Dear NMusers,
This is a great reminder for us to consider the reliability of standard
errors in our models, thanks Ken & Alan. The more non-linear the models
become, the less reliable and the more important other perspectives on
parameter values such as sensitivity analysis and prior knowledge.
The nmusers archive has many great threads on the topic that are
available to review such as
https://www.mail-archive.com/nmusers@globomaxnm.com/msg05423.html and
related
https://www.mail-archive.com/nmusers@globomaxnm.com/msg05419.html . In
summary, log-transformation only can get you so far but can perhaps be
seen as a sort of minimal effort.
To add to the Lewis's quote about SEs - "they are not worth the
electrons used to compute them" (see the links), Pyry had some very
interesting observations he shared with me about the SE of the CV of a
log-normal omega: it inflates with higher values of omega compared to
the SE of omega itself.
Best regards,
Jeroen
http://pd-value.com
jer...@pd-value.com
@PD_value
+31 6 23118438
-- More value out of your data!
On 29-07-2024 14:41, kgkowalsk...@gmail.com wrote:
Dear NMusers,
It was recently pointed out to me by a statistical colleague that my
recent NMusers post about the inverse Hessian (R matrix) evaluated at
the maximum likelihood estimates is a consistent estimator of the
covariance matrix (i.e., converges to the true value with large N) is
only true for linear models. For nonlinear models, the standard
errors produced by NONMEM and other nonlinear estimation software are
not only asymptotic but also approximate. Moreover, how well that
approximation works will also depend on the parameterization. This I
believe is one of the motivations behind “mu referencing” in NONMEM
and the use of log transformations of the parameters to help improve
Wald-based approximations. I thank Alan Maloney for pointing this out
to me.
Kind regards,
Ken
*From:*kgkowalsk...@gmail.com <kgkowalsk...@gmail.com>
*Sent:* Saturday, July 27, 2024 12:36 PM
*To:* 'Santosh' <santosh2...@gmail.com>; nmusers@globomaxnm.com
*Subject:* RE: [NMusers] Obtaining RSE%
Dear Santosh,
There is a good reason for this. Wald (1943) has shown that the
inverse of the Hessian (R matrix) evaluated at the maximum likelihood
estimates is a consistent estimator of the covariance matrix. It is
based on Wald’s approximation that the likelihood surface locally near
the maximum likelihood estimates can be approximated by a quadratic
function in the parameters. This theory does not hold for any set of
parameter estimates along the algorithm’s search path prior to
convergence to the maximum likelihood estimates. Moreover, inverting
the Hessian evaluated at an interim step prior to convergence would
likely be a poor approximation especially early in the search path
where the gradients are large (i.e., large changes in OFV for a given
change in the parameters would probably have substantial curvature and
not be well approximated by a quadratic model in the parameters).
Thus, the COV step in NONMEM is only applied once convergence is
obtained during the EST step.
Wald, A. “Tests of statistical hypotheses concerning several
parameters when the number of observations is large.” /Trans. Amer.
Math. Soc./ 1943;54:426.
Best,
Ken
Kenneth G. Kowalski
President
Kowalski PMetrics Consulting, LLC
Email: kgkowalsk...@gmail.com <mailto:kgkowalsk...@gmail.com>
Cell: 248-207-5082
*From:*owner-nmus...@globomaxnm.com
<mailto:owner-nmus...@globomaxnm.com><owner-nmus...@globomaxnm.com
<mailto:owner-nmus...@globomaxnm.com>> *On Behalf Of *Santosh
*Sent:* Friday, July 26, 2024 3:38 AM
*To:* nmusers@globomaxnm.com <mailto:nmusers@globomaxnm.com>
*Subject:* [NMusers] Obtaining RSE%
Dear esteemed experts!
When using one or more estimation methods & covariance step in a
NONMEM control stream, the resulting ext file contains final estimate
(for all estimation steps) & standard error (only for the last
estimation step).
Is there a way that standard error is generated for every estimation step?
TIA
Santosh
