                              BMD Analysis (AUC)

                                 Iris Camacho
                                Paul Schlosser

Saillenfait et al. 2003, fetal body weights

                              Internal Dose, AUC 
                                   (mg-hr/L)
                                   # litters
                                 Fetal BW (g)
                              Standard Deviation
                                       0
                                      24
                                     5.671
                                     0.37
                                     93.84
                                      20
                                     5.623
                                     0.358
                                    190.67
                                      19
                                     5.469
                                     0.252
                                    397.28
                                      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.
                                    0.9361
                                    -83.661
                                      384
                                      242
                                      788
                                      497
                                       
                               Exponential (M3)
                                   Rel. Dev.
                                    0.7571
                                    -81.697
                                      389
                                      242
                                      728
                                      411
                                       
                               Exponential (M4)
                                   Rel. Dev.
                                    0.9361
                                    -83.661
                                      384
                                      166
                                      788
                                      497
               Same best fit, but more fitted parameters than M2
                               Exponential (M5)
                                   Rel. Dev.
                                    0.7571
                                    -79.793
                                      382
                                      99
                                     error
                                     error
                    Model did not compute BMD for BMR = 10%
                                     Hill
                                   Rel. Dev.
                                      NA
                                    -79.793
                                      388
                                      103
                                     error
                                     error
                     Chi-square test for fit is not valid.
                    Model did not compute BMD for BMR = 10%
                                     Power
                                   Rel. Dev.
                                    0.7539
                                    -81.695
                                      390
                                      246
                                      784
                                      466
                                       
                                Polynomial 3°
                                   Rel. Dev.
                                    0.7254
                                    -79.793
                                      336
                                      156
                                     error
                                     error
                     Chi-square test for fit is not valid.
                    Model did not compute BMD for BMR = 10%
                                Polynomial 2°
                                   Rel. Dev.
                                    0.7362
                                    -81.680
                                      388
                                      156
                                      697
                                      462
                                       
                                    Linear
                                   Rel. Dev.
                                    0.9386
                                    -83.666
                                      384
                                      246
                                      768
                                      491
                                       

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=246 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.2_Setting.(d)  
  	  Gnuplot Plotting File:  
 							Thu Sep 12 12:20:01 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           5.41074           5.41074             5.95455           5.95455
           b       0.000130807       0.000130807          0.00335347        0.00335347
           c                --                --               0.862566          0.862566
           d                --                 1                  --                 1

                               Parameter Estimates by Model
     Variable          Model 2           Model 3           Model 4           Model 5
     --------          -------           -------           -------           -------
     lnalpha        10.5079           11.4881             10.5079           9.88714
         rho          -7.33335          -7.90647            -7.33335           -6.9715
           a           5.66744           5.65942             5.66744           5.65863
           b       0.000133676       0.000194725         0.000133676        0.00437576
           c             --                --                      0          0.947666
           d             --           1.15149                  --           2.20686
            


Table of Stats From Input Data
     Dose      N         Obs Mean     Obs Std Dev
     -----    ---       ----------   -------------
         0     24        5.671         0.37
     93.84     20        5.623        0.358
     190.7     19        5.469        0.252
     397.3     25        5.393        0.446

                      Estimated Values of Interest
      Model      Dose      Est Mean      Est Std     Scaled Residual
     -------    ------    ----------    ---------    ----------------
          2         0         5.667       0.3306          0.05272
                93.84         5.597       0.3462           0.3385
                190.7         5.525        0.363          -0.6702
                397.3         5.374       0.4017           0.2326
          3         0         5.659       0.3302           0.1718
                93.84         5.603       0.3434           0.2565
                190.7         5.533       0.3609          -0.7761
                397.3          5.37       0.4063           0.2833
          4         0         5.667       0.3306          0.05272
                93.84         5.597       0.3462           0.3385
                190.7         5.525        0.363          -0.6702
                397.3         5.374       0.4017           0.2326
          5         0         5.659       0.3336           0.1816
                93.84          5.62       0.3417          0.04083
                190.7         5.514       0.3651          -0.5367
                397.3         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        45.83051            4     -83.66102
                         3         45.8487            5     -81.69739
                         4        45.83051            4     -83.66102
                         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                         0.132           2              0.9361
    Test 5a                       0.09566           1              0.7571
    Test 5b                       0.03638           1              0.8487
    Test 6a                         0.132           2              0.9361
    Test 6b                    7.105e-014           0                 N/A
    Test 7a                    -5.37e-011           0                 N/A
    Test 7b                       0.09566           1              0.7571
    Test 7c                         0.132           2              0.9361

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             383.715             241.79
        3             389.358            242.411
        4             383.715            165.531
        5             382.145            98.9937









































==================================================================== 
   	  Hill Model. (Version: 2.17;  Date: 01/28/2013) 
  	  Input Data File: C:/Bmds/BMDS240/Data/hil_NMP_Saillenfait_BW.2_Opt.(d)  
  	  Gnuplot Plotting File:  C:/Bmds/BMDS240/Data/hil_NMP_Saillenfait_BW.2_Opt.plt
 							Thu Sep 12 13:04:24 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 =      3.74441
                              k =      151.058
           Asymptotic Correlation Matrix of Parameter Estimates
                 lalpha          rho    intercept            v            n            k
    lalpha            1           -1        -0.35        -0.23        -0.19         0.43
       rho           -1            1         0.36         0.23         0.19        -0.43
 intercept        -0.35         0.36            1        -0.32        -0.34       -0.021
         v        -0.23         0.23        -0.32            1         0.93        -0.92
         n        -0.19         0.19        -0.34         0.93            1        -0.85
         k         0.43        -0.43       -0.021        -0.92        -0.85            1

                                 Parameter Estimates
                                                         95.0% Wald Confidence Interval
       Variable         Estimate        Std. Err.     Lower Conf. Limit   Upper Conf. Limit
         lalpha          9.88714          14.1954            -17.9353             37.7096
            rho         -6.97151          8.29351            -23.2265             9.28348
      intercept          5.65863         0.071347              5.5188             5.79847
              v        -0.356526         0.463103            -1.26419            0.551139
              n             2.43          4.62921            -6.64308             11.5031
              k          223.068          318.717            -401.606             847.742

     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
93.84    20       5.62         5.62        0.358        0.342         0.0408
190.7    19       5.47         5.51        0.252        0.365         -0.537
397.3    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        -5.51097e-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 =        388.251
            BMDL =       103.239

 ==================================================================== 
   	  Power Model. (Version: 2.17;  Date: 01/28/2013) 
  	  Input Data File: C:/Bmds/BMDS240/Data/pow_NMP_Saillenfait_BW.2_Opt.(d)  
  	  Gnuplot Plotting File:  C:/Bmds/BMDS240/Data/pow_NMP_Saillenfait_BW.2_Opt.plt
 							Thu Sep 12 12:51:23 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 =     -3189.28
                          power =     -1.56195

           Asymptotic Correlation Matrix of Parameter Estimates
                 lalpha          rho      control        slope        power
    lalpha            1           -1        -0.41         0.42          0.4
       rho           -1            1         0.41        -0.42         -0.4
   control        -0.41         0.41            1        -0.68        -0.64
     slope         0.42        -0.42        -0.68            1            1
     power          0.4         -0.4        -0.64            1            1

                                 Parameter Estimates
                                                         95.0% Wald Confidence Interval
       Variable         Estimate        Std. Err.     Lower Conf. Limit   Upper Conf. Limit
         lalpha          11.5119          13.3283            -14.6112             37.6349
            rho         -7.92031          7.78614            -23.1809             7.34024
        control          5.65939        0.0692321              5.5237             5.79508
          slope     -0.000326476       0.00170611         -0.00367039          0.00301744
          power          1.13409         0.856156            -0.54394             2.81213


     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
93.84    20       5.62          5.6        0.358        0.343           0.26
190.7    19       5.47         5.53        0.252        0.361         -0.779
397.3    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           45.847392            5     -81.694784
              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            0.0982629          1          0.7539

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 = 389.506       
            BMDL = 246.217       
==================================================================== 
   	   Polynomial Model  -  3[rd] degree. (Version: 2.17;  Date: 01/28/2013) 
  	  Input Data File: C:/Bmds/BMDS240/Data/ply_NMP_Saillenfait_BW2_Opt.(d)  
  	  Gnuplot Plotting File:  C:/Bmds/BMDS240/Data/ply_NMP_Saillenfait_BW2_Opt.plt
 							Thu Sep 12 13:33:47 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 =  0.000455788
                         beta_2 = -1.25962e-005
                         beta_3 = 2.43847e-008

           Asymptotic Correlation Matrix of Parameter Estimates
                 lalpha          rho       beta_0       beta_1       beta_2       beta_3
    lalpha            1           -1        -0.17        -0.18         0.34        -0.41
       rho           -1            1         0.17         0.18        -0.34         0.41
    beta_0        -0.17         0.17            1        -0.44         0.24        -0.16
    beta_1        -0.18         0.18        -0.44            1        -0.95         0.91
    beta_2         0.34        -0.34         0.24        -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          9.88714          15.0342            -19.5793             39.3536
            rho         -6.97151          8.77078            -24.1619             10.2189
         beta_0          5.65863        0.0697728             5.52188             5.79539
         beta_1      0.000143739       0.00262457         -0.00500033          0.00528781
         beta_2    -7.09479e-006     2.18836e-005       -4.99859e-005        3.57963e-005
         beta_3     1.23843e-008     4.00862e-008       -6.61832e-008        9.09518e-008

     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
93.84    20       5.62         5.62        0.358        0.342         0.0408
190.7    19       5.47         5.51        0.252        0.365         -0.537
397.3    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        -5.48823e-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 level.  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 =        336.067
            BMDL =        156.263

==================================================================== 
   	  Polynomial Model  -  2[nd] Order. (Version: 2.17;  Date: 01/28/2013) 
  	  Input Data File: C:/Bmds/BMDS240/Data/ply_NMP_Saillenfait_BW2_Opt.(d)  
  	  Gnuplot Plotting File:  C:/Bmds/BMDS240/Data/ply_NMP_Saillenfait_BW2_Opt.plt
 							Thu Sep 12 13:16:42 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 =      5.68648
                         beta_1 =  -0.00121037
                         beta_2 = 1.15923e-006

           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         0.53
    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         0.53         0.59        -0.97            1

                                 Parameter Estimates
                                                         95.0% Wald Confidence Interval
       Variable         Estimate        Std. Err.     Lower Conf. Limit   Upper Conf. Limit
         lalpha          11.3659          13.6638            -15.4147             38.1465
            rho          -7.8349          7.97466             -23.465             7.79515
         beta_0          5.66143         0.068493             5.52719             5.79568
         beta_1     -0.000624567        0.0010711         -0.00272388          0.00147474
         beta_2    -2.68501e-007     2.63938e-006       -5.44159e-006        4.90459e-006

     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          0.142
93.84    20       5.62          5.6        0.358        0.344          0.293
190.7    19       5.47         5.53        0.252        0.361         -0.767
397.3    25       5.39         5.37        0.446        0.406          0.272



 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           45.839777            5     -81.679555
              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             0.113493          1          0.7362

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 =        388.382
            BMDL =        155.967




==================================================================== 
   	  Linear Model (Polynomial Model, 1[st] order). (Version: 2.17;  Date: 01/28/2013) 
  	  Input Data File: C:/Bmds/BMDS240/Data/lin_NMP_Saillenfait_BW2_Opt.(d)  
  	  Gnuplot Plotting File:  C:/Bmds/BMDS240/Data/lin_NMP_Saillenfait_BW2_Opt.plt
 							Thu Sep 12 13:44:42 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 =      5.66365
                         beta_1 = -0.000731332

           Asymptotic Correlation Matrix of Parameter Estimates
                 lalpha          rho       beta_0       beta_1
    lalpha            1           -1        0.079        -0.11
       rho           -1            1       -0.079         0.11
    beta_0        0.079       -0.079            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          10.6377          11.5633            -12.0259             33.3013
            rho         -7.40923          6.75646            -20.6517              5.8332
         beta_0          5.66658         0.055234             5.55832             5.77484
         beta_1     -0.000737251      0.000261912         -0.00125059        -0.000223912

     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         0.0655
93.84    20       5.62          5.6        0.358        0.346          0.331
190.7    19       5.47         5.53        0.252        0.363         -0.685
397.3    25       5.39         5.37        0.446        0.402           0.24

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           45.833158            4     -83.666316
              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             0.126732          2          0.9386

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 =        384.305
            BMDL =        245.743

