Re: WRES AND OUTLIER IDENTIFICATION/EXCLUSION

From: Nick Holford n.holford@auckland.ac.nz Subject: Re: [NMusers] WRES AND OUTLIER IDENTIFICATION/EXCLUSION Date: Monday, October 02, 2006 5:59 PM Ken, Thanks for confirming the distinction between outlier subjects (i.e. large ETA) and subjects with outliers (i.e. large EPS). While attempting to contribute to this thread I have realized the importance of distinguishing between 'outlier subjects' and 'subjects with outliers'. The $MIX code I posted earlier attempts to separate subjects with outliers from those without outliers based on the size of EPS (see again below). This is in the spirit of the discussion to identify subjects who do not fit well and therefore may be subjects with outliers regardless of any ETA adjustment. The fractional likelihood method you and Matt H. propose estimates some average fraction of outlier observations. I am still not very comfortable about the fractional likelihood method because it assumes a similar fraction of outlier and non-outlier observations in each subject (which might be addressed by having an ETA on this fraction) but more importantly (for this thread) it doesn't distinguish in a discrete way between outlier and non-outlier observations. I think we need to have something like $MIX that works at the observation level (instead of only at the subject level) because it produces a discrete classification. You refer to a "two stage approach" in your earlier emails but I am afraid I didnt really follow all the details. Here is a specific proposal using NONMEM: Step 1: Use the simple $MIX method (See Step 1 code below) to identify subjects with outliers from subjects without outliers. Save posthoc estimates of the ETA specific parameters for the next step. This step estimates 'P' (your nomenclature) i.e. the proportion of subjects with outliers and also discretely identifies (using MIXEST) which subject belongs to each group (ME1). Step 2: Code the data with ID changing for each observation and use the individual posthoc parameters (FIXED) to make subject specific predictions. Set up a $MIX model to identify the overall proportion of observation level outliers in the total set of observations (See Step 2 code below). You can estimate different residual error model parameters for outlier and non-outlier observations ('K') during this second step but the main outcome would to use MIXEST at the observation level to distinguish outlier observations (ME2). The second step can be used to to test the hypothesis that there are indeed two populations of observations i.e. outliers and non-outliers by fitting with and without $MIX and 'K'. Overall this 2 step procedure is not as elegant as doing it in one step but it could maybe achieve the goal of this thread i.e. to identify outlier observations with a somewhat objective and automatable procedure. The main problem I think is in the posthoc estimates of the parameters in Step 1 which will be somewhat contaminated with the misspecification of the outlier vs nonoutlier residual error. Perhaps it is here that the fractional likelihood method might help. Best wishes, Nick Caution: The following code has not been tested. It is intended primarily to demonstrate the ideas being discused. It might easily contain errors! ************* STEP 1 ****************** $THETA (0,0.1,1) ; P prob of being subject with outlier 1 (0,3,); Pop clearance 2 (0,10,); Pop volume 3 (0,1,) ; SD of additive residual error 4 (0,1,) ; K fractional difference in SD in subjects with outliers 5 $OMEGA 0.5 ; between subject variability in CL 0.5 ; between subject variability in V $SIGMA 1 FIX ; unit random effect $MIX NSPOP=2 P(1)=THETA(1) ; P prob of being subject with outlier P(2)=1-P(1) $PRED IF (NEWIND.EQ.0) OBID=0 ; counter for observations CL=THETA(2)*EXP(ETA(1)) V=THETA(3)*EXP(ETA(2)) MYPRED=DOSE/V*EXP(-CL/V*TIME) ;SD= ... any residual error model you want expressed with THETAs e.g SD=THETA(4) ; additive residual error SD IF (MIXNUM.EQ.1) THEN ; subject with outliers KOUT=THETA(3) ELSE KOUT=1 ENDIF Y=MYPRED+KOUT*SD*EPS(1) IF (MDV.EQ.0) OBID=OBID+1 ; create observation level ID ME1=MIXEST $TABLE ID ME1 OBID TIME DOSE CL V DV MDV NOAPPEND NOPRINT ONEHEADER FILE=step1.fit ************* STEP 2 ****************** $DATA step1.fit IGNORE=@ ;OBID is an observation level ID type of data item ;ICL and IV are the posthoc individual estimates of CL and V from Step 1 $INPUT SID ME1 ID=OBID TIME DOSE ICL IV DV MDV $THETA (0,0.1,1) ; proportion of obs which are outliers 1 (0,1,) ; SD of additive residual error 2 (0,1,) ; K fractional difference in SD in outlier population 3 $OMEGA 0 FIX ; keep NM-TRAN happy (no OMEGAs needed in this step $SIGMA 1 FIX ; unit random effect $MIX NSPOP=2 P(1)=THETA(1) ; proportion of obs which are outliers P(2)=1-P(1) $PRED MYPRED=DOSE/IV*EXP(-ICL/IV*TIME) SD=THETA(2) ; additive residual error SD IF (MIXNUM.EQ.1) THEN ; outlier observation KOUT=THETA(3) ELSE KOUT=1 ENDIF Y=MYPRED+KOUT*SD*EPS(1) ME2=MIXEST $TABLE SID ID ME1 ME2 -- Nick Holford, Dept Pharmacology & Clinical Pharmacology University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand email:n.holford@auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556 http://www.health.auckland.ac.nz/pharmacology/staff/nholford/