Re: FW: Block versus diagonal omega
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
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
Quoted reply history
On 8/30/2010 1:41 PM, Hu, Chuanpu [CNTUS] wrote:
> *From:* Hu, Chuanpu [CNTUS]
> *Sent:* Monday, August 30, 2010 8:46 AM
> *To:* 'Mark Sale'
> *Cc:* 'nmusers'
> *Subject:* RE: [NMusers] Block versus diagonal omega
>
> Mark,
>
> Nice thought – the test can be conducted, but the devil is in the
> details. This has to do with the intricacies of the role alternative
> hypothesis plays in hypothesis testing:
>
> For the original parameterization testing OMEGA, the hypothesis test is
>
> H0: OMEGA=0, vs. H1: OMEGA>0
>
> For the THETA parameterization testing OMEGA, the hypothesis test is
>
> H0: THETA=0, vs. H1: THETA<>0
>
> So without getting into the math, the intuitive argument is that the
> alternative hypotheses in the 2 situations are different, therefore it
> is logical that the testing criteria must change. The world of math does
> not contain contradictions even though it may appear so at times. J
>
> Chuanpu
>
> *From:* Mark Sale [mailto:[email protected]]
> *Sent:* Sunday, August 29, 2010 9:19 AM
> *To:* Hu, Chuanpu [CNTUS]
> *Cc:* nmusers
> *Subject:* RE: [NMusers] Block versus diagonal omega
>
> Chuanpu,
> Do I extrapolate correctly then that:
>
> V = THETA(1)*EXP(THETA(2)*ETA(1))
> .
> .
> .
> $OMEGA
> (1,FIXED).
>
> Can be tested (THETA(2) <> 0), since it is not a truncated distribution?
> might be an interesting exercise to do this with LRT and compare to the
> randomization test with the usual specification.
>
> Mark
>
> --- On *Fri, 8/27/10, Hu, Chuanpu [CNTUS] /<[email protected]>/* wrote:
>
> From: Hu, Chuanpu [CNTUS] <[email protected]>
> Subject: RE: [NMusers] Block versus diagonal omega
> To: "Mark Sale - Next Level Solutions" <[email protected]>,
> "Eleveld,DJ" <[email protected]>
> Cc: "nmusers" <[email protected]>
> Date: Friday, August 27, 2010, 4:33 PM
>
> Theoretically, the NONMEM objective function drop for adding a diagonal
> element follows a mixture chi-square distribution, from which follows
> that using the “usual” chi-square distribution would be conservative.
> This has to do with 0 being on the boundary of possible values. (See
> Pinheiro and Bates, Mixed Effects Models in S and S-PLUS, Springer,
> 2000.) As this boundary issue does not apply to off-diagonal elements,
> the “usual” chi-square distribution should be fine (with the usual
> statistical asymptotic caveats).
>
> I’d like to mention that, while the “find the best fit” mindset may be
> suitable for the typical exploratory setting, the p-values from repeated
> (e.g., stepwise) tests are not statistically interpretable. To have
> valid p-values, confirmatory analyses would be needed, which in my mind
> deserves a wider use. J
>
> Chuanpu
>
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>
> Chuanpu Hu, Ph.D.
>
> Director, Pharmacometrics
>
> Pharmacokinetics
>
> Biologics Clinical Pharmacology
>
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