The NLIN Procedure

Example 67.7 Parameter Profiling and Bootstrapping

This example, which uses a model from Clarke (1987), shows why it is difficult to intuitively reason about the characteristics of the parameters that occur in a nonlinear model. As a consequence, the example demonstrates the indispensability of parameter profiling and confidence curves to identify the nonlinear characteristics of model parameters. The parameter profiling is also augmented with a plot that shows the influence of each observation on the parameter estimates. Furthermore, the BOOTSTRAP statement in PROC NLIN is invoked in order to study the sampling distribution of parameter estimates and make more accurate statistical inferences.

The model takes the form

\[  \mr {E}[Y|x] = \theta _3 + \theta _2 \exp \{ \theta _1 x \}   \]

The data set in this example is from Clarke (1987). The following DATA step creates this data set:

data clarke1987a;
   input  x y;
1    3.183
2    3.059
3    2.871
4    2.622
5    2.541
6    2.184
7    2.110
8    2.075
9    2.018
10   1.903
11   1.770
12   1.762
13   1.550

The model is fit by using the following statements in the NLIN procedure:

ods graphics on;
proc nlin data=clarke1987a plots(stats=none)=diagnostics;
   parms theta1=-0.15
   profile theta1 theta3 / range = -6 to 2 by 0.2 all;
   bootstrap / nsamples = 2000 seed=123 bootplots bootci bootcov;
   model y = theta3 + theta2*exp(theta1*x);
ods graphics off;

Output 67.7.1: Parameter Estimates and Bootstrap Estimate of the Covariance Matrix

The NLIN Procedure

Parameter Estimate Approx
Std Error
Approximate 95% Confidence
Std Dev
Bootstrap Bias-Corrected
95% Confidence Limits
theta1 -0.1031 0.0255 -0.1599 -0.0462 0.0253 -0.1505 -0.0531
theta2 2.5190 0.2658 1.9268 3.1112 0.4450 2.1716 3.5325
theta3 0.9631 0.3216 0.2466 1.6797 0.4907 -0.2565 1.3697

Bootstrap Covariance Matrix Estimate
  theta1 theta2 theta3
theta1 0.00063801 0.00834138 -.01006296
theta2 0.00834138 0.19802119 -.21570467
theta3 -.01006296 -.21570467 0.24082392

By default, the BOOTCI option in the BOOTSTRAP statement adds a column that contains the bias-corrected bootstrap confidence limits to the Parameter Estimates table (Output 67.7.1). The BOOTCOV option produces the Bootstrap Covariance Matrix table (Output 67.7.1). The discrepancy between the Wald-based confidence limits and the bias-corrected bootstrap confidence limits is much greater for $\theta _3$ than for $\theta _1$. The profile t plot, confidence curve, and histograms of the bootstrap estimates show why this is the case.

The profile t plot for parameter $\theta _3$ in Output 67.7.3 shows a definite deviation from the linear reference line that has a slope of 1 and passes through the origin. Hence, Wald-based inference for $\theta _3$ is not appropriate. In contrast, the profile t plot for parameter $\theta _1$ in Output 67.7.2 shows that Wald-based inference for $\theta _1$ might be sufficient.

Output 67.7.4 and Output 67.7.5 show the confidence curves for $\theta _1$ and $\theta _3$, respectively. For $\theta _3$, you can see a significant difference between the Wald-based confidence interval and the corresponding likelihood-based interval. In such cases, the likelihood-based intervals are preferred because their coverage rate is much closer to the nominal values than the coverage rate of the Wald-based intervals (Donaldson and Schnabel, 1987; Cook and Weisberg, 1990).

Output 67.7.6 depicts the influence of each observation on the value of $\theta _3$. Observations 6 and 13 have the most influence on the value of this parameter. The plot is generated using the leave-one-out method and should be contrasted with the local influence plot in Output 67.7.7, which is based on assessing the influence of an additive perturbation of the response variable.

Output 67.7.8 and Output 67.7.9 are histograms that show the distribution of the bootstrap parameter estimates for $\theta _1$ and $\theta _3$, respectively. These histograms complement the information that is obtained about $\theta _1$ and $\theta _3$ from the profile t plots. Specifically, they show that the bootstrap parameter estimate of $\theta _1$ has a distribution close to normal, whereas that of $\theta _3$ has a distribution that deviates significantly from normal. Again, this leads to the conclusion that inferences based on linear approximations, such as Wald-based confidence intervals, work better for $\theta _1$ than for $\theta _3$.

Finally, this example shows that the adequacy of a linear approximation with regard to a certain parameter cannot be inferred directly from the model. If it could, then $\theta _3$, which enters the model linearly, would have a completely linear behavior, whereas $\theta _1$ would have a highly nonlinear behavior. However, the diagnostics that are based on the profile t plot and confidence curves, and the histograms of the bootstrap parameter estimates, show that the opposite holds. For a detailed discussion about this issue, see Cook and Weisberg (1990).

Output 67.7.2: Profile t Plot

Output 67.7.3: Profile t Plot

Output 67.7.4: Confidence Curve

Output 67.7.5: Confidence Curve

Output 67.7.6: Observation Influence on Parameter Estimate

Output 67.7.7: Local Influence Plot

Output 67.7.8: Histogram of Bootstrap Parameter Estimates

Output 67.7.9: Histogram of Bootstrap Parameter Estimates