+--------------------------------------------------------------------+
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| $SIGMA |
| |
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MEANING: Supplies initial estimates for the NONMEM SIGMA Matrix
CONTEXT: NM-TRAN Control Record
USAGE:
$SIGMA [DIAGONAL(n) | BLOCK(n) | BLOCK(n) SAME(m) | BLOCK SAME(m)]
[[value1] [value2] [value3] ...
[(value,value...) xn]
[BLOCK(n) VALUES(diag,odiag)]
[label=value] ...
[BLOCK(n) [NAMES (label1,...,labeln)] [VALUES (diag,odiag)]
[FIXED] [UNINT]
[VARIANCE|STANDARD] [COVARIANCE|CORRELATON] [CHOLESKY]
SAMPLE:
$SIGMA BLOCK(3) 6. .005 .3 .0002 .006 .4
DISCUSSION:
Gives initial estimates and constraints for elements of one or several
blocks of the SIGMA matrix, i.e., the matrix of variances and covari-
ances of the eps variables in the statistical model. This record
should appear only if the statistical model contains eps variables.
Multiple $SIGMA records may be used to define multiple blocks of
SIGMA. The order of the appearance of all blocks over all records
corresponds to the order of the blocks in SIGMA.
If the initial estimates are omitted for any element(s) of SIGMA, then
NONMEM will try to obtain the initial estimates.
OPTIONS:
There are six forms:
1. $SIGMA [DIAGONAL(n)] [ v11 v22 v33 ... vnn ]
This gives the initial estimates of the diagonal elements of a
diagonal block of SIGMA. E.g.,
$SIGMA .04 .12
Initial estimate of variance of eps(1) = .04
Initial estimate of variance of eps(2) = .12
Each initial estimate may optionally be coded with one of the
forms:
init options (init options) (options init)
With NONMEM 7.3 (value,value...)xn is permitted, so that repeated
inputs of $SIGMA may be entered easily. Any initial value or
group of initial values may be enclosed in parentheses and fol-
lowed by "xn", which means to replicate the values within paren-
theses n times ("repeated value").
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.
Option FIXED indicates that the variance is to be constrained to
be fixed to the given initial estimate. (When FIXED appears any-
where, then the block is described by NONMEM as consisting of
separate blocks, each of dimension one.)
Option UNINT is used with NONMEM 7.5. UNINT is used during the
Optimal Design Step to identify an eps as uninteresting. UNINT
may be used anywhere that FIXED may be used.
Option VARIANCE indicates that the initial estimate is understood
to be a variance of the eps. This is the default.
Option STANDARD indicates that the initial estimate is understood
to be a standard deviation of the eps. May also be coded SD.
An initial estimate may be 0 only if the variance or standard
deviation is fixed to this estimate.
2. $SIGMA BLOCK(n) [ v11 v21 v22 v31 v32 v33 ... vn1 vn2 ... vnn ]
This gives the initial estimates of all the elements of a nondi-
agonal ("full") block of SIGMA. E.g.,
$SIGMA BLOCK(2) .04 .002 .12
Initial estimate of variance of eps(1) = .04
Initial estimate of covariance of eps(2), eps(1) = .002
Initial estimate of variance of eps(2) = .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.
Option UNINT is used with NONMEM 7.5. UNINT is used during the
Optimal Design Step to identify an eps as uninteresting. UNINT
may be used anywhere that FIXED may be used.
VARIANCE indicates that all initial estimates given for diagonal
elements are understood to be initial estimates of variances of
eps. This is the default.
STANDARD indicates that all initial estimates given for diagonal
elements are understood to be initial estimates of standard devi-
ations of epsilons. May also be coded SD.
COVARIANCE indicates that all initial estmates given for off-
diagonal elements are understood to be initial estimates of
covariances of epsilons. This is the default.
CORRELATON indicates that all initial estmates given for off-
diagonal elements are understood to be initial estimates of cor-
relations of epsilons.
CHOLESKY indicates that the block is specified in its Cholesky
form.
Options VARIANCE or STANDARD may be combined with COVARIANCE or
CORRELATON.
Note that NONMEM converts all initial estimates to variance and
covariances. The values desplayed in the NONMEM report and in
the raw and additional output files are always variances and
covariances.
Examples:
The following describe the same block (within rounding errors):
$SIGMA BLOCK(2) ; or $SIGMA VARIANCE COVARIANCE BLOCK(2)
0.64
-0.24 0.58
$SIGMA STANDARD BLOCK(2)
0.8
-0.24 0.762
$SIGMA STANDARD CORRELATION BLOCK(2)
0.8
-0.394 0.762
$SIGMA VARIANCE CORRELATION BLOCK(2)
0.64
-0.394 0.58
$SIGMA CHOLESKY BLOCK(2)
0.8
-0.3 0.7
The (entire) initial estimate of the block must be positive defi-
nite. 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 con-
tiguous subdiagonals symmetrically ocurring across the diagonal,
the elements off both the diagonal and the subdiagonals are con-
strained 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 struc-
tures for $SIGMA BLOCK(3), the 0's are preserved:
x
0x
00x
x
xx
0xx
With NONMEM 7.3 if the initial estimate of a block is not posi- |
tive 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., |
DIAGONAL SHIFT OF 1.1000E-03 WAS IMPOSED TO ENSURE POSITIVE DEF- |
INITENESS
3. $SIGMA 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.
With NONMEM 7.3 (m) is permitted. If (m) is present, then this
record is equivalent to m identical records without (m). E.g.,
$SIGMA BLOCK(2) SAME(3)
is equivalent to
$SIGMA BLOCK(2) SAME
$SIGMA BLOCK(2) SAME
$SIGMA BLOCK(2) SAME
4.
$SIGMA 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, VARI-
ANCE,STANDARD,COVARIANCE,CORRELATON, UNINT) may follow VALUES or
be placed between BLOCK and VALUES.
E.g.,
$SIGMA BLOCK(6) VALUES(0.1,0.01)
is the same as
$SIGMA 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 sigma priors):
$SIGMA BLOCK(6) FIX VALUES(0.15,0.0)
5.
$SIGMA label=value (NM75)
The symbolic label substitution feature is new with NONMEM 7.5.
This is a compact method of defining an EPS (an element of SIGMA)
specifying its initial estimate, and specifying a label for the
subscript for this element of SIGMA. The label may be used as a
subscript for EPS in abbreviated code, and will also identify
this element of SIGMA in the NONMEM output. If new $SIGMA
records change the ordering, the abbreviated code does not have
to be changed. For example, suppose the first element of SIGMA
that is defined happens to be
$SIGMA RSW=.06
The NONMEM report will describe the relationship, e.g.,
LABELS FOR EPS
EPS(1)=EPS(RSW)
and EPS(RSW) rather than EPS1 will appear in the NONMEM report.
The abbreviated code can use this symbolic subscript instead of
the numeric subscript.
As with $OMEGA, $SIGMA and $THETA records (if elements of THETA
are used) must be placed ahead of any records that use the sym-
bolic label.
Another example defines symbolic labels for a block of SIGMA:
$SIGMA BLOCK(2)
RSW= 0.3
EX= 0.01 0.35
Or, for diagonals,
$SIGMA
RSW= 0.3
EX= 0.35
6.
$SIGMA BLOCK(n) NAMES (label1,...,labeln) VALUES (odiag,diag) (NM75)
With NONMEM 7.5, Symbolic label substitution may be specified for
an entire block using the NAMES option. This is a compact way of
defining one or more epsilons with labels and, when combined with
VALUES, with initial values. For example
$SIGMA BLOCK(2) NAMES(RSW,EX) VALUES(0.03,0.01)
This is equivalent to
$SIGMA BLOCK(2)
RSW= 0.03
EX= 0.01 0.03
If both are present, VALUES() must come after NAMES().
REFERENCES: Guide I Section C.3.4.6, D.5.2, D.5.3
REFERENCES: Guide IV Section III.B.10, V.C.6
REFERENCES: Guide V Section 9.3
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Created by nmhelp2html v. 1.0 written by Niclas Jonsson (Modified by AJB 5/2006,11/2007,10/2012)