+--------------------------------------------------------------------+
| |
| $SIGMA |
| |
+--------------------------------------------------------------------+
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)]
[BLOCK(n)VALUES(diag,odiag)]
[VARIANCE|STANDARD] [COVARIANCE|CORRELATON] [CHOLESKY]
[[value1] [value2] [value3] ... [FIXED]]
[(value)xn]
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 epsilon variables in the statistical model. This record
should appear only if the statistical model contains epsilon vari-
ables, i.e., the data are population. 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.
OPTIONS:
There are four 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)
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"). |
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 VARIANCE indicates that the initial estimate is understood
to be a variance of the epsilons. This is the default.
Option STANDARD indicates that the initial estimate is understood
to be a standard deviation of the epsilons. 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.
VARIANCE indicates that all initial estimates given for diagonal
elements are understood to be initial estimates of variances of
epsilons. 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.
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.
"(m) may be omitted. If m is present, then this record is equiva- |
lent 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". VALUES must follow |
BLOCK. Other options (such as FIXED, CHOLESKY, VARIANCE,STAN- |
DARD,COVARIANCE,CORRELATON) 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 |
If the values are omitted in forms 1 or 2, then NONMEM will try to
obtain initial estimates.
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
REFERENCES: Guide I Section E.5.1
REFERENCES: Guide IV Section III.B.11
REFERENCES: Guide V Section 9.3
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