Re: FW: 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
Quoted reply history
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
>
> *From:* [email protected]
> [mailto:[email protected]] *On Behalf Of *Mark Sale - Next
> Level Solutions
> *Sent:* Monday, August 30, 2010 3:55 PM
> *Cc:* [email protected]
> *Subject:* RE: FW: [NMusers] Block versus diagonal omega
>
> Chuanpu,
>
> My experience is the same as Leonid's. I get the same OBJ, the same
> (transformed) parameters, with H(null) and H(alt). Hence my confusion, I
> get the same numbers, but one test of hypothesis is not valid (or
> perhaps conservative) and the other may be (noting that the distribution
> for THETA only makes sense if THETA is constrainted to be >0 or < 0, a
> distribution that crosses 0 seems meaningless to me).
>
> But, I'm usually not all that interested in testing hypotheses WRT
> OMEGA, and I'm pleased to learn that:
>
> 1. If you do a test of hypothesis it is conservative
>
> 2. AIC/BIC seem to still be valid indicators of "preference" WRT
> -2Loglikelihood change.
>
> Mark Sale MD
> Next Level Solutions, LLC
> www.NextLevelSolns.com http://www.NextLevelSolns.com
> 919-846-9185
>
> A carbon-neutral company
>
> See our real time solar energy production at:
>
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>
> -------- Original Message --------
> Subject: Re: FW: [NMusers] Block versus diagonal omega
> From: Leonid Gibiansky <[email protected]
> <mailto:[email protected]>>
> Date: Mon, August 30, 2010 2:47 pm
> To: "Hu, Chuanpu [CNTUS]" <[email protected] <mailto:[email protected]>>
> Cc: [email protected] <mailto:[email protected]>
>
> Chuanpu,
>
> In all stable problems that I tried, parametrization
>
> ETA()
> $OMEGA
> 0.1 ; estimated
>
> was equivalent (in terms of the estimated value and objective
> function) to
>
> THETA(*)*ETA()
> $OMEGA
> 1 FIXED
>
> Also,
>
> H0: THETA=0, vs. H1: THETA<>0
>
> is the same as
>
> H0: OMEGA=0, vs. H1: OMEGA>0
>
> since OMEGA=THETA^2
>
> In theta-form, the problem has two identical solution
>
> THETA()=SQRT(OMEGA) and THETA()= -SQRT(OMEGA)
>
> Leonid
>
> --------------------------------------
> Leonid Gibiansky, Ph.D.
> President, QuantPharm LLC
> web: www.quantpharm.com http://www.quantpharm.com
> e-mail: LGibiansky at quantpharm.com http://quantpharm.com
> tel: (301) 767 5566