THE IMPORTANCE OF THE SIMULATION EXPECTATION AS A GOODNESS OF FIT DIAGNOSTIC FOR CATEGORICAL POPULATION PHARMACODYNAMIC MODELS
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1 THE IMPORTANCE OF THE SIMULATION EXPECTATION AS A GOODNESS OF FIT DIAGNOSTIC FOR CATEGORICAL POPULATION PHARMACODYNAMIC MODELS Marc R. Gastonguay, Jeffrey T. Hane Metrum Research Group LLC, Avon, CT 06001
2 Which Model is the Base Model? MODEL A MODEL B Observed Scores Predicted Conditional Expectation Copyright 004 metrum research group LLC Observed Scores Predicted Simulation Expectation
3 Answer: Both are the Base Model MODEL A MODEL B Observed Scores Conditional Expectation Observed Scores Simulation Expectation 3 Copyright 004 metrum research group LLC
4 Introduction Most goodness-of-fit (GOF) diagnostics for analysis of categorical population (POP) pharmacodynamic (PD) data using nonlinear mixed effects models (NLMEM) involve comparisons of observed and predicted probabilities under the conditional expectation (CE) from the model. The simulation expectation (SE) has also been proposed as a GOF diagnostic. 4 Copyright 004 metrum research group LLC
5 CE vs. SE Conditional Expectation (CE): predictions based on conditional estimates of individual random effects (ETAs) given population model and individual data Simulation Expectation (SE): predictions based on randomly simulated ETAs from estimated population inter-individual variance distribution and population model 5 Copyright 004 metrum research group LLC
6 Objective To compare the performance of CE and SE GOF diagnostics for ordered-categorical POP PD models Copyright 004 metrum research group LLC 6
7 Methods Simulations of a hypothetical POP PD database (400 individuals, 930 total PD observations) for an ordered categorical response were conducted under the assumptions of true drug and covariate effects. Simulated data were subsequently analyzed using both a null effect (base model) and a full covariate model with NLMEM. For each simulation, GOF diagnostic plots were created using the CE and the SE. Copyright 004 metrum research group LLC 7
8 Estimation Methods Population PD Modeling via Nonlinear Mixed- Effects (NONMEM software with Laplacian estimation method) Proportional odds model for ordinal categorical data implemented in mixed-effects modeling context Cumulative probabilities across score levels were described with cumulative logit functions. M-1 cumulative logits for an M-category response 8 Copyright 004 metrum research group LLC
9 Cumulative Logit Model for Ordered Categorical Response g{ P(Y t logit(p) = m η)} = logit(p) Log[p/(1 p)] = f m + η p = fm exp 1+ exp + η f m + η where : η~n( 0,ω ) f m = m L= 1 α L i + f (AUC,Treatment, covariates, etc.) 9 Copyright 004 metrum research group LLC
10 Results The CE diagnostics consistently revealed an overly-optimistic GOF (Figure 1). When the base model was used for estimation, bias in GOF description was up to +300% for the lowest categorical score. The SE resulted in a more realistic depiction of model GOF. Copyright 004 metrum research group LLC 10
11 Figure 1: Base Model Diagnostics Observed Scores Conditional Expectation Observed Scores Simulation Expectation 11 Copyright 004 metrum research group LLC
12 Figure : Covariate Effects-Base Model 3 ETA1 ETA AUC (ng*hr/ml) Biomarker (ng/ml) ETA1 ETA Age (years) 0 1 Treatment (0=Placebo, 1=Active) 1 Copyright 004 metrum research group LLC
13 Figure 3: Final Model Diagnostics Observed Scores Conditional Expectation Observed Scores Simulation Expectation 13 Copyright 004 metrum research group LLC
14 Figure 4: Covariate Effects-Final Model ETA1 ETA Drug AUC (ng*hr/ml) Biomarker AUC (ng*hr/ml) 4 ETA1 ETA Age (years) 0 1 Treatment (0=Placebo, 1=Active) 14 Copyright 004 metrum research group LLC
15 Figure 5: Random Effects Distributions ETA1 (Base Model) ETA1 (Final Model) 15 Copyright 004 metrum research group LLC
16 Conclusions For ordered categorical POP PD models, the SE is a more reliable and accurate descriptor of GOF than the CE; the CE should not be used for this purpose. The CE is useful for identifying covariate-related trends in random effects (Figures & 4) and for checking assumptions about distributions of random effects (Figure 5). Similar results are expected with other categorical or odd-type POP PD models. Copyright 004 metrum research group LLC 16
17 NMTRAN Control Stream: Simulation $PROB Outcome Score PD Model: FULL MODEL SIM $INPUT C DV VISD ID TRT SEX WT AGE BLIN BIO AUC OBS $DATA PD5.CSV IGNORE=C $PRED ; ORDINAL CATEGORICAL RESPONSE ; CUMULATIVE ODDS RATIO MODEL ; (SEE AGRESTI, A. CATEGORICAL ; DATA ANALYSIS WILEY. PP ; ) ; SCALE = 0, 1,, 3, 4 (OUTCOME ; SCORE RANGING FROM GOOD [0] TO ; WORST ; POSSIBLE [4]) ; BENEFICIAL DRUG EFFECT ; INCREASES PROBABILITY OF LOW ; SCORES ; cutpoint values (alpha[j] in Agresti ; notation) AL0 = THETA(1) ;SCORE <=0 AL1 = AL0+THETA() ;SCORE <=1 AL = AL1+THETA(3) ;SCORE <= AL3 = AL+THETA(4) ;SCORE <=3 ; CONCENTRATION-RESPONSE ; relationship (beta*x in Agresti notation) ; TRT 1=DRUG, 0=PLACEBO ; BLIN 0 =LOW, 1=HIGH BASELINE ; SYMPTOMS ; BIO IS BIOMARKER AUC ; AGE (YEARS) ; AUC = DRUG AUC E= THETA(5)*BLIN + THETA(6)*BIO + THETA(7)*AGE +THETA(8)*AUC 17 Copyright 004 metrum research group LLC
18 NMTRAN Control Stream: Simulation ; logits for cumulative probabilities ; X IS PREDICTOR VARIABLE ; J = 0,1,,3 AND IS DENOTED BY [J] LGT0 = AL0+E+ETA(1) ;LGT[0](X) LGT1 = AL1+E+ETA(1) ;LGT[1](X) LGT = AL+E+ETA(1) ;LGT[](X) LGT3 = AL3+E+ETA(1) ;LGT[3](X) ; exponentiate logit C0 = EXP(LGT0) C1 = EXP(LGT1) C = EXP(LGT) C3 = EXP(LGT3) ; CUMULATIVE PROBABILITIES P0 = C0/(1+C0) ;P(Y<=0 X) P1 = C1/(1+C1) ;P(Y<=1 X) P = C/(1+C) ;P(Y<= X) P3 = C3/(1+C3) ;P(Y<=3 X) ; LIKELIHOOD FOR CATEGORY=J Y0=P0 Y1=P1-P0 Y=P-P1 Y3=P3-P Y4=1-P3 ; SIMULATION EXPECTATION IF(ICALL.EQ.4) THEN ;simulation CALL RANDOM (,R) IF(R.LE.P0) DV=0 IF(R.GT.P0.AND.R.LE.P1) DV=1 IF(R.GT.P1.AND.R.LE.P) DV= IF(R.GT.P.AND.R.LE.P3) DV=3 IF(R.GT.P3) DV=4 X=R ENDIF 18 Copyright 004 metrum research group LLC
19 NMTRAN Control Stream: Simulation IND0=0 IND1=0 IND=0 IND3=0 IND4=0 IF (DV.EQ.0) IND0=1 IF (DV.EQ.1) IND1=1 IF (DV.EQ.) IND=1 IF (DV.EQ.3) IND3=1 IF (DV.EQ.4) IND4=1 IF(BIO.LE.50) BIOM=1 IF(BIO.GT.50.AND.BIO.LE.100) BIOM= IF(BIO.GT.100.AND.BIO.LE.150) BIOM=3 IF(BIO.GT.150) BIOM=4 IF(AUC.EQ.0) ACAT=1 IF(AUC.GT.0.AND.AUC.LE.0000) ACAT= IF(AUC.GT.0000.AND.AUC.LE.40000) ACAT=3 IF(AUC.GT AND.AUC.LE.60000) ACAT=4 IF(AUC.GT AND.AUC.LE.80000) ACAT=5 IF(AUC.GT.80000) ACAT=6 Y=Y0*IND0+Y1*IND1+Y*IND+Y3*IND3+ Y4*IND4 $MSFI 007.msf ; from estimation run $SIM (95465) (546 UNIFORM) ONLY SUB=100 NOPRED $TABLE NOPRINT NOHEADER FILE=008.TAB ID ETA(1) P0 P1 P P3 Y0 Y1 Y Y3 Y4 VISD BIO TRT BLIN AGE AUC BIOM ACAT OBS 19 Copyright 004 metrum research group LLC
20 References Agresti, A. Categorical Data Analysis. John Wiley & Sons Inc., New York, Kalbfleisch J.G., Probability and Statistical Inference, Volume 1: Probability. nd Edition. Springer-Verlag, New York, Sheiner, L.B. A New Approach to the Analysis of Analgesic Drug Trials, Illustrated with Bromfenac Data. Clin. Pharmacol. Ther. 1994; 56: Ware L.H., and Lipsitz, S. Issues in the Analysis of Repeated Categorical Outcomes. Statistics in Medicine. Vol. 7: , For copies of this poster, please visit Copyright 004 metrum research group LLC 0
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