1 Modeling and Estimation of Benchmark Dose (BMD) for Binary Response Data Wei Xiong

2 Outline Benchmark dose (BMD) and datasets Statistical models logistic probit multi–stage gamma multi–hit Model fitting and analyses Conclusions

3 where, ADI: acceptable daily intake SF: safety factor In environment risk assessment, NOAEL (no-observed-adverse-effect level) is used to derive a safe dose,

4 (Filipsson et al., 2003) Problem:

5 BMD: Point estimate of the dose which induces a given response (e.g. 10%) above unexposed controls BMDL: 1–sided 95% confidence lower limit for BMD Benchmark dose (BMD)

6 Fit a model to all data Estimate the BMD from a given BMR (10%) Derive “safe dose” from BMD (Filipsson et al., 2003) Advantage: BMD uses all the data information by fitting a model

7 Non–cancer data Ryan and Van (1981) ididi riri nini mice in each dose group drug: botulinum toxin in 10 –15 gram response: death (Y or N) within 24 hrs

8 Plot of non-cancer Data

9 Cancer data Bryan and Shimkin (1943) id i r i n i to 21 mice in each dose group drug: carcinogenic methylcholanthrene in 10 –6 gram response: tumor (Y or N)

10 Plot of cancer data

11 How to estimate BMD ? What models to be used ? Need to use different models for the cancer and non-cancer data How to fit the model curve

12 Statistical models Logistic Probit Multi–stage Gamma multi–hit Model form: where, 1>  >=0 is the background response as dose  0 F is the cumulative dist’n function

13 Probit model Assuming: log(d) is approx. normally distributed

14 Logistic model Assuming: log(d) has a logistic distribution

15 Multi–stage model (Crump, 1981) Assuming: 1.Ordered stages of mutation, initiation or transformation for a cell to become a tumor 2.Probability of tumor occurrence at j th stage is proportional to dose by  j d j

16 Gamma multi–hit model (Rai and Van, 1981) Assuming: a tumor incidence is induced by at least   1 hits of units of dose and  follows a Poisson distribution The gamma model is derived from the Poisson dist’n of 

17 Model fitting Models are fit by maximum likelihood method Model fitting tested by Pearson’s  2 statistic If p-value  10%, the model fits the data well and the mle of BMD is obtained from the fitted model where, is estimated from the fitted model

18 BMDL by LRT (Crump and Howe, 1985) where,  and  are model parameters  P is the log(BMD) at response = p

19 The BMDL is the value  P, which is lower than the mle, so that,

20 BMDL by Fieller’s Theorem (Morgan, 1992) Fieller’s Theom constructs CI for the ratio of R.V. For logistic model, the BMDL is derived as, where,

21 BMDL computation BMDS (benchmark dose software, US EPA) provides the 4 models for BMDL using LRT S–Plus calculates BMDL using LRT and Fieller’s Theorem

22 BMDS logistic modeling for non–cancer data (Pearson’s  2, p = > 0.1)

23 BMDS multi–stage modeling for non–cancer data (Pearson’s  2, p = )

24 BMDS two–stage modeling for cancer data (Pearson’s  2, p = 0.556)

25 BMDL=1.536 by LRT (Probit model for cancer data)

26 SoftwareModel LogisticProbit BMDS S–plus MLE of BMD (non–cancer data) ( p–value by Pearson’s  2 ) (0.325)(0.386)

27 Summary of BMDL (non–cancer data) MethodsSoftwareModel LogisticProbit LRT BMDS S–plus Fieller’s Theorem S–plus

28 SoftwareModel LogisticProbitTwo– stage Multi– hit BMDS S–plus MLE of BMD (cancer data) # p–value by Pearson’s  2 # 0.585# 0.666# 0.556# 0.602

29 MethodsSoftware Model LogisticProbit Two– stage Multi– hit LRT BMDS S–plus Fieller’s Theorem S–plus Summary of BMDL (cancer data)

30 Conclusions Non–cancer data, BMD = (logistic) and (probit) in 10 –15 gram; cancer data, BMD = (logistic), (probit), (multi–stage) and (multi–hit). Logistic and probit model fit both data sets well, multi– stage and multi–hit fit only the cancer data well. BMDL obtained by Fieller’s Theorem seems to be smaller than that by LRT, why ?

31 Questions ?

32 A note on qchisq( ) of 1 – sided 95% > (qnorm( ))^2 [1] > qchisq(1 - 2 * 0.05, 1) [1]

33 95% CI for proportion in slides 21 & 22 When n is large, nP  5 and n(1-P)  5, the sample proportion p is used to infer underlying proportion P. p is approximately normal with mean P and s.e.=sqrt(P(1-P)/n) Solving the following equation,

34 Fitted and re–parameterized model Fitted logistic model Re-parameterized logistic model where,

35 Abbott’s Formula where, P – observed response c – response at dose zero BMR – benchmark response with default value 10%