RE: FW: [NMusers] 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

 

 

-----Original Message-----

From: owner-nmusers_at_globomaxnm.com [mailto:owner-nmusers_at_globomaxnm.com] On Behalf Of Leonid Gibiansky

Sent: Monday, August 30, 2010 8:05 PM

To: Hu, Chuanpu [CNTUS]

Cc: Mark Sale - Next Level Solutions; nmusers_at_globomaxnm.com

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

 

Received on Tue Aug 31 2010 - 10:00:03 EDT