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 Exponential Model. (Version: 1.9; Date: 01/29/2013) 

 Input Data File: C:/USEPA/BMDS240/Data/exp_Ray 14wksubC in rats - 49d RBC 

M_Setting.(d) 

 Gnuplot Plotting File: 

 Sun May 25 17:54:13 2014 

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 BMDS Model Run - Rat 98d subchronic - 49d Mrat RBC 

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 The form of the response function by Model: 

 Model 2: Y[dose] = a * exp{sign * b * dose} 

 Model 3: Y[dose] = a * exp{sign * (b * dose)^d} 

 Model 4: Y[dose] = a * [c-(c-1) * exp{-b * dose}] 

 Model 5: Y[dose] = a * [c-(c-1) * exp{-(b * dose)^d}] 

 Note: Y[dose] is the median response for exposure = dose; 

 sign = +1 for increasing trend in data; 

 sign = -1 for decreasing trend. 

 Model 2 is nested within Models 3 and 4. 

 Model 3 is nested within Model 5. 

 Model 4 is nested within Model 5. 

 Dependent variable = Mean 

 Independent variable = Dose 

 Data are assumed to be distributed: normally 

 Variance Model: exp(lnalpha +rho *ln(Y[dose])) 

 The variance is to be modeled as Var(i) = exp(lalpha + log(mean(i)) * rho) 

 Total number of dose groups = 4 

 Total number of records with missing values = 0 

 Maximum number of iterations = 500 

 Relative Function Convergence has been set to: 1e-008 

 Parameter Convergence has been set to: 1e-008 

 MLE solution provided: Exact 

 Initial Parameter Values 

 Variable Model 2 Model 3 Model 4 Model 5 

 -------- ------- ------- ------- ------- 

 lnalpha -2.98213 -2.98213 -2.98213 -2.98213 

 rho 1.91103 1.91103 1.91103 1.91103 

 a 803.413 803.413 1254.75 1254.75 

 b 0.0386515 0.0386515 0.208832 0.208832 

 c -- -- 0.477424 

0.477424 

 d -- 1 -- 1 

 Parameter Estimates by Model 

 Variable Model 2 Model 3 Model 4 Model 5 
 -------- ------- ------- ------- ------- 

 lnalpha -6.33806 -6.33806 -4.07553 -4.25438 

 rho 2.42906 2.42906 2.06272 2.08851 

 a 1111.36 1111.36 1195.44 1183.41 

 b 0.0386655 0.0386655 0.493411 0.561031 

 c -- -- 0.526705 0.532594 

 d -- 1 -- 1.87706 

 Table of Stats From Input Data 

 Dose N Obs Mean Obs Std Dev 

 ----- --- ---------- ------------- 

 0 10 1195 163.1 

 0.1 10 1166 244.3 

 1.5 10 903 138 

 15 10 629 109.3 

 Estimated Values of Interest 

 Model Dose Est Mean Est Std Scaled Residual 

 ------- ------ ---------- --------- ---------------- 

 2 0 1111 210.4 1.257 

 0.1 1107 209.4 0.89 

 1.5 1049 196.1 -2.351 

 15 622.3 104 0.2049 

 3 0 1111 210.4 1.257 

 0.1 1107 209.4 0.89 

 1.5 1049 196.1 -2.351 

 15 622.3 104 0.2049 

 4 0 1195 194.6 -0.007155 

 0.1 1168 190 -0.03663 

 1.5 899.6 145.1 0.07492 

 15 630 100.5 -0.03115 

 5 0 1183 192.9 0.19 

 0.1 1181 192.5 -0.2454 

 1.5 898.6 144.7 0.09594 

 15 630.3 99.91 -0.04048 

 Other models for which likelihoods are calculated: 

 Model A1: Yij = Mu(i) + e(ij) 

 Var{e(ij)} = Sigma^2 

 Model A2: Yij = Mu(i) + e(ij) 

 Var{e(ij)} = Sigma(i)^2 

 Model A3: Yij = Mu(i) + e(ij) 

 Var{e(ij)} = exp(lalpha + log(mean(i)) * rho) 

 Model R: Yij = Mu + e(i) 

 Var{e(ij)} = Sigma^2 

 Likelihoods of Interest 

 Model Log(likelihood) DF AIC 
 ------- ----------------- ---- ------------ 

 A1 -223.6124 5 457.2247 

 A2 -220.0339 8 456.0678 

 A3 -221.0093 6 454.0186 

 R -245.5007 2 495.0014 

 2 -226.1592 4 460.3184 

 3 -226.1592 4 460.3184 

 4 -221.0528 5 452.1056 

 5 -221.0093 6 454.0186 

 Additive constant for all log-likelihoods = -36.76. This constant added to the 

 above values gives the log-likelihood including the term that does not 

 depend on the model parameters. 

 Explanation of Tests 

 Test 1: Does response and/or variances differ among Dose levels? (A2 vs. R) 

 Test 2: Are Variances Homogeneous? (A2 vs. A1) 

 Test 3: Are variances adequately modeled? (A2 vs. A3) 

 Test 4: Does Model 2 fit the data? (A3 vs. 2) 

 Test 5a: Does Model 3 fit the data? (A3 vs 3) 

 Test 5b: Is Model 3 better than Model 2? (3 vs. 2) 

 Test 6a: Does Model 4 fit the data? (A3 vs 4) 

 Test 6b: Is Model 4 better than Model 2? (4 vs. 2) 

 Test 7a: Does Model 5 fit the data? (A3 vs 5) 

 Test 7b: Is Model 5 better than Model 3? (5 vs. 3) 

 Test 7c: Is Model 5 better than Model 4? (5 vs. 4) 

 Tests of Interest 

 Test -2*log(Likelihood Ratio) D. F. p-value 

 -------- ------------------------ ------ -------------- 

 Test 1 50.93 6 < 0.0001 

 Test 2 7.157 3 0.06706 

 Test 3 1.951 2 0.377 

 Test 4 10.3 2 0.0058 

 Test 5a 10.3 2 0.0058 

 Test 5b -2.217e-012 0 N/A 

 Test 6a 0.08697 1 0.7681 

 Test 6b 10.21 1 0.001395 

 Test 7a -2.842e-013 0 N/A 

 Test 7b 10.3 2 0.0058 

 Test 7c 0.08697 1 0.7681 

 The p-value for Test 1 is less than .05. There appears to be a 

 difference between response and/or variances among the dose 

 levels, it seems appropriate to model the data. 

 The p-value for Test 2 is less than .1. A non-homogeneous 

 variance model appears to be appropriate. 

 The p-value for Test 3 is greater than .1. The modeled 

 variance appears to be appropriate here. 
 The p-value for Test 4 is less than .1. Model 2 may not adequately 

 describe the data; you may want to consider another model. 

 The p-value for Test 5a is less than .1. Model 3 may not adequately 

 describe the data; you may want to consider another model. 

 Degrees of freedom for Test 5b are less than or equal to 0. 

 The Chi-Square test for fit is not valid. 

 The p-value for Test 6a is greater than .1. Model 4 seems 

 to adequately describe the data. 

 The p-value for Test 6b is less than .05. Model 4 appears 

 to fit the data better than Model 2. 

 Degrees of freedom for Test 7a are less than or equal to 0. 

 The Chi-Square test for fit is not valid. 

 The p-value for Test 7b is less than .05. Model 5 appears 

 to fit the data better than Model 3. 

 The p-value for Test 7c is greater than .05. Model 5 does 

 not seem to fit the data better than Model 4. 

 Benchmark Dose Computations: 

 Specified Effect = 0.100000 

 Risk Type = Relative deviation 

 Confidence Level = 0.950000 

 BMD and BMDL by Model 

 Model BMD BMDL 

 ------- ------------ ---------- 

 2 2.72492 2.25722 

 3 2.72493 2.25722 

 4 0.481039 0.318256 

 5 0.834692 0.320258 
 500

 600 700 800 900

 1000 1100 1200 1300 1400

 0 2 4 6 8 10 12 14

Mean Response

dose

Exponential Model 2, with BMR of 0.1 Rel. Dev. for the BMD and 0.95 Lower Confidence Level for BMDL

17:54 05/25 2014

BMDL BMD

Exponential

 500 600 700 800 900

 1000 1100 1200 1300 1400

 0 2 4 6 8 

Mean Response

dose

Exponential Model 3, with BMR of 0.1 Rel. Dev. for the BM

17:54 05/25 2014

BMDL BMD

Exponential

 500 600 700 800 900

 1000 1100 1200 1300 1400

 0 2 4 6 8 10 12 14

Mean Response

dose

Exponential Model 4, with BMR of 0.1 Rel. Dev. for the BMD and 0.95 Lower Confidence Level for BMDL

17:54 05/25 2014

BMDL BMD

Exponential

 500 600 700 800 900

 1000 1100 1200 1300 1400

 0 2 4 6 8 

Mean Response

dose

Exponential Model 5, with BMR of 0.1 Rel. Dev. for the BM

17:54 05/25 2014

BMDL BMD

Exponential