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 Covariate 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) 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 covariate 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.

Dear Anna, The times I have used SCM in combination with a frequentist prior, it has been only to test if a new population is different (i.e. the data used to generate the prior were all of a different population, e.g. healthy volunteers and other indications than the currently-investigated population. Or something similar where the covariate value is a nominal level which was not present in the data used to generate the prior). To use SCM in combination with NWPRI in that situation is straightforward. With regards to your first question. If you use a prior only to support the estimation of this specific covariate coefficient (no prior on other parameters), then you do not need to center around the same covariate value. The point estimate for the covariate coefficient will be the same, regardless of how you center these covariate models (but standard errors, correlation between estimates and the covariance matrix as a whole will be different depending on how you center). However, with prior across all model parameters, it would be important to center across the same covariate value, since the population typical value is with regards to this (e.g. the population typical clearance for a subject with 70 kg body weight). Maybe this is what Gisleskog et al. were referring to? With regards to model selection with a prior on all available parameters this is not as straightforward. If your prior is a full model (including all covariates that you want to test in SCM), then in principle using prior on all fixed effects would be possible. But due to correlation between the estimates in your prior you could end up leaving a covariate out of the model, in a way that would not have happened in a combined analysis. An alternative may be to have separate priors for each model you test, but I do not think there is any automated software to support that procedure. If you have access to the data used to generate the prior, it may be easier to combine all data, rather than using a frequentist prior. I suspect this is not possible in your case, since you ask these questions? Best regards Jakob Jakob Ribbing, Ph.D. Senior Consultant, Pharmetheus AB Cell/Mobile: +46 (0)70 514 33 77 [email protected] www.pharmetheus.com http://www.pharmetheus.com/ Phone, Office: +46 (0)18 513 328 Uppsala Science Park, Dag Hammarskjölds väg 36B SE-752 37 Uppsala, Sweden This communication is confidential and is only intended for the use of the individual or entity to which it is directed. It may contain information that is privileged and exempt from disclosure under applicable law. If you are not the intended recipient please notify us immediately. Please do not copy it or disclose its contents to any other person.

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) in question? A little more info around what your “base” model is and why you use a prior could help answering. You write “"OBJECTIVE FUNCTION VALUE WITHOUT CONSTANT" (without PRIOR penalty). ”, but the CONSTANT mentioned is not related to the prior, 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 “randtest”. Best regards, Mats

Dear Jakob and Mats, Thank you very much for your kind answers. 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 covariate” 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 population) 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 OBJECTIVE FUNCTION), - second the objective function on the data (OBJECTIVE FUNCTION VALUE WITHOUT CONSTANT), - third the SUM of the “PRIOR penalty” and the objective 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 covariate” 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 population) 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 <[email protected]> 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) > in question? A little more info around what your “base” model is and why > you use a prior could help answering. > > > > You write “"OBJECTIVE FUNCTION VALUE WITHOUT CONSTANT" (without PRIOR > penalty). ”, but the CONSTANT mentioned is not related to the prior, 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 “randtest”. > > > > Best regards, > > Mats > > *From:* [email protected] <[email protected]> *On > Behalf Of *Anna Chan Kwong > *Sent:* den 16 maj 2019 10:46 > *To:* [email protected] > *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 Covariate > 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=aHR0cHM6Ly91dXBoYXJtYWNvbWV0cmljcy5naXRodWIuaW8vUHNOL2RvY3MuaHRtbA%3D%3D&_s=bWF0cy5rYXJsc3NvbkBmYXJtYmlvLnV1LnNl&_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 covariate > 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. > > > > > > > > > När du har kontakt med oss på Uppsala universitet med e-post så innebär > det att vi behandlar dina personuppgifter. För att läsa mer om hur vi gör > det kan du läsa här: http://www.uu.se/om-uu/dataskydd-personuppgifter/ > > E-mailing Uppsala University means that we will process your personal > data. For more information on how this is performed, please read here: > http://www.uu.se/en/about-uu/data-protection-policy >

Dear Anna, Just to answer quickly on the question directed to me: Yes, you understood it correctly. To implement this in practice, since your NONMEM dataset only includes one level for the covariate (all subjects have the same covariate value), you will need to make a small addition to the SCM configuration file. Lets assume DIS=1 is the covariate, where 1 codes for a new indication (which was not present in the legacy data used to generate the prior), or a new subpopulation, e.g. children/adolescents, where only adults where present in the legacy data. You would then need to add this to the scm configuration file: [code] *:DIS-2=PARCOV=(1+THETA(1)) Other than that, it is also a little tricky to generate the NONMEM control stream with NWPRI on all parameters (of the base model). PsN has an automatic function for generating this code (option to update_inits), but it may be good to check manually. Best regards Jakob Jakob Ribbing, Ph.D. Senior Consultant, Pharmetheus AB Cell/Mobile: +46 (0)70 514 33 77 [email protected] www.pharmetheus.com http://www.pharmetheus.com/ Phone, Office: +46 (0)18 513 328 Uppsala Science Park, Dag Hammarskjölds väg 36B SE-752 37 Uppsala, Sweden