The TRANSREG Procedure

Example 101.1 Transformation Regression of Exhaust Emissions Data

In this example, the data are from an experiment in which nitrogen oxide emissions from a single cylinder engine are measured for various combinations of fuel, compression ratio, and equivalence ratio. The data are provided by Brinkman (1981). This gas data set is available from the Sashelp library.

The equivalence ratio and nitrogen oxide variables are continuous and numeric, so spline transformations of these variables are requested. The spline transformation of the dependent variable is restricted to be monotonic. Each spline is degree three with nine knots (one at each decile) in order to give PROC TRANSREG a great deal of freedom in finding transformations. The compression ratio variable has only five discrete values, so an optimal scoring is requested with monotonicity constraints. The character variable Fuel is nominal, so it is optimally scored without any monotonicity constraints. Observations with missing values are excluded with the NOMISS a-option.

ods graphics on;

title 'Gasoline Example';
title2 'Iteratively Estimate NOx, CpRatio, EqRatio, and Fuel';

* Fit the Nonparametric Model;
proc transreg data=sashelp.Gas solve test nomiss plots=all;
   ods exclude where=(_path_ ? 'MV');
   model mspline(NOx / nknots=9) = spline(EqRatio / nknots=9)
                                   monotone(CpRatio) opscore(Fuel);
run;

Output 101.1.1: Transformation Regression Example: The Nonparametric Model

Gasoline Example
Iteratively Estimate NOx, CpRatio, EqRatio, and Fuel

The TRANSREG Procedure


Dependent Variable Mspline(NOx)
Nitrogen Oxide

Number of Observations Read 171
Number of Observations Used 169

TRANSREG MORALS Algorithm Iteration History for Mspline(NOx)
Iteration
Number
Average
Change
Maximum
Change
R-Square Criterion
Change
Note
0 0.41900 3.80550 0.05241    
1 0.11984 0.83327 0.91028 0.85787  
2 0.03727 0.17688 0.93981 0.02953  
3 0.02795 0.10880 0.94969 0.00987  
4 0.02088 0.07279 0.95382 0.00413  
5 0.01530 0.05031 0.95582 0.00201  
6 0.01130 0.03922 0.95688 0.00106  
7 0.00852 0.03197 0.95748 0.00060  
8 0.00657 0.02531 0.95783 0.00035  
9 0.00510 0.01975 0.95805 0.00022  
10 0.00398 0.01534 0.95818 0.00013  
11 0.00314 0.01200 0.95827 0.00009  
12 0.00250 0.00953 0.95832 0.00005  
13 0.00199 0.00752 0.95836 0.00003  
14 0.00159 0.00594 0.95838 0.00002  
15 0.00127 0.00470 0.95839 0.00001  
16 0.00102 0.00373 0.95840 0.00001  
17 0.00081 0.00297 0.95841 0.00001  
18 0.00065 0.00237 0.95841 0.00000  
19 0.00052 0.00189 0.95841 0.00000  
20 0.00042 0.00151 0.95842 0.00000  
21 0.00033 0.00120 0.95842 0.00000  
22 0.00027 0.00096 0.95842 0.00000  
23 0.00021 0.00077 0.95842 0.00000  
24 0.00017 0.00061 0.95842 0.00000  
25 0.00014 0.00049 0.95842 0.00000  
26 0.00011 0.00039 0.95842 0.00000  
27 0.00009 0.00031 0.95842 0.00000  
28 0.00007 0.00025 0.95842 0.00000  
29 0.00006 0.00020 0.95842 0.00000  
30 0.00005 0.00016 0.95842 0.00000 Not Converged

WARNING: Failed to converge, however criterion change is less than 0.0001.


The TRANSREG Procedure Hypothesis Tests for Mspline(NOx)
Nitrogen Oxide

Univariate ANOVA Table Based on the Usual Degrees of Freedom
Source DF Sum of Squares Mean Square F Value Liberal p
Model 21 326.0176 15.52465 161.35 >= <.0001
Error 147 14.1443 0.09622    
Corrected Total 168 340.1619      
The above statistics are not adjusted for the fact that the dependent variable was transformed and so are generally liberal.

Root MSE 0.31019 R-Square 0.9584
Dependent Mean 2.34593 Adj R-Sq 0.9525
Coeff Var 13.22262    


The squared multiple correlation for the initial model is approximately 0.05. PROC TRANSREG increases the R square to over 0.95 by transforming the variables. The transformation plots show how each variable is transformed. The transformation of compression ratio (TCpRatio) is nearly linear. The transformation of equivalence ratio (TEqRatio) is nearly parabolic. It can be seen from this plot that the optimal transformation of equivalence ratio is nearly uncorrelated with the original scoring. This suggests that the large increase in R square is due to this transformation. The transformation of nitrogen oxide (TNOx) is similar to a log transformation. The final plot shows the transformed dependent variable plotted as a function of the predicted values. This plot is reasonably linear, showing that the nonlinearities in the data are being accounted for fairly well by the TRANSREG model.

These results suggest the parametric model

\begin{eqnarray*}  \log (\mbox{NOX}) &  = &  b_0 + b_1 \times \mbox{\Variable{EqRatio}} + b_2 \times \mbox{\Variable{EqRatio}}^2 + b_3 \times \mbox{\Variable{CpRatio}} \\[0.10in]& &  ~  + \sum _ j b_ j \mbox{class}_ j (\mbox{\Variable{Fuel}}) + \mbox{error} \\ \end{eqnarray*}

You can perform this analysis with PROC TRANSREG. The following statements produce Output 101.1.2:

title2 'Now fit log(NOx) = b0 + b1*EqRatio + b2*EqRatio**2 +';
title3 'b3*CpRatio + Sum b(j)*Fuel(j) + Error';

*-Fit the Parametric Model Suggested by the Nonparametric Analysis-;
proc transreg data=sashelp.Gas solve ss2 short nomiss plots=all;
   model log(NOx) = pspline(EqRatio / deg=2) identity(CpRatio)
                    opscore(Fuel);
run;

Output 101.1.2: Transformation Regression Example: The Parametric Model

Gasoline Example
Now fit log(NOx) = b0 + b1*EqRatio + b2*EqRatio**2 +
b3*CpRatio + Sum b(j)*Fuel(j) + Error

The TRANSREG Procedure


Dependent Variable Log(NOx)
Nitrogen Oxide

Number of Observations Read 171
Number of Observations Used 169

Log(NOx)
Algorithm converged.


The TRANSREG Procedure Hypothesis Tests for Log(NOx)
Nitrogen Oxide

Univariate ANOVA Table Based on the Usual Degrees of Freedom
Source DF Sum of Squares Mean Square F Value Pr > F
Model 8 79.33838 9.917298 213.09 <.0001
Error 160 7.44659 0.046541    
Corrected Total 168 86.78498      

Root MSE 0.21573 R-Square 0.9142
Dependent Mean 0.63130 Adj R-Sq 0.9099
Coeff Var 34.17294    

Univariate Regression Table Based on the Usual Degrees of Freedom
Variable DF Coefficient Type II
Sum of
Squares
Mean Square F Value Pr > F Label
Intercept 1 -15.274649 57.1338 57.1338 1227.60 <.0001 Intercept
Pspline.EqRatio_1 1 35.102914 62.7478 62.7478 1348.22 <.0001 Equivalence Ratio 1
Pspline.EqRatio_2 1 -19.386468 64.6430 64.6430 1388.94 <.0001 Equivalence Ratio 2
Identity(CpRatio) 1 0.032058 1.4445 1.4445 31.04 <.0001 Compression Ratio
Opscore(Fuel) 5 0.158388 5.5619 1.1124 23.90 <.0001 Fuel


The LOG transformation computes the natural log. The PSPLINE expansion expands EqRatio into a linear term, EqRatio, and a squared term, $\mbox{\Variable{EqRatio}}^2$. An identity transformation of CpRatio and an optimal scoring of Fuel is requested. These should provide a good parametric operationalization of the optimal transformations. The final model has an R square of 0.91 (smaller than before since the model has fewer parameters, but still quite good).