Example 10. Log-Linear Regression: Lumber Data

The weight of lumber is to be estimated from external measurements of a sample of pine trees. The variables are described below:
Y: Weight of lumber from a tree
X1: Height of the tree (in feet)
X2: Diameter of the tree at a height of about 4 feet
X3: Age of the tree (in years)
X4: Specific gravity of the tree

Source: Freund, R.J. and Littell, R.C. (1991). SAS System for Regression. SAS Institute Inc.

Table 10: Lumber Data

   Y     X1   X2   X3     X4
 174    5.7   34   10  0.409
 745    8.1   68   17  0.501
 814    8.3   70   17  0.445
 408    7.0   54   17  0.442
 226    6.2   37   12  0.353
1675   11.4   79   27  0.429
1491   11.6   70   26  0.497
 121    4.5   37   12  0.380
  58    3.5   32   15  0.420
 278    6.2   45   15  0.449
 220    5.7   48   20  0.471
 342    6.0   57   20  0.447
 209    5.6   40   20  0.439
  84    4.0   44   27  0.394
 313    6.7   52   21  0.422
  60    4.0   38   27  0.496
1692   12.1   74   27  0.476
  74    4.5   37   12  0.382
 515    8.6   60   23  0.502
 766    9.3   63   18  0.458
 345    6.5   57   18  0.474
 210    5.6   46   12  0.413
 100    4.3   41   12  0.382
 122    4.5   42   12  0.457
 539    7.7   64   19  0.478
 815    8.8   70   22  0.496
 194    5.0   53   23  0.485
 280    5.4   61   23  0.488
 296    6.0   56   23  0.435
 462    7.4   52   14  0.474
 200    5.6   48   19  0.441
 229    5.5   50   19  0.506
 125    4.3   50   19  0.410

Questions:

1. Describe the characteristics of a multiple regression model, regressing Y on the predictor variables. Discuss the model coefficients.
   
   
2. Describe a linear regression in the logs of all the variables. Show that this is the same as a multiplicative (nonlinear) model in the original variables.
   
   
3. Fit a log-linear multiplicative model. Compare the model fits.

Keywords: Log-linear models, intrinsically linear models, transformations