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 parametereffects curvature. This example demonstrates how the diagnostics features of PROC NLIN can be used to perform postconvergence diagnostics.
The model takes the form
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 67.6.1: Bias, Skewness, and Global Nonlinearity Measures
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 ParameterEffects Curvature  3.5719 
RMS ParameterEffects 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 67.6.1). 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 67.6.2, Output 67.6.3, and Output 67.6.4. 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 closetolinear, perhaps necessitating the construction of another model.