Re: $MIX and $CONTR

On Tue, 2007-07-31 at 16:10 +1200, Nick Holford wrote: > Leonid, > > The model you propose is based on the response variable being a binary > categorical variable - responder or non-responder. Such a likelihood based > model is straightforward and can easily use age or other time varying > covariates to help make the prediction. > > But the problem I am proposing is based on a continuous scale observation -- > it is not categorical and I do not want to waste information by making some > kind of categories out of the continuous variable. > > You do not describe why "the mixture model is not an appropriate tool to > describe this model". You also say "The biological problem that you describe > simply does not fit in the framework of the mixture model". You just say it > isn't appropriate or does not fit the framework but don't give any reasons > for your assertions. > > I would be interested in you trying to describe exactly why the biological > problem I proposed is not suitable for a mixture model. > > Nick > > Leonid Gibiansky wrote: > > > > Hi Nick, > > I fully agree with #1. It is a very reasonable model biologically, but the > > mixture model is not an > > appropriate tool to describe this model. You may want to describe it with > > the different procedure: > > describe probability (not jump, but continuous function of age or something > > else) of being a > > responder, and on top of it define different conditional probability of a > > response for responders > > and non responders. > > > > P(AGE): probability of being a responder > > > > Presp: probability of response for responders > > Pnresp: probability of response for non-responders > > > > Y=P*Presp+(1-P)Pnresp > > > > Of course, each of P, Presp, Pnresp should be expressed via THETAs, ETAs, > > whatever we need; may > > depend on gene expression if you like. > > > > I am not sure that I agree with #2: this is not a nonmem implementation > > deficiency. The biological > > problem that you describe simply does not fit in the framework of the > > mixture model, it should be > > described differently, may be as above (this was only a quick sketch, may > > not work for you), may be > > in a more complicated way, but not as a mixture model. > > > > Alternatively, you may assign different IDs to the same subject at age 10 > > and 50 (it is naive to > > expect that we stay unchanged for a lifetime): if the person changes so > > much that it jumps from > > responders to non-responders, you may expect other changes as well. > > > > Please, let us know if you come up with NONMEM description of your problem. > > > > Leonid > > > > Nick Holford wrote: > > > Leonid, > > > > > > Thanks for your thoughtful comments. > > > > > > I think there are two things to keep separate: > > > > > > 1. Imagine a drug response can be described by a bi-modal distribution > > > when one examines a sample of subjects who are all children. A mixture > > > model can be used to identify the parameters of this mixed distribution. > > > One can then categorize people as being more likely to be in distribution > > > ('responders') than in the other distribution ('non-responders'). > > > > > > Supposing that we now observe that in a different sample of older > > > subjects that the mixing probability is different but the two > > > distributions have similar parameters. We would therefore conclude that > > > the mixing probability changes with age. > > > > > > The underlying basis for the distinction between responder and > > > non-responder may be due to the expression of a gene. The expression of > > > some genes is well known to vary during human development. Imagine that > > > this gene is either turned on ('responder') or is turned off > > > ('non-responder'). This would be a biological explanation of the age > > > dependence of the mixing parameter. If the genotype is unknown then we > > > would need to use a mixture model to try to decide which genotype is most > > > likely in an individual. But because the probability of expression of the > > > gene is also unknown but varies with age then a mixture model with age as > > > a covariate seems reasonable. If we observed the same subjects as > > > children and later as adults then we should be able to describe the age > > > related shift in the probability of being a 'responder' or a > > > 'non-responder'. > > > > > > It is this kind of situation that I tried to describe using the NM-TRAN > > > code. > > > > > > 2. NONMEM is a particular implementation of a method for describing and > > > estimating the parameters of mixture model. It seems that NONMEM may only > > > have a rather simple view of mixtures i.e. an individual always belongs > > > to the same distribution and this cannot change. > > > > > > I am not sure yet if it is true that NONMEM assumes a fixed (life long) > > > within subject assignment to a distribution but I dont see any a priori > > > reason why an individual should not shift from one distribution to > > > another (as explained in the example above). > > > > > > Nick > > > > > > Leonid Gibiansky wrote: > > >> Nick, > > >> > > >> I think you are reading it incorrectly. I would guess that for a > > >> long-term study you may relate > > >> probability of response to either baseline age or to the end-of-study > > >> age, or to any other age in > > >> between, but this value should be one per subject, not one per record. > > >> Probability of a subject be > > >> in one group is assigned based on the entire data set for this subject, > > >> and not extracted from one > > >> record. Probability of being a responder may depend on age, but it > > >> should be 1 value for a subject, > > >> not 1 value for a record. If you want to simulate subject with same > > >> covariates but different age, > > >> give them separate IDs. You may need to simulate ETAs separately (if you > > >> want patients with the same > > >> random effects but different ages). > > >> > > >> I do not have sufficient experience or knowledge of the nonmem code to > > >> be more assertive. This is my > > >> guess only, but guess confirmed by a special treatment of a time > > >> dependency in the $MIX block (as > > >> you described). > > >> > > >> Leonid > > >> > > >> Nick Holford wrote: > > >>> Leonid, > > >>> > > >>> I dont see why a proportion should not change with time. The > > >>> documentation for $MIX says this: > > >>> > > >>> "Then AGE may be used on the right in $MIX. ... > > >>> AGE(i) refers to the value of AGE on the i-th. observation record of > > >>> the individual record." > > >>> > > >>> This clearly shows that it was anticipated that one might wish to use > > >>> AGE from any observation record in the individual record. If the > > >>> records represent a time series (which is 100% necessary if AGE > > >>> varies!) then this implies that a time varying covariate can be used to > > >>> model the mixing probability. > > >>> > > >>> In my example I propose that the probability of being a responder > > >>> changes with age. E.g. children tend to respond to amphetamine like > > >>> drugs by being less active while adults tend to be stimulated. > > >>> > > >>> This seems like a very plausible way to describe the response in a > > >>> population. I dont think there is any a priori reason why there should > > >>> not be a within subject variation in a mixing fraction. > > >>> > > >>> My view of a mixing fraction is that it is a substitute for a missing > > >>> covariate. If one has a time varying covariate then the mixing fraction > > >>> would be time varying. In the example proposed in the NONMEM help the > > >>> covariate proposed for use in $MIX is AGE. If AGE is properly recorded > > >>> in the data set then AGE is guaranteed to be a time varying covariate! > > >>> > > >>> Nick > > >>> > > >>> Leonid Gibiansky wrote: > > >>>> Nick, > > >>>> I am not sure that the entire idea is correct: subject can only belong > > >>>> to one population, it cannot > > >>>> jump from population to population. Therefore, time dependent P() > > >>>> should not be allowed. > > >>>> Record-number dependence in $MIX was probably invented to have an > > >>>> option of defining the Ps either > > >>>> by the baseline values, or by the values at the end of the study, but > > >>>> not for time-dependence. This > > >>>> could explains differences with the PK block approach. > > >>>> Thanks > > >>>> Leonid > > >>>> > > >>>> > First question: Why is the proportion of simulated subjects > > >>>> different from what I expected? It > > >>>> seems like all the values are being simulated with AGE=50 instead of > > >>>> AGE=0. > > >>>> > > > >>>> > Second question: More generally, if we used AGE in other > > >>>> subroutines (e.g. $PK, $PRED) then AGE > > >>>> would change depending on the value in the current event record. Why > > >>>> doesn't this happen with $MIX? > > >>>> > > > >>>> > Third question: Is there a way to know the index of the observation > > >>>> record that is being used by > > >>>> $MIX? If I wanted to use AGE like I do in $PK it seems I must give the > > >>>> index of the current > > >>>> observation record. > > >>>> > > >>>> Nick Holford wrote: > > >>>>> I wonder if someone can explain this item in the online NONMEM help > > >>>>> guide for $MIX. > > >>>>> > > >>>>> $INPUT ... AGE ... > > >>>>> $CONTR DATA=(AGE) > > >>>>> Then AGE may be used on the right in $MIX. AGE and AGE(1) > > >>>>> both > > >>>>> refer to the value of AGE on the first observation record of > > >>>>> the > > >>>>> individual record. AGE(i) refers to the value of AGE on the > > >>>>> i- > > >>>>> th. observation record of the individual record. > > >>>>> > > >>>>> Assume there are 2 records for each subject like this > > >>>>> > > >>>>> ID AGE DV > > >>>>> 1 0 10.506 > > >>>>> 1 50 10.331 > > >>>>> 2 0 10.039 > > >>>>> 2 50 10.99 > > >>>>> 3 0 9.3782 > > >>>>> 3 50 9.9395 > > >>>>> 4 0 98.438 > > >>>>> 4 50 99.411 > > >>>>> 5 0 10.598 > > >>>>> 5 50 9.6335 > > >>>>> > > >>>>> and this code is used to simulate with a different mixing fraction if > > >>>>> AGE is less than 25 compared with AGE greater than or equal to 25: > > >>>>> > > >>>>> $PROB MIX > > >>>>> $DATA agemix.csv > > >>>>> $INPUT ID AGE DV > > >>>>> $SIM (20070730) ONLYSIM NSUB=1 > > >>>>> $THETA > > >>>>> 0.25 ; PLT25 25% of young are non-responder > > >>>>> 0.75 ; PGE25 75% of older are non-responder > > >>>>> 10 ; NONRESPONDER > > >>>>> 100 ; RESPONDER > > >>>>> $OMEGA 0.01 > > >>>>> $OMEGA 0.1 > > >>>>> $SIGMA 1 > > >>>>> > > >>>>> $PRED > > >>>>> IF (MIXNUM.EQ.1) THEN ; non-responder > > >>>>> Y=THETA(3) + ETA(1) + EPS(1) > > >>>>> ELSE ; responder > > >>>>> Y=THETA(4) + ETA(2) + EPS(1) > > >>>>> ENDIF > > >>>>> > > >>>>> $CONTR DATA=(AGE) > > >>>>> $MIX > > >>>>> NSPOP=2 > > >>>>> IF (AGE.LT.25) THEN > > >>>>> P(1)=THETA(1) ; young non-responder > > >>>>> P(2)=1-THETA(1) > > >>>>> ELSE > > >>>>> P(1)=THETA(2) ; older non-responder > > >>>>> P(2)=1-THETA(2) > > >>>>> ENDIF > > >>>>> > > >>>>> $TABLE ID AGE DV > > >>>>> NOAPPEND NOPRINT ONEHEADER FILE=agemix.fit > > >>>>> > > >>>>> I choose to define a response > 50 as a responder and <=50 as a > > >>>>> non-responder. > > >>>>> The simulated DV values (10,000 subjects) had 75% of non-responders > > >>>>> (with the same proportion at both ages). I had expected 25% because > > >>>>> AGE in $MIX is supposed to be the AGE on the first obs record i.e. > > >>>>> AGE=0. I got identical results with NONMEM VI and NONMEM V. > > >>>>> > > >>>>> First question: Why is the proportion of simulated subjects different > > >>>>> from what I expected? It seems like all the values are being > > >>>>> simulated with AGE=50 instead of AGE=0. > > >>>>> > > >>>>> Second question: More generally, if we used AGE in other subroutines > > >>>>> (e.g. $PK, $PRED) then AGE would change depending on the value in the > > >>>>> current event record. Why doesn't this happen with $MIX? > > >>>>> > > >>>>> Third question: Is there a way to know the index of the observation > > >>>>> record that is being used by $MIX? If I wanted to use AGE like I do > > >>>>> in $PK it seems I must give the index of the current observation > > >>>>> record. > > >>>>> > > >>>>> Nick > > >>>>> > > >>>>> -- > > >>>>> Nick Holford, Dept Pharmacology & Clinical Pharmacology > > >>>>> University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New > > >>>>> Zealand > > >>>>> [EMAIL PROTECTED] tel:+64(9)373-7599x86730 fax:+64(9)373-7090 > > >>>>> www.health.auckland.ac.nz/pharmacology/staff/nholford > > >>>>> > > >>>>> > > >>> -- > > >>> Nick Holford, Dept Pharmacology & Clinical Pharmacology > > >>> University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New > > >>> Zealand > > >>> [EMAIL PROTECTED] tel:+64(9)373-7599x86730 fax:+64(9)373-7090 > > >>> www.health.auckland.ac.nz/pharmacology/staff/nholford > > >>> > > >>> > > > > > > -- > > > Nick Holford, Dept Pharmacology & Clinical Pharmacology > > > University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New > > > Zealand > > > [EMAIL PROTECTED] tel:+64(9)373-7599x86730 fax:+64(9)373-7090 > > > www.health.auckland.ac.nz/pharmacology/staff/nholford > > > > > > > > -- > Nick Holford, Dept Pharmacology & Clinical Pharmacology > University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand > [EMAIL PROTECTED] tel:+64(9)373-7599x86730 fax:+64(9)373-7090 > www.health.auckland.ac.nz/pharmacology/staff/nholford