The NLIN Procedure

Example 81.6 ODS Graphics and Diagnostics

The model in this example, taken from St. Laurent and Cook (1993), shows an unusual behavior in that the intrinsic curvature is substantially larger than the parameter-effects curvature. This example demonstrates how the diagnostics features of PROC NLIN can be used to perform postconvergence diagnostics.

The model takes the form

$\mr{E}[Y|x_1, x_2] = \alpha x_1 + \exp \{ \gamma x_2 \}$

The following DATA step creates a small data set to be used in this example:

data contrived;
   input x1 x2 y;
   datalines;
-4.0   -2.5  -10.0   
-3.0   -2.0   -5.0
-2.0   -1.5   -2.0   
-1.0   -1.0   -1.0
 0.0    0.0    1.5    
 1.0    1.0    4.0
 2.0    1.5    5.0    
 3.0    2.0    6.0
 4.0    2.5    7.0 
-3.5   -2.2   -7.1
-3.5   -1.7   -5.1
 3.5    0.7    6.1
 2.5    1.2    7.5
;

The model is fit with the following statements in the NLIN procedure:

ods graphics on;
proc nlin data=contrived bias hougaard
        NLINMEASURES plots(stats=all)=(diagnostics);
   parms alpha=2.0
         gamma=0.0;
   model y = alpha*x1 + exp(gamma*x2);
run;
ods graphics off;

Output 81.6.1: Bias, Skewness, and Global Nonlinearity Measures

The NLIN Procedure

Parameter	Estimate	Approx Std Error	Approximate 95% Confidence Limits		Skewness	Bias	Percent Bias
alpha	1.9378	0.4704	0.9024	2.9733	6.6491	0.5763	29.7
gamma	0.0718	0.7923	-1.6720	1.8156	-7.5596	-0.9982	-1390

Global Nonlinearity Measures
Max Intrinsic Curvature	6.6007
RMS Intrinsic Curvature	4.0421
Max Parameter-Effects Curvature	3.5719
RMS Parameter-Effects Curvature	2.1873
Curvature Critical Value	0.5011
Raw Residual Variance	2.1722
Projected Residual Variance	1.4551

The bias, skewness, and both the maximum and RMS intrinsic curvatures, compared to the critical curvature value, show that the model is highly nonlinear (Output 81.6.1). As such, performing diagnostics with the raw residuals can be problematic because they might have undesirable statistical properties: a nonzero mean and a negative semidefinite (instead of zero) covariance with the predicted values and different variances. In addition, the use of tangential leverage is questionable in this case.

The partial results from this NLIN run are shown in Output 81.6.2, Output 81.6.3, and Output 81.6.4. The diagnostics plots corroborate the previously mentioned expectations: highly correlated raw residuals (with the predicted values), significant differences between tangential and Jacobian leverages and projected residuals which overcome some of the shortcomings of the raw residuals. Finally, considering the large intrinsic curvature, reparameterization might not make the model close-to-linear, perhaps necessitating the construction of another model.

Output 81.6.2: Diagnostics Panel

Output 81.6.3: Leverage Plots

Output 81.6.4: Local Influence Plot