EFSA Guidance for BMD analysis Fitting Models & Goodness of Fit

Transcription

1 EFSA Guidace for BMD aalysis Fittig Models & Goodess of Fit 1 st March 2017

2 OUTLINE Geeral priciples of model fittig & goodess of fit Cotiuous dose-respose data Cell proliferatio (CP) data Cadidate models Illustratio Quatal respose data Thyroid epithelial cell vacuolisatio (TECV) data Cadidate models Illustratio 2

3 GENERAL PRINCIPLES Data vary accordig to a distributio Cotiuous: log-ormal with mea ad variace Quatal: biomial with probability parameter Model the mea parameter as fuctio of dose Fit the model to data ad get estimates (ML) Compare differet cadidate models usig a goodess-of-fit criterio (AIC) 3

4 CONTINOUS RESPONSE DATA Cell Proliferatio Example 4

5 CELL PROLIFERATION Local Lymph Node Assay (LLNA) test Cell proliferatio as a idicator of sesitizatio Related to 15 rubber chemicals, focus o oe chemical Dose levels ad umber of observatios per level 5

6 CELL PROLIFERATION Data of dose ad cell proliferatio o origial scale 6

7 CELL PROLIFERATION Data of cell proliferatio o log scale 7

8 CELL PROLIFERATION Data of dose ad cell proliferatio both o log scale 8

9 CELL PROLIFERATION Null model of o effect Full model: each dose level its ow mea 9

10 CELL PROLIFERATION Equal variaces: Levee s test (p=0.89) Normal distributio: Shapiro-Wilk (p=0.42) 10

11 CELL PROLIFERATION Cadidate models for cotiuous data are models for the mea of the ormal distributio o log-scale 11

12 CELL PROLIFERATION 3 parameter Expoetial model 12

13 MAXIMUM LIKELIHOOD ESTIMATION & INFERENCE Maximum Likelihood (ML) estimatio & iferece Values for a, b ad d that maximize the likelihood The likelihood of the data you observed Maximize likelihood = maximize log-likelihood 13

14 CELL PROLIFERATION Maximum Likelihood (ML) estimatio & iferece ML (most likely) values for the parameters Less likely values for the parameters 14

15 MAXIMUM LIKELHOOD ESTIMATION & INFERENCE ML is idetical to least squares estimatio i case of the ormal distributio (possibly o log-scale) Difficulty for o-liear models: multiple local maxima ad covergece issues 15

16 CELL PROLIFERATION 4 parameter Expoetial model 16

17 CELL PROLIFERATION 3 parameter Hill model 17

18 CELL PROLIFERATION 4 parameter Hill model 18

19 AKAIKE INFORMATION CRITERION AIC stems from iformatio theory Measures the iformatio lost by replacig the true ukow data geeratig model by the model used Estimates the Kullback-Leibler distace The smaller AIC the better the model Defied as AIC = -2 loglikelihood + 2 #par The lower -2 loglikelihood the closer the fit to the data Pealizatio for overfittig, for too complex models AIC balaces accuracy with complexity 19

20 CELL PROLIFERATION Model Log-likelihood # par Null Full EXP EXP HILL HILL # par = umber of parameters i mea model + variace parameter 20

21 CELL PROLIFERATION Model Log-likelihood # par AIC Null Full EXP EXP HILL HILL AIC Mi = < AIC FULL + 2 = All AIC < AIC NULL 2 =

22 CELL PROLIFERATION Model averaged fitted curve Model AIC Weight EXP EXP HILL HILL

23 CELL PROLIFERATION Just oe compoud, but there are 15 compouds Approach: iclude compoud as covariate, leadig to a 4 parameter expoetial model, with compoud specific values for a ad b ad commo values for c ad d 23

24 QUANTAL RESPONSE DATA thyroid epithelial cell vacuolisatio 24

25 THYROID EPITHELIAL CELL VACUOLISATION 2 year study i rats Three doses of a substace Chages i thyroid epithelial cell vacuolisatio 25

26 THYROID EPITHELIAL CELL VACUOLISATION Null model of o effect Full model: each dose level its ow probability 26

27 THYROID EPITHELIAL CELL VACUOLISATION Biomial distributio essetially the oly choice Extesios: beta-biomial model & hierarchical data models 27

28 THYROID EPITHELIAL CELL VACUOLISATION Cadidate models for quatal data 28

29 THYROID EPITHELIAL CELL VACUOLISATION Logistic model 29

30 THYROID EPITHELIAL CELL VACUOLISATION Probit model 30

31 THYROID EPITHELIAL CELL VACUOLISATION Log-logistic model 31

32 THYROID EPITHELIAL CELL VACUOLISATION Log-probit model 32

33 THYROID EPITHELIAL CELL VACUOLISATION Weibull model 33

34 THYROID EPITHELIAL CELL VACUOLISATION Gamma model 34

35 THYROID EPITHELIAL CELL VACUOLISATION LMS model 35

36 THYROID EPITHELIAL CELL VACUOLISATION Model Log-likelihood # par AIC Null Full Logistic Probit Log-logistic Log-probit Weibull Gamma LMS AIC Mi = < AIC FULL + 2 = All AIC < AIC NULL 2 =

37 THYROID EPITHELIAL CELL VACUOLISATION Next presetatio Model Averagig ad BMD estimatio 37

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