RE: Obtaining RSE%
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
Quoted reply history
From: [email protected] <[email protected]>
Sent: Saturday, July 27, 2024 12:36 PM
To: 'Santosh' <[email protected]>; [email protected]
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: <mailto:[email protected]> [email protected]
Cell: 248-207-5082
From: <mailto:[email protected]> [email protected] <
<mailto:[email protected]> [email protected]> On Behalf
Of Santosh
Sent: Friday, July 26, 2024 3:38 AM
To: <mailto:[email protected]> [email protected]
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