Dear Aziz - The approximate likelihood methods in NONMEM such as FO, FOCE,and LAPLACE optimize an objective function than is parameterized internally by the Cholesky factor L of Omega, regardless of whether the matrix is diagonal (the EM -based methods do something considerably different and work directly with Omega rather than the Cholesky factor.) Thus for the approximate likelihood methods, the SE's computed internally by $COV from the Hessian or Sandwich or Fisher score methods are first computed with respect to these Cholesky parameters , and then the corresponding SE's of the full Omega=LL' are computed by a 'propagation of errors' approach which skews the results, particularly if the SE's are large. Thus in a sense regarding your dilemma about whether Model 1 or Model 2 is better with respect to applicability of $COV results, one answer is that both are fundamentally distorted by the propagation of errors method with respect to the Omega elements.
But regarding your fundamental question 'can we trust the output of $COV '- all of this makes very little difference. Standard errors computed by $COV are inherently dubious - the applicability of the usual asymptotic arguments is very questionable for the types/sizes of data sets we often deal with. As Lewis Sheiner used to say of these results, 'they are not worth the electrons used to compute them'. They are the best we can do for the level of computational investment put into them - If you want something better, try a bootstrap or profiling method. ________________________________________ NOTICE: The information contained in this electronic mail message is intended only for the personal and confidential use of the designated recipient(s) named above. This message may be an attorney-client communication, may be protected by the work product doctrine, and may be subject to a protective order. As such, this message is privileged and confidential. If the reader of this message is not the intended recipient or an agent responsible for delivering it to the intended recipient, you are hereby notified that you have received this message in error and that any review, dissemination, distribution, or copying of this message is strictly prohibited. If you have received this communication in error, please notify us immediately by telephone and e-mail and destroy any and all copies of this message in your possession (whether hard copies or electronically stored copies). Thank you. buSp9xeMeKEbrUze
