NIST - Ratkowsky3 Dataset

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NIST - Ratkowsky3 Dataset

   1: /*
   2:  * Statistical Reference Datasets  (Nonlinear Regression)
   3:  * Statistical Engineering Division
   4:  * National Institute of Standards and Technology
   5:  * http://www.nist.gov/itl/div898/strd/
   6:  *
   7:  * Dataset Name:  Ratkowsky3        (Ratkowsky3.dat)
   8:  * 
   9:  * Description:   This model and data are an example of fitting  
  10:  *                sigmoidal growth curves taken from Ratkowsky (1983).  
  11:  *                The response variable is the dry weight of onion bulbs 
  12:  *                and tops, and the predictor variable is growing time. 
  13:  * 
  14:  * Reference:     Ratkowsky, D.A. (1983).  
  15:  *                Nonlinear Regression Modeling.
  16:  *                New York, NY:  Marcel Dekker, pp. 62 and 88.
  17:  * 
  18:  * Data:          1 Response  (y = onion bulb dry weight)
  19:  *                1 Predictor (x = growing time)
  20:  *                15 Observations
  21:  *                Higher Level of Difficulty
  22:  *                Observed Data
  23:  * 
  24:  * Model:         Exponential Class
  25:  *                4 Parameters (b1 to b4)
  26:  * 
  27:  *                y = b1 / ((1+exp[b2-b3*x])**(1/b4))  +  e
  28:  * 
  29:  *           Starting Values                  Certified Values
  30:  *  
  31:  *         Start 1     Start 2           Parameter     Standard Deviation
  32:  *   b1 =   100         700           6.9964151270E+02  1.6302297817E+01
  33:  *   b2 =    10           5           5.2771253025E+00  2.0828735829E+00
  34:  *   b3 =     1           0.75        7.5962938329E-01  1.9566123451E-01
  35:  *   b4 =     1           1.3         1.2792483859E+00  6.8761936385E-01
  36:  *  
  37:  * Residual Sum of Squares:                    8.7864049080E+03
  38:  * Residual Standard Deviation:                2.8262414662E+01
  39:  * Degrees of Freedom:                                9
  40:  * Number of Observations:                           15 
  41:  */
  42: Title "Ratkowsky3";
  43: Variables y,x;
  44: Parameter b1 = 100;
  45: Parameter b2 = 10;
  46: Parameter b3 = 1;
  47: Parameter b4 = 1;
  48: Function y = b1 / ((1+exp(b2-b3*x))**(1/b4));
  49: Plot;
  50: Data;

Beginning computation...
Stopped due to: Relative function convergence.


   ----  Final Results  ----

NLREG version 4.0
Copyright (c) 1992-1997 Phillip H. Sherrod.  All rights reserved.
This is a registered copy of NLREG that may not be redistributed.

Ratkowsky3
Number of observations = 15
Maximum allowed number of iterations = 500
Convergence tolerance factor = 1.000000E-010
Stopped due to: Relative function convergence.
Number of iterations performed = 12
Final sum of squared deviations = 8.7864049E+003
Final sum of deviations = -7.0961484E+000
Standard error of estimate = 28.2624
Average deviation = 16.5605
Maximum deviation for any observation = 59.5153
Proportion of variance explained (R^2) = 0.9918  (99.18%)
Adjusted coefficient of multiple determination (Ra^2) = 0.9896  (98.96%)
Durbin-Watson test for autocorrelation = 2.187


             ----  Descriptive Statistics for Variables  ----

 Variable    Minimum value   Maximum value    Mean value     Standard dev.
----------  --------------  --------------  --------------  --------------
         y           16.08          724.93        423.2953        277.2907
         x               1              15               8        4.472136


                   ----  Calculated Parameter Values  ----

 Parameter  Initial guess   Final estimate   Standard error      t      Prob(t)
----------  -------------  ----------------  --------------  ---------  -------
        b1            100        699.641517         16.3023      42.92  0.00001
        b2             10        5.27712468        2.082875       2.53  0.02780
        b3              1       0.759629318       0.1956614       3.88  0.00255
        b4              1        1.27924823       0.6876196       1.86  0.08975


                  ----  Analysis of Variance  ----

  Source     DF   Sum of Squares    Mean Square    F value   Prob(F)
----------  ----  --------------  --------------  ---------  -------
Regression     3         1067675        355891.7     445.55  0.00001
Error         11        8786.405        798.7641
Total         14         1076462