# Re: inclusion of covariates with \$PRIOR

From: Anna Chan Kwong <anna.chankwong>
Date: Fri, 17 May 2019 14:39:47 +0200

Dear Jakob and Mats,

My questions were general, for different applications. After reading the
article by Gisleskog at al. (in particular, the last part of the
discussion), I thought it was possible to build a “Model without co=
variate”
with \$PRIOR and then to add covariates with SCM (with the 4 conditions I
listed in my first email). Moreover, the building of a model using SCM in
combination with a frequentist prior is sometimes reported in the
literature, without precision on its implementation.

Thus, I wanted to add a PRIOR on the theta of the parameter (not on the
theta of the covariate), that is : PAR = PARp * PARCOV

PARp = parameter with prior (\$THETAP = parameter in the prior populatio=
n)

PARCOV estimated on the new dataset only, and centered around prior median
of the covariate, e.g. with the equation (PARCOV =
(COV/medianCOV)**THETA(COV), medianCOV is the median in the prior dataset).

I wrote ”"OBJECTIVE FUNCTION VALUE WITHOUT CONSTANT" (without PRIOR
penalty).” because this is what is reported in the output of NONMEM=
with
\$PRIOR:

N*LOG(2PI) CONSTANT TO OBJECTIVE FUNCTION: 2152.15

OBJECTIVE FUNCTION VALUE WITHOUT CONSTANT: 3633.00

OBJECTIVE FUNCTION VALUE WITH CONSTANT: 5785.15

[...]

PRIOR CONSTANT TO OBJECTIVE FUNCTION: 1297.35

OBJECTIVE FUNCTION VALUE WITHOUT CONSTANT: 3633.00

OBJECTIVE FUNCTION VALUE WITH CONSTANT: 4930.35

REPORTED OBJECTIVE FUNCTION DOES NOT CONTAIN CONSTANT

The first block, with the N*LOG(2PI) constant, is common to all the outputs
(also without PRIOR).

I’m interested in the second block, which reports

- first the “PRIOR penalty” (PRIOR CONSTANT TO OBJECTIV=
E FUNCTION),

- second the objective function on the data (OBJECTIVE FUNCTION VALUE
WITHOUT CONSTANT),

- third the SUM of the “PRIOR penalty” and the objectiv=
e function on
the data (OBJECTIVE FUNCTION VALUE WITH CONSTANT)

From the article by Gisleskog et al, I understood that the third term (�=
��PRIOR
penalty” + objective function on the data (OBJECTIVE FUNCTION VALUE=
WITH
CONSTANT)) should be used to perform hypothesis tests. However, when I
tried to perform an automated SCM in PsN on a “Model without covari=
ate”
with \$PRIOR NWPRI (which I thought was possible because of the \$PRIOR NWPRI
mention in the SCM user guide), it was the second term (objective function
on the data (OBJECTIVE FUNCTION VALUE WITHOUT CONSTANT)) that was compared.

Jakob, if I understand correctly the beginning of your email, you use
\$PRIOR NWPRI with SCM to test if a parameter is different in a new
population. That is, testing if DIFF is different from zero if we code the
parameter PARn = PARp * (1 + DIFF),

PARn = parameter in the new population

PARp = parameter with prior (\$THETAP = parameter in the prior populatio=
n)

Then comparing two models, one with PARn = PARp (DIFF=0), one with PARn=
=
PARp * (1 + DIFF), with the Likelihood Ratio Test.

Is that what you meant? I was not aware of this method.

Again, thank you very much for your suggestions,

Best regards,

Anna

Le jeu. 16 mai 2019 à 14:22, Mats Karlsson <Mats.Karlsson
e> a
écrit :

> Hi Anna,
>
>
>
> That you want to explore covariate relationships on a parameter suggests
> that you believe your data contain plenty of information about the
> parameter. Therefore do you really need to use a prior on the parameter(s=
)
odel is and why
> you use a prior could help answering.
>
>
>
> You write “"OBJECTIVE FUNCTION VALUE WITHOUT CONSTANT" (without P=
RIOR
> penalty). ”, but the CONSTANT mentioned is not related to the pri=
or, but
> rather the term of the OFV that is related to the number of observations
> only (Nobs*LN(2*PI)).
>
> For obtaining a reference distribution for the likelihood ratio test a
> randomization (permutation) test is often useful as it uses the real data
> as opposed to simulated data. PsN functionality for this is “rand=
test”.
>
>
>
> Best regards,
>
> Mats
>
> *From:* owner-nmusers
> Behalf Of *Anna Chan Kwong
> *Sent:* den 16 maj 2019 10:46
> *To:* nmusers
> *Subject:* [NMusers] inclusion of covariates with \$PRIOR
>
>
>
> Dear NMusers
>
> I am wondering about the inclusion of covariates with the \$PRIOR
> subroutine.
>
> The article "Use of Prior Information to Stabilize a Population Data
> Analysis" (Gisleskog, Karlsson, Beal 2002) states that Stepwise Covariat=
e
> Modelling (SCM) is possible on a parameter estimated with prior
> information, under conditions :
>
> 1) Population parameters have to be centered around the prior geometric
> mean (often the median) of the covariate (for example, if the power
> function is used: (COV/medianCOV)**THETA(COV), medianCOV is the median in
> the prior dataset)
> Is it correct to use functions like linear function
> (1+THETA(COV)*(COV-medianCOV) or exponential function
> (exp(THETA(COV)*(COV-medianCOV) ?
>
> 2) the SUM of the objective function and the PRIOR penalty should be used
> to perform hypothesis tests.
>
> Could you confirm I have properly understood this condition??
> I am in doubt because automated SCM with \$PRIOR in PsN (
> https://uupharmacometrics.github.io/PsN/docs.html
> <https://urlproxy.sunet.se/canit/urlproxy.php?_q=aHR0cHM6Ly91dXBoYXJtYW=
NvbWV0cmljcy5naXRodWIuaW8vUHNOL2RvY3MuaHRtbA%3D%3D&_s=bWF0cy5rYXJsc3NvbkB=
mYXJtYmlvLnV1LnNl&_c=ef5b1054&_r=dXUtc2U%3D>)
> compares the "OBJECTIVE FUNCTION VALUE WITHOUT CONSTANT" (without PRIOR
> penalty).
>
>
> 3) hypothesis tests such as the Likelihood Ratio Test needs to be
> performed with the ACTUAL significance level
>
> Is there a way to determine the actual significance level faster than
> Stochastic Simulation and Estimation?
>
> 4) the prior omega of the parameter on which the covariate impacts should
> be decreased by the product of THETA(COV)² and the prior population
> variance of log(COV).
> Does that mean we should manually adjust the \$OMEGAP value of a parameter
> on which we test the covariate ? OMEGAP(adjusted) = OMEGAP -
> (THETA(COV))²*var
>
> with OMEGAP = prior OMEGA estimate of the parameter on which the covari=
ate
> is added ; var = prior population variance of log COV
>
> Thank you very much for your understanding,
>
> Sincerely yours,
>
> Anna Chan Kwong
> PhD sudent in Pharmacometrics, Marseille University.
>
>
>
>
>
>
>
>
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