RE: FW: 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 919-846-9185 A carbon-neutral company See our real time solar energy production at: http://enlighten.enphaseenergy.com/public/systems/aSDz2458
Quoted reply history
-------- Original Message --------
Subject: Re: FW: [NMusers] Block versus diagonal omega
From: Leonid Gibiansky < [email protected] >
Date: Mon, August 30, 2010 2:47 pm
To: "Hu, Chuanpu [CNTUS]" < [email protected] >
Cc: [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
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
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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