Document ID: EPA-HQ-OPPT-2012-0725-0053
Agency: epa
Document Type: Supporting & Related Material
Title: 
Posted Date: 2013-11-14T05:00Z

BMD Analysis (Cmax)

                                 Iris Camacho
                                Paul Schlosser

Saillenfait et al. 2003, fetal body weights

                             Internal Dose, Cmax 
                                    (mg/L)
                                   # litters
                                 Fetal BW (g)
                              Standard Deviation
                                       0
                                      24
                                     5.671
                                     0.37
                                     11.90
                                      20
                                     5.623
                                     0.358
                                     24.05
                                      19
                                     5.469
                                     0.252
                                     49.57
                                      25
                                     5.393
                                     0.446

                                  Model Type
                                   Risk Type
                                    P-Value
                                      AIC
                                 BMD (BMR= 5%)
                                BMDL (BMR= 5%)
                                      BMD
                                  (BMR= 10%)
                                     BMDL
                                  (BMR= 10%)
                                   Comments
                               Exponential (M2)
                                   Rel. Dev.
                                    0.9337
                                    -83.656
                                     47.8
                                     30.2
                                     98.3
                                     62.0
                                       
                               Exponential (M3)
                                   Rel. Dev.
                                    0.7594
                                    -81.699
                                     48.6
                                     30.3
                                     90.1
                                     51.3
                                       
                               Exponential (M4)
                                   Rel. Dev.
                                    0.8351
                                    -83.656
                                     47.8
                                     20.9
                                     98.3
                                     62.0
               Same best fit, but more fitted parameters than M2
                               Exponential (M5)
                                   Rel. Dev.
                                    0.9337
                                    -79.793
                                     47.8
                                     12.6
                                     error
                                     error
                    Model did not compute BMD for BMR = 10%
                                     Hill
                                   Rel. Dev.
                                      NA
                                    -79.793
                                     48.5
                                     13.1
                                     error
                                     error
                     Chi-square test for fit is not valid.
                    Model did not compute BMD for BMR = 10%
                                     Power
                                   Rel. Dev.
                                    0.7562
                                    -81.697
                                     48.6
                                     30.7
                                     88.9
                                     51.2
                                       
                                Polynomial 3°
                                   Rel. Dev.
                                      NA
                                    -79.793
                                     42.5
                                     19.7
                                     error
                                     error
                     Chi-square test for fit is not valid.
                    Model did not compute BMD for BMR = 10%
                                Polynomial 2°
                                   Rel. Dev.
                                    0.7367
                                    -81.680
                                     48.5
                                     19.7
                                     85.9
                                     57.6
                                       
                                    Linear
                                   Rel. Dev.
                                    0.9366
                                    -83.662
                                     47.9
                                     30.7
                                     95.8
                                     61.3
                                       

Among models with valid fits, the linear model has the lowest AIC and the BMDL values varied less than three fold.   Therefore the linear model is selected.

BMDL05=30.7 mg-hr/L was selected for the analysis.

Model output files for BMR=5%

==================================================================== 
   	  Exponential Model. (Version: 1.9;  Date: 01/29/2013) 
 	  Input Data File: C:/Bmds/BMDS240/Data/exp_NMP_Saillenfait_BW_Cmax_Setting.(d)  
  	  Gnuplot Plotting File:  
 							Thu Sep 12 14:49:40 2013
 ==================================================================== 
 BMDS Model Run 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
   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         -0.761471         -0.761471           -0.761471         -0.761471
         rho         -0.784145         -0.784145           -0.784145         -0.784145
           a           5.40995           5.40995             5.95455           5.95455
           b        0.00105023        0.00105023            0.026838          0.026838
           c                --                --               0.862566          0.862566
           d                --                 1                  --                 1

                               Parameter Estimates by Model
     Variable          Model 2           Model 3           Model 4           Model 5
     --------          -------           -------           -------           -------
     lnalpha        10.4028           11.4716             10.4028           9.88712
         rho          -7.27186          -7.89679            -7.27186          -6.97149
           a           5.66815           5.65943             5.66815           5.65863
           b        0.00107208        0.00161294          0.00107208          0.034696
           c             --                --                      0          0.947571
           d             --           1.16651                  --           2.22275

            Table of Stats From Input Data
     Dose      N         Obs Mean     Obs Std Dev
     -----    ---       ----------   -------------
         0     24        5.671         0.37
      11.9     20        5.623        0.358
     24.05     19        5.469        0.252
     49.57     25        5.393        0.446

                      Estimated Values of Interest
      Model      Dose      Est Mean      Est Std     Scaled Residual
     -------    ------    ----------    ---------    ----------------
          2         0         5.668       0.3307          0.04223
                 11.9         5.596       0.3464           0.3447
                24.05         5.524       0.3632          -0.6585
                49.57         5.375       0.4012           0.2269
          3         0         5.659       0.3302           0.1716
                 11.9         5.603       0.3434           0.2543
                24.05         5.533        0.361          -0.7737
                49.57          5.37       0.4062           0.2832
          4         0         5.668       0.3307          0.04223
                 11.9         5.596       0.3464           0.3447
                24.05         5.524       0.3632          -0.6585
                49.57         5.375       0.4012           0.2269
          5         0         5.659       0.3336           0.1816
                 11.9          5.62       0.3417          0.04083
                24.05         5.514       0.3651          -0.5367
                49.57         5.372       0.3997           0.2565

   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        45.53621            5     -81.07243
                        A2        48.82817            8     -81.65634
                        A3        45.89652            6     -79.79305
                         R        41.22097            2     -78.44193
                         2        45.82797            4     -83.65594
                         3        45.84964            5     -81.69927
                         4        45.82797            4     -83.65594
                         5        45.89652            6     -79.79305

   Additive constant for all log-likelihoods =     -80.87.  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                         15.21           6             0.01865
     Test 2                         6.584           3             0.08641
     Test 3                         5.863           2             0.05331
     Test 4                        0.1371           2              0.9337
    Test 5a                       0.09377           1              0.7594
    Test 5b                       0.04333           1              0.8351
    Test 6a                        0.1371           2              0.9337
    Test 6b                     2.26e-012           0                 N/A
    Test 7a                   -2.127e-011           0                 N/A
    Test 7b                       0.09377           1              0.7594
    Test 7c                        0.1371           2              0.9337

     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 less than .1.  You may want to consider a different variance model.

     The p-value for Test 4 is greater than .1.  Model 2 seems to adequately describe the data.

     The p-value for Test 5a is greater than .1.  Model 3 seems to adequately describe the data.

     The p-value for Test 5b is greater than .05.  Model 3 does not seem to fit the data better than Model 2.

     The p-value for Test 6a is greater than .1.  Model 4 seems to adequately describe the data.

     Degrees of freedom for Test 6b are less than or equal to 0.  The Chi-Square test for fit is not valid.

     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 greater than .05.  Model 5 does not seem 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 = 0.050000
            Risk Type = Relative deviation
     Confidence Level = 0.950000
                BMD and BMDL by Model
      Model             BMD                BMDL
     -------        ------------        ----------
        2             47.8448            30.1693
        3             48.5931            30.2615
        4             47.8448            20.9141
        5             47.7531            12.5535

==================================================================== 
   	  Hill Model. (Version: 2.17;  Date: 01/28/2013) 
  	  Input Data File: C:/Bmds/BMDS240/Data/hil_NMP_Saillenfait_BW_Cmax_Opt.(d)  
  	  Gnuplot Plotting File:  C:/Bmds/BMDS240/Data/hil_NMP_Saillenfait_BW_Cmax_Opt.plt
 							Thu Sep 12 15:07:41 2013
 ==================================================================== 
 BMDS Model Run 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
   The form of the response function is: 
 	  Y[dose] = intercept + v*dose^n/(k^n + dose^n)
   Dependent variable = Mean
   Independent variable = Dose
   Power parameter restricted to be greater than 1
   The variance is to be modeled as Var(i) = exp(lalpha  + rho * ln(mean(i)))

   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

                  Default Initial Parameter Values  
                         lalpha =     -1.98839
                            rho =            0
                      intercept =        5.671
                              v =       -0.278
                              n =      3.77069
                              k =      19.0795

           Asymptotic Correlation Matrix of Parameter Estimates
                 lalpha          rho    intercept            v            n            k
    lalpha            1           -1        -0.35        -0.24        -0.19         0.43
       rho           -1            1         0.36         0.24         0.19        -0.43
 intercept        -0.35         0.36            1        -0.31        -0.34       -0.017
         v        -0.24         0.24        -0.31            1         0.94        -0.92
         n        -0.19         0.19        -0.34         0.94            1        -0.85
         k         0.43        -0.43       -0.017        -0.92        -0.85            1

                                 Parameter Estimates
                                                         95.0% Wald Confidence Interval
       Variable         Estimate        Std. Err.     Lower Conf. Limit   Upper Conf. Limit
         lalpha          9.88715          14.1954            -17.9352             37.7096
            rho         -6.97151          8.29351            -23.2265             9.28347
      intercept          5.65863         0.071347              5.5188             5.79847
              v        -0.358204         0.474051            -1.28733            0.570918
              n          2.44477          4.67247             -6.7131             11.6026
              k          28.2007          40.5975            -51.3689              107.77

     Table of Data and Estimated Values of Interest
 Dose       N    Obs Mean     Est Mean   Obs Std Dev  Est Std Dev   Scaled Res.
------     ---   --------     --------   -----------  -----------   ----------
    0    24       5.67         5.66         0.37        0.334          0.182
 11.9    20       5.62         5.62        0.358        0.342         0.0408
24.05    19       5.47         5.51        0.252        0.365         -0.537
49.57    25       5.39         5.37        0.446          0.4          0.257
 
 Warning: Likelihood for fitted model larger than the Likelihood for model A3.

 Model Descriptions for likelihoods 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 + rho*ln(Mu(i)))
Model A3 uses any fixed variance parameters that 
were specified by the user

 Model  R:         Yi = Mu + e(i)
            Var{e(i)} = Sigma^2

                       Likelihoods of Interest
            Model      Log(likelihood)   # Param's      AIC
             A1           45.536214            5     -81.072428
             A2           48.828168            8     -81.656336
             A3           45.896524            6     -79.793047
         fitted           45.896524            6     -79.793047
              R           41.220967            2     -78.441934

                   Explanation of Tests  
 Test 1:  Do responses and/or variances differ among Dose levels? 
          (A2 vs. R)
 Test 2:  Are Variances Homogeneous? (A1 vs A2)
 Test 3:  Are variances adequately modeled? (A2 vs. A3)
 Test 4:  Does the Model for the Mean Fit? (A3 vs. fitted)
 (Note:  When rho=0 the results of Test 3 and Test 2 will be the same.)

                     Tests of Interest    
   Test    -2*log(Likelihood Ratio)  Test df        p-value    
   Test 1              15.2144          6         0.01865
   Test 2              6.58391          3         0.08641
   Test 3              5.86329          2         0.05331
   Test 4        -5.35465e-011          0              NA

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 less than .1.  You may want to consider a different variance model

NA - Degrees of freedom for Test 4 are less than or equal to 0.  The Chi-Square test for fit is not valid
 

        Benchmark Dose Computation
Specified effect =          0.05
Risk Type        =     Relative deviation 
Confidence level =           0.95
             BMD =        48.4703
            BMDL =       13.0919

 ==================================================================== 
   	  Power Model. (Version: 2.17;  Date: 01/28/2013) 
	  Input Data File: C:/Bmds/BMDS240/Data/pow_NMP_Saillenfait_BW_Cmax_Opt.(d)  
  	  Gnuplot Plotting File:  C:/Bmds/BMDS240/Data/pow_NMP_Saillenfait_BW_Cmax_Opt.plt
 							Thu Sep 12 15:14:26 2013
 ==================================================================== 
 BMDS Model Run 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   The form of the response function is: 
   Y[dose] = control + slope * dose^power
   Dependent variable = Mean
   Independent variable = Dose
   The power is restricted to be greater than or equal to 1
   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

                  Default Initial Parameter Values  
                         lalpha =     -1.98839
                            rho =            0
                        control =        5.671
                          slope =     -129.432
                          power =     -1.57384
           Asymptotic Correlation Matrix of Parameter Estimates
                 lalpha          rho      control        slope        power
    lalpha            1           -1        -0.41         0.43          0.4
       rho           -1            1         0.41        -0.43         -0.4
   control        -0.41         0.41            1        -0.69        -0.64
     slope         0.43        -0.43        -0.69            1         0.99
     power          0.4         -0.4        -0.64         0.99            1

                                 Parameter Estimates
                                                         95.0% Wald Confidence Interval
       Variable         Estimate        Std. Err.     Lower Conf. Limit   Upper Conf. Limit
         lalpha          11.4954          13.3254            -14.6218             37.6126
            rho         -7.91072          7.78441            -23.1679             7.34643
        control          5.65941        0.0692366              5.5237             5.79511
          slope      -0.00326494        0.0114188          -0.0256454           0.0191155
          power          1.14888         0.868344           -0.553038             2.85081

     Table of Data and Estimated Values of Interest
 Dose       N    Obs Mean     Est Mean   Obs Std Dev  Est Std Dev   Scaled Res.
------     ---   --------     --------   -----------  -----------   ----------
    0    24       5.67         5.66         0.37         0.33          0.172
 11.9    20       5.62          5.6        0.358        0.343          0.257
24.05    19       5.47         5.53        0.252        0.361         -0.777
49.57    25       5.39         5.37        0.446        0.406          0.283
 Model Descriptions for likelihoods 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 + rho*ln(Mu(i)))
     Model A3 uses any fixed variance parameters that were specified by the user
 Model R:         Yi = Mu + e(i)
            Var{e(i)} = Sigma^2

                       Likelihoods of Interest
            Model      Log(likelihood)   # Param's      AIC
             A1           45.536214            5     -81.072428
             A2           48.828168            8     -81.656336
             A3           45.896524            6     -79.793047
         fitted           45.848341            5     -81.696681
              R           41.220967            2     -78.441934

                   Explanation of Tests  
 Test 1:  Do responses and/or variances differ among Dose levels? 
          (A2 vs. R)
 Test 2:  Are Variances Homogeneous? (A1 vs A2)
 Test 3:  Are variances adequately modeled? (A2 vs. A3)
 Test 4:  Does the Model for the Mean Fit? (A3 vs. fitted)
 (Note:  When rho=0 the results of Test 3 and Test 2 will be the same.)

                     Tests of Interest    
   Test    -2*log(Likelihood Ratio)  Test df        p-value    
   Test 1              15.2144          6         0.01865
   Test 2              6.58391          3         0.08641
   Test 3              5.86329          2         0.05331
   Test 4             0.096366          1          0.7562

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 less than .1.  You may want to consider a different variance model

The p-value for Test 4 is greater than .1.  The model chosen seems to adequately describe the data
 
               Benchmark Dose Computation
Specified effect =          0.05
Risk Type        =     Relative deviation 
Confidence level =          0.95
             BMD = 48.6112       
            BMDL = 30.7309  
      

==================================================================== 
   	   Polynomial Model  -  3[rd] degree. (Version: 2.17;  Date: 01/28/2013) 

  	  Input Data File: C:/Bmds/BMDS240/Data/ply_NMP_Saillenfait_BW_Cmax_Opt.(d)  
  	  Gnuplot Plotting File:  C:/Bmds/BMDS240/Data/ply_NMP_Saillenfait_BW_Cmax_Opt.plt
 							Thu Sep 12 15:20:55 2013
 ==================================================================== 
 BMDS Model Run 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
   The form of the response function is: 
	   Y[dose] = beta_0 + beta_1*dose + beta_2*dose^2 + ...
   Dependent variable = Mean
   Independent variable = Dose
   Signs of the polynomial coefficients are not restricted
   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

                  Default Initial Parameter Values  
                         lalpha =     -1.98839
                            rho =            0
                         beta_0 =        5.671
                         beta_1 =   0.00380278
                         beta_2 = -0.000806573
                         beta_3 = 1.24414e-005

           Asymptotic Correlation Matrix of Parameter Estimates
                 lalpha          rho       beta_0       beta_1       beta_2       beta_3
    lalpha            1           -1        -0.17        -0.18         0.35        -0.41
       rho           -1            1         0.17         0.18        -0.35         0.41
    beta_0        -0.17         0.17            1        -0.44         0.23        -0.16
    beta_1        -0.18         0.18        -0.44            1        -0.95         0.91
    beta_2         0.35        -0.35         0.23        -0.95            1        -0.99
    beta_3        -0.41         0.41        -0.16         0.91        -0.99            1

                                 Parameter Estimates
                                                         95.0% Wald Confidence Interval
       Variable         Estimate        Std. Err.     Lower Conf. Limit   Upper Conf. Limit
         lalpha          9.88711          15.0342            -19.5793             39.3535
            rho         -6.97149          8.77078            -24.1619             10.2189
         beta_0          5.65863        0.0697728             5.52188             5.79539
         beta_1       0.00124345         0.020928          -0.0397746           0.0422615
         beta_2     -0.000452908       0.00139175         -0.00318069          0.00227488
         beta_3     6.28146e-006     2.03946e-005       -3.36912e-005        4.62541e-005

     Table of Data and Estimated Values of Interest
 Dose       N    Obs Mean     Est Mean   Obs Std Dev  Est Std Dev   Scaled Res.
------     ---   --------     --------   -----------  -----------   ---------
    0    24       5.67         5.66         0.37        0.334          0.182
 11.9    20       5.62         5.62        0.358        0.342         0.0408
24.05    19       5.47         5.51        0.252        0.365         -0.537
49.57    25       5.39         5.37        0.446          0.4          0.257
 Warning: Likelihood for fitted model larger than the Likelihood for model A3.
 Model Descriptions for likelihoods 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 + rho*ln(Mu(i)))
Model A3 uses any fixed variance parameters 
that were specified by the user

 Model  R:         Yi = Mu + e(i)
            Var{e(i)} = Sigma^2

                       Likelihoods of Interest
            Model      Log(likelihood)   # Param's      AIC
             A1           45.536214            5     -81.072428
             A2           48.828168            8     -81.656336
             A3           45.896524            6     -79.793047
         fitted           45.896524            6     -79.793047
              R           41.220967            2     -78.441934

                   Explanation of Tests  
 Test 1:  Do responses and/or variances differ among Dose levels? 
          (A2 vs. R)
 Test 2:  Are Variances Homogeneous? (A1 vs A2)
 Test 3:  Are variances adequately modeled? (A2 vs. A3)
 Test 4:  Does the Model for the Mean Fit? (A3 vs. fitted)
 (Note:  When rho=0 the results of Test 3 and Test 2 will be the same.)

                     Tests of Interest    
   Test    -2*log(Likelihood Ratio)  Test df        p-value    
   Test 1              15.2144          6         0.01865
   Test 2              6.58391          3         0.08641
   Test 3              5.86329          2         0.05331
   Test 4        -4.84164e-011          0              NA

The p-value for Test 1 is less than .05.  There appears to be a difference between response and/or variances among the dose levelsIt 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 less than .1.  You may want to consider a different variance model

NA - Degrees of freedom for Test 4 are less than or equal to 0.  The Chi-Square test for fit is not valid
 
             Benchmark Dose Computation
Specified effect =          0.05
Risk Type        =     Relative deviation 
Confidence level =          0.95
             BMD =        42.4735
            BMDL =        19.7354

==================================================================== 
   	  Polynomial Model  -  2[nd] Order. (Version: 2.17;  Date: 01/28/2013) 
 	  Input Data File: C:/Bmds/BMDS240/Data/ply_NMP_Saillenfait_BW_Cmax_Opt.(d)  
  	  Gnuplot Plotting File:  C:/Bmds/BMDS240/Data/ply_NMP_Saillenfait_BW_Cmax_Opt.plt
 							Thu Sep 12 15:31:05 2013
 ==================================================================== 
 BMDS Model Run 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
   The form of the response function is: 

   Y[dose] = beta_0 + beta_1*dose + beta_2*dose^2 + ...

   Dependent variable = Mean
   Independent variable = Dose
   Signs of the polynomial coefficients are not restricted
   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

                  Default Initial Parameter Values  
                         lalpha =     -1.98839
                            rho =            0
                         beta_0 =      5.68636
                         beta_1 =  -0.00953107
                         beta_2 = 7.10263e-005

           Asymptotic Correlation Matrix of Parameter Estimates
                 lalpha          rho       beta_0       beta_1       beta_2
    lalpha            1           -1        -0.26         0.49        -0.53
       rho           -1            1         0.26        -0.49         0.52
    beta_0        -0.26         0.26            1        -0.72         0.59
    beta_1         0.49        -0.49        -0.72            1        -0.97
    beta_2        -0.53         0.52         0.59        -0.97            1

                                 Parameter Estimates
                                                         95.0% Wald Confidence Interval
       Variable         Estimate        Std. Err.     Lower Conf. Limit   Upper Conf. Limit
         lalpha          11.3774          13.6414            -15.3592             38.1139
            rho         -7.84161          7.96162            -23.4461             7.76288
         beta_0          5.66132        0.0685418             5.52698             5.79566
         beta_1      -0.00486813       0.00853414          -0.0215947           0.0118585
         beta_2    -1.99964e-005      0.000168622        -0.000350489         0.000310497

     Table of Data and Estimated Values of Interest
 Dose       N    Obs Mean     Est Mean   Obs Std Dev  Est Std Dev   Scaled Res.
------     ---   --------     --------   -----------  -----------   ----------
    0    24       5.67         5.66         0.37         0.33          0.144
 11.9    20       5.62          5.6        0.358        0.344          0.291
24.05    19       5.47         5.53        0.252        0.361         -0.769
49.57    25       5.39         5.37        0.446        0.406          0.273

 Model Descriptions for likelihoods 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 + rho*ln(Mu(i)))
     Model A3 uses any fixed variance parameters that
     were specified by the user

 Model  R:         Yi = Mu + e(i)
            Var{e(i)} = Sigma^2

                       Likelihoods of Interest
            Model      Log(likelihood)   # Param's      AIC
             A1           45.536214            5     -81.072428
             A2           48.828168            8     -81.656336
             A3           45.896524            6     -79.793047
         fitted           45.840013            5     -81.680026
              R           41.220967            2     -78.441934

                   Explanation of Tests  
 Test 1:  Do responses and/or variances differ among Dose levels? 
          (A2 vs. R)
 Test 2:  Are Variances Homogeneous? (A1 vs A2)
 Test 3:  Are variances adequately modeled? (A2 vs. A3)
 Test 4:  Does the Model for the Mean Fit? (A3 vs. fitted)
 (Note:  When rho=0 the results of Test 3 and Test 2 will be the same.)

                     Tests of Interest    
   Test    -2*log(Likelihood Ratio)  Test df        p-value    
   Test 1              15.2144          6         0.01865
   Test 2              6.58391          3         0.08641
   Test 3              5.86329          2         0.05331
   Test 4             0.113021          1          0.7367

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 less than .1.  You may want to consider a different variance model

The p-value for Test 4 is greater than .1.  The model chosen seems to adequately describe the data
 
             Benchmark Dose Computation
Specified effect =          0.05
Risk Type        =     Relative deviation 
Confidence level =          0.95
             BMD =        48.4891
            BMDL =         19.737

==================================================================== 
   	  Linear Model (Polynomial Model, 1[st] order). (Version: 2.17;  Date: 01/28/2013) 
  	  Input Data File: C:/Bmds/BMDS240/Data/lin_NMP_Saillenfait_BW_Cmax_Opt.(d)  
  	  Gnuplot Plotting File:  C:/Bmds/BMDS240/Data/lin_NMP_Saillenfait_BW_Cmax_Opt.plt
 							Thu Sep 12 15:36:49 2013
 ==================================================================== 
 BMDS Model Run 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~                                                                                                                                                                                                                                                                                                                                                                                                                                    
   The form of the response function is: 
	   Y[dose] = beta_0 + beta_1*dose + beta_2*dose^2 + ...
   Dependent variable = Mean
   Independent variable = Dose
   Signs of the polynomial coefficients are not restricted
   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

                  Default Initial Parameter Values  
                         lalpha =     -1.98839
                            rho =            0
                         beta_0 =      5.66456
                         beta_1 =   -0.0058728

           Asymptotic Correlation Matrix of Parameter Estimates
                 lalpha          rho       beta_0       beta_1
    lalpha            1           -1        0.077        -0.11
       rho           -1            1       -0.077         0.11
    beta_0        0.077       -0.077            1        -0.72
    beta_1        -0.11         0.11        -0.72            1

                                 Parameter Estimates
                                                         95.0% Wald Confidence Interval
       Variable         Estimate        Std. Err.     Lower Conf. Limit   Upper Conf. Limit
         lalpha          10.5323          11.5553            -12.1157             33.1803
            rho          -7.3476          6.75178            -20.5809             5.88566
         beta_0           5.6673        0.0554004             5.55871             5.77588
         beta_1      -0.00591414       0.00209746          -0.0100251         -0.00180319

     Table of Data and Estimated Values of Interest
 Dose       N    Obs Mean     Est Mean   Obs Std Dev  Est Std Dev   Scaled Res.
------     ---   --------     --------   -----------  -----------   ----------
    0    24       5.67         5.67         0.37        0.331         0.0549
 11.9    20       5.62          5.6        0.358        0.346          0.337
24.05    19       5.47         5.53        0.252        0.363         -0.673
49.57    25       5.39         5.37        0.446        0.402          0.235

 Model Descriptions for likelihoods 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 + rho*ln(Mu(i)))
Model A3 uses any fixed variance parameters 
that were specified by the user

 Model  R:         Yi = Mu + e(i)
            Var{e(i)} = Sigma^2

                       Likelihoods of Interest
            Model      Log(likelihood)   # Param's      AIC
             A1           45.536214            5     -81.072428
             A2           48.828168            8     -81.656336
             A3           45.896524            6     -79.793047
         fitted           45.831033            4     -83.662065
              R           41.220967            2     -78.441934

                   Explanation of Tests  
 Test 1:  Do responses and/or variances differ among Dose levels? 
          (A2 vs. R)
 Test 2:  Are Variances Homogeneous? (A1 vs A2)
 Test 3:  Are variances adequately modeled? (A2 vs. A3)
 Test 4:  Does the Model for the Mean Fit? (A3 vs. fitted)
 (Note:  When rho=0 the results of Test 3 and Test 2 will be the same.)

                     Tests of Interest    
   Test    -2*log(Likelihood Ratio)  Test df        p-value    
   Test 1              15.2144          6         0.01865
   Test 2              6.58391          3         0.08641
   Test 3              5.86329          2         0.05331
   Test 4             0.130982          2          0.9366

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 less than .1.  You may want to consider a different variance model

The p-value for Test 4 is greater than .1.  The model chosen seems to adequately describe the data
 
             Benchmark Dose Computation
Specified effect =          0.05
Risk Type        =     Relative deviation 
Confidence level =          0.95
             BMD =        47.9131
            BMDL =        30.6591