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| COVARIANCE MATRIX OF ESTIMATE |
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MEANING: NONMEM's estimate of the precision of its parameter estimates
CONTEXT: NONMEM output
DISCUSSION:
From asymptotic statistical theory, the distribution of the parameter
estimates is multivariate normal, with a variance-covariance matrix
that can be estimated from the data. Such an estimate forms the basic
output of NONMEM's Covariance Step. The variance-covariance matrix is
not to be confused with either SIGMA, the covariance matrix for the
second level random effects, or with OMEGA, the covariance matrix for
the first level random effects. These two matrices describe the vari-
ability of epsilons or etas, respectively, about their means. The
variance-covariance matrix of the (distribution of) parameter esti-
mates, on the other hand, describes the variability under the assumed
model of the parameter estimates across (imagined) replicated data
sets, using the design of the real data set. The following is an
example of the NONMEM output giving the estimate of the variance-
covariance matrix.
**************** COVARIANCE MATRIX OF ESTIMATE ********************
TH 1 TH 2 OM11 OM12 OM22 SG11
TH 1 1.02E+00
TH 2 -7.34E-03 6.50E-04
OM11 1.50E+01 -2.88E-01 2.78E+02
OM12 ......... ......... ......... .........
OM22 3.73E-04 -3.33E-05 1.47E-02 ......... 1.72E-06
SG11 -7.79E-02 1.26E-03 -1.45E+00 ......... -5.59E-05 2.55E-02
The matrix (which is symmetric) is given in lower triangular form. In
this example, the 2x2 matrix, OMEGA, was constrained to be diagonal;
the omitted entries above (.........) indicate that OM12 is not esti-
mated, and consequently has no corresponding row/column in the vari-
ance-covariance matrix. When the size of the array exceeds 75x75, a
compressed form is printed in which the omitted entries (.........)
are not printed. The compressed form may also be requested for arrays
smaller than 75x75 (See $covariance).
The (estimated) variance-covariance matrix is computed from the R and
S matrices; it is Rinv*S*Rinv, where Rinv is the inverse of the R
matrix. The R matrix is the Hessian matrix of the objective function,
evaluated at the parameter estimates. The S matrix is obtained by
summing the cross-product gradient vectors of the individual-based
objective functions, evaluated at the parameter estimates. The indi-
vidual-based objective functions are the separate terms contributed by
each individual's data to the overall objective function, and the
cross-product gradient vectors are summed across the individuals in
the data set.
The inverse variance-covariance matrix R*Sinv*R is also output
(labeled as the Inverse Covariance Matrix), where Sinv is the inverse
of the S matrix. If S is judged to be singular, a pseudo-inverse of S
is used, and since a pseudo-inverse is not unique, the inverse vari-
ance-covariance matrix is really not unique. In either case, the
inverse variance-covariance matrix can be used to develop a joint con-
fidence region for the complete set of population parameters. As we
usually develop a confidence region for a very limited set of popula-
tion parameters, this use of the inverse variance-covariance matrix is
somewhat limited.
An error message from the Covariance Step stating that the R matrix is
not positive semidefinite suggests that the parameter estimate does
not correspond to a true (local) minimum and is not to be trusted.
(It may be a saddle point.) An error message stating that the R
matrix is positive semidefinite, but singular, indicates that the
objective function is flat in a neighborhood of the parameter esti-
mate, and so the minimum is not really unique, and there is probably
some overparametrization. With both error messages, neither a vari-
ance-covariance matrix nor inverse variance-covariance matrix is out-
put. An error message stating that the S matrix is singular indicates
strong overparameterization. However, provided the R matrix is judged
to be positive semidefnite and nonsingular (i.e. positive definite),
both the variance-covariance and inverse variance-covariance matrices
are output.
When the R matrix is judged to be singular, but positive semidefinite,
then the T matrix, R*Sinv*R, where Sinv is the inverse (or a pseudo-
inverse) of the S matrix, is output. This cannot be called the
inverse covariance matrix, as the covariance matrix does not exist.
However, as with the inverse variance-covariance matrix, T can be used
to develop a joint confidence region for the complete set of popula-
tion parameters.
There are options that allow the variance-covariance matrix to be com-
puted as either 2*Rinv or 4*Sinv. Asymptotic statistical theory sug-
gests that these matrices are appropriate under the additional assump-
tion that the objective function is indeed additively proportional to
minus twice the log likelihood function for the data.
Unless the reported number of significant digits in the final parame-
ter estimate is at least as large as the requested number of signifi-
cant digits, the Covariance Step will not be implemented.
(See sig digits). Sometimes, the number of significant digits is not
reportable. However, when it is and the user thinks this number to be
adequate, and a model specification file was output (See model speci-
fication file), NONMEM may be run again where the Covariance Step is
implemented, while the Estimation step is is not repeated (i.e. the
MAXEVAL option is set to 0). With the subsequent run, the model spec-
ification file should be input, and the requested number of signifi-
cant digits should be set to a value less than the reported number of
significant digits from the first run (presumably, this value would be
the reported number rounded down to the highest integral value).
(See standard error, correlation matrix of estimate).
REFERENCES: Guide I Section C.3.5.2
REFERENCES: Guide II Section D.2.5
REFERENCES: Guide V Section 5.4, 13.3
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