RE: FW: Block versus diagonal omega
Hi Leonid,
(I suspected this point might get belabored.) Applying the theory to the
THETA()>=0 parameterization obtains the mixture chi-square distribution, for
the same reason as the OMEGA case, hence results are indeed the same. Problem
arises only with interpreting THETA()<>0 and considering the LRT as a
chi-square distribution without mixture.
There is usually no need to worry about “regularity conditions”; they tend to
be easily satisfied for most practical scenarios. Although the boundary does
cause the issue (I hope it is clear now), other than this diagonal variance
testing case I really can see no reasons to be concerned. In addition, this
problem does not occur to off-diagonal elements (unless for some strange reason
you want to test a perfect correlation).
Whether parameters, OMEGA elements or not, “should” be included is a different
matter. The issues you mention, which I think in essence are numerical
approximation and how much “learning” to do, are obviously interesting, but
that is probably another topic..
Chuanpu
Quoted reply history
-----Original Message-----
From: [email protected] [mailto:[email protected]] On
Behalf Of Leonid Gibiansky
Sent: Monday, August 30, 2010 8:05 PM
To: Hu, Chuanpu [CNTUS]
Cc: Mark Sale - Next Level Solutions; [email protected]
Subject: Re: FW: [NMusers] Block versus diagonal omega
Chuanpu,
These two problems (in OMEGA and THETA parametrizations) are identical (in a
sense that they provide same parameter values and OF). Moreover, one can
propose the third parametrization:
SQRT(THETA())*ETA()
$OMEGA
1 FIXED
with THETA() > 0 being the variance of the random effect (rather than SD). The
tests based on them are either all valid, or all invalid.
I have not seen anybody going into that level of "rigorousness" as to analyze
conditions when the OF follows theoretical chi^2 distribution (for each
specific model parameter).
If so, we can use the same test for variances as well. The fact that we can
does not imply that we should: in my experience, the number and the structure
of random effects is defined mostly by the amount of individual data and
stability of the problem (that is related to the amount of data). It may also
be defined by the estimation method: newer methods allow (or even require) more
complex OMEGA structure.
Thanks
Leonid
--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
On 8/30/2010 4:22 PM, Hu, Chuanpu [CNTUS] wrote:
> Mark, Leonid et al,
>
> I guess my previous message was not clear. (And by the way, I have
> used something similar to the THETA-parameterization and observed the
> same NONMEM OBJF values, as should be.) The question is what
> distribution the NONMEM objective function difference follows. The
> proof of it being chi-square with 1 df requires certain mathematical
> “regularity conditions” that the THETA parameterization would violate
> (otherwise its distribution would not be a mixture chi-squire!). So,
> the hypothesis test based on OMEGA-parameterization is valid (with
> mixture chi-squared), and the test based on THETA-parameterization is invalid.
>
> Chuanpu