) may be omitted.

NONMEM v7.3+ allows (m) so that the record is equivalent to m identical records, e.g.,

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  $OMEGA BLOCK(2) SAME(3)

is equivalent to

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  $OMEGA BLOCK(2) SAME
  $OMEGA BLOCK(2) SAME
  $OMEGA BLOCK(2) SAME

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  $OMEGA BLOCK(n) VALUES(diag,odiag)

This supplies initial values for a block such that the initial estimates of the diagonal elements are all the same, specified by "diag", and the initial estimates of the off-diagonal elements are all the same, specified by "odiag". If present, VALUES must follow BLOCK. Other options (such as FIXED, CHOLESKY, VARIANCE,STANDARD,COVARIANCE,CORRELATON, UNINT) may follow VALUES or be placed between BLOCK and VALUES. E.g.,

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  $OMEGA BLOCK(6) VALUES(0.1,0.01)
  is the same as
  $OMEGA BLOCK(6)
  0.1
  0.01 0.1
  (0.01)x2 0.1
  (0.01)x3 0.1
  (0.01)x4 0.1
  (0.01)x5 0.1

For fixed block (such as for omega priors):

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  $OMEGA BLOCK(6) FIX VALUES(0.15,0.0)

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  $OMEGA label=value (NM75)

The symbolic label substitution feature is new with NONMEM 7.5. This is a compact method of defining an ETA (an element of OMEGA) specifying its initial estimate, and specifying a label for the index for this element of OMEGA. The label may be used as a index for ETA in abbreviated code, and will also identify this element of OMEGA in the NONMEM output. If new $OMEGA records change the ordering, the abbreviated code does not have to be changed. For example, suppose the first element of OMEGA that is defined happens to be

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  $OMEGA ECL=.4

The NONMEM report will describe the relationship, e.g.,

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  LABELS FOR ETAS
  ETA(1)=ETA(ECL)

and ETA(CL) rather than ETA1 will appear in the NONMEM report. The abbreviated code can use this symbolic index instead of the index. Then, these take effect on both ETA's and MU_'s.

For example, suppose the following code is present for the first elements of THETA and ETA. Note that $OMEGA and $THETA records must be placed ahead of any records that use the symbolic label.

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$THETA CL=(0.0,7.0)
$OMEGA ECL= 0.3
$PK
MU_ECL=THETA(CL)
CL=EXP(MU_ECL+ETA(ECL))

This is equivalent to

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$THETA (0.0,7.0)
$OMEGA .3
$PK
MU_1+THETA(1)
CL=EXP(MU_1+ETA(1))

Another example defines symbolic labels for a block of OMEGA:

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$OMEGA BLOCK(4)
ECL=  0.3
EV1=  0.01 0.35
EQ=   0.01 0.01 0.54
EV2=  0.01 0.01 0.01 0.67

Or, for diagonals,

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$OMEGA
ECL= 0.3
EV1= 0.35
EQ=  0.54
EV2= 0.67

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  $OMEGA BLOCK(n) NAMES  (label1,...,labeln)  VALUES  (odiag,diag)

Symbolic label substitution may be specified for an entire block using the NAMES option. This is a compact way of defining one or more ETAs with labels and, when combined with VALUES, with initial values. For example

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$OMEGA BLOCK(4) NAMES(ECL,EV1,EQ,EV2) VALUES(0.03,0.01)

This is equivalent to

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$OMEGA BLOCK(4)
ECL=  0.03
EV1=  0.01 0.03
EQ=   0.01 0.01 0.03
EV2=  0.01 0.01 0.01 0.03

If both are present, VALUES() must come after NAMES().

In NM73+, if all diagonal elements of \(\Omega\) are "1.0E+06 FIXED", then NONMEM outputs

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ANALYSIS TYPE: POPULATION WITH UNCONSTRAINED ETAS

excludes the population density component of the total likelihood is not included. This allows a population data set to be analyzed as if the data from each individual were single-subject data. Furthermore, some theta parameters could be shared across subjects ("pooled fit parameters"), whereas ETAs are free to fit each individual without any population constraint. Parallelization is possible.

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$OMEGA  SCALE(scalevalue)

One may have your OMEGA or SIGMA values multiplied by a SCALE value. The SCALE value takes into affect on any OMEGA or SIGMA values that come after it in the record:

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$OMEGA 0.5 0.7 SCALE(2.0) 0.8 0.9 SCALE(1.5) 0.1 0.2

is equivalent to

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$OMEGA 0.5 0.7 1.6 1.8 0.15 0.3

Another example:

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  $OMEGAP SCALE(0.7) BLOCK(4) FIX VALUES(0.5,0.01)

is equivalent to:

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  $OMEGAP BLOCK(4) FIX VALUES(0.35,0.07)

Another example:

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  $OMEGAP SCALE(0.7) BLOCK(4) FIX
  0.5
  0.01 0.5
  SCALE(2.5)
  0.02 0.02 0.003

is equivalent to

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  $OMEGAP SCALE(0.7) BLOCK(4) FIX
  0.5
  0.01 0.5
  0.05 0.05 0.0075

The SCALE is most useful when it is desired to scale an informed prior OMEGA from a previous analysis that differs from NONMEMs normal treatment. For example, you may wish to scale the omega prior by (df-p-1)/df, where df=degrees of freedom, and p=dimension of omega block. Rather than always re-calculate by hand, you can change the scale on the fly.