The HPSEVERITY Procedure

An Example of Modeling Regression Effects

Consider a scenario in which the magnitude of the response variable might be affected by some regressor (exogenous or independent) variables. The HPSEVERITY procedure enables you to model the effect of such variables on the distribution of the response variable via an exponential link function. In particular, if you have k random regressor variables denoted by $x_ j$ ( $j=1,\dotsc ,k$ ), then the distribution of the response variable Y is assumed to have the form

$Y \sim \exp (\sum _{j=1}^{k} \beta _ j x_ j) \cdot \mathcal{F}(\Theta )$

where $\mathcal{F}$ denotes the distribution of Y with parameters $\Theta$ and $\beta _ j (j=1,\dotsc ,k)$ denote the regression parameters (coefficients).

For the effective distribution of Y to be a valid distribution from the same parametric family as $\mathcal{F}$ , it is necessary for $\mathcal{F}$ to have a scale parameter. The effective distribution of Y can be written as

$Y \sim \mathcal{F}(\theta , \Omega )$

where $\theta$ denotes the scale parameter and $\Omega$ denotes the set of nonscale parameters. The scale $\theta$ is affected by the regressors as

$\theta = \theta _0 \cdot \exp (\sum _{j=1}^{k} \beta _ j x_ j)$

where $\theta _0$ denotes a base value of the scale parameter.

Given this form of the model, PROC HPSEVERITY allows a distribution to be a candidate for modeling regression effects only if it has an untransformed or a log-transformed scale parameter.

All the predefined distributions, except the lognormal distribution, have a direct scale parameter (that is, a parameter that is a scale parameter without any transformation). For the lognormal distribution, the parameter $\mu$ is a log-transformed scale parameter. This can be verified by replacing $\mu$ with a parameter $\theta = e^\mu$ , which results in the following expressions for the PDF f and the CDF F in terms of $\theta$ and $\sigma$ , respectively, where $\Phi$ denotes the CDF of the standard normal distribution:

$f(x; \theta , \sigma ) = \frac{1}{x \sigma \sqrt {2 \pi }} e^{-\frac{1}{2}\left(\frac{\log (x) - \log (\theta )}{\sigma }\right)^2} \quad \text {and} \quad F(x; \theta , \sigma ) = \Phi \left(\frac{\log (x) - \log (\theta )}{\sigma }\right)$

With this parameterization, the PDF satisfies the $f(x;\theta ,\sigma ) = \frac{1}{\theta } f(\frac{x}{\theta }; 1, \sigma )$ condition and the CDF satisfies the $F(x;\theta ,\sigma ) = F(\frac{x}{\theta }; 1, \sigma )$ condition. This makes $\theta$ a scale parameter. Hence, $\mu = \log (\theta )$ is a log-transformed scale parameter and the lognormal distribution is eligible for modeling regression effects.

The following DATA step simulates a lognormal sample whose scale is decided by the values of the three regressors X1, X2, and X3 as follows:

$\mu = \log (\theta ) = 1 + 0.75 \; \text {X1} - \text {X2} + 0.25 \; \text {X3}$

/*----------- Lognormal Model with Regressors ------------*/
data test_sev3(keep=y x1-x3
               label='A Lognormal Sample Affected by Regressors');
   array x{*} x1-x3;
   array b{4} _TEMPORARY_ (1 0.75 -1 0.25);
   call streaminit(45678);
   label y='Response Influenced by Regressors';
   Sigma = 0.25;
   do n = 1 to 100;
      Mu = b(1); /* log of base value of scale */
      do i = 1 to dim(x);
         x(i) = rand('UNIFORM');
         Mu = Mu + b(i+1) * x(i);
      end;
      y = exp(Mu) * rand('LOGNORMAL')**Sigma;
      output;
   end;
run;

The following PROC HPSEVERITY step fits the lognormal, Burr, and gamma distribution models to this data. The regressors are specified in the SCALEMODEL statement.

proc hpseverity data=test_sev3 crit=aicc print=all;
   loss y;
   scalemodel x1-x3;

   dist logn burr gamma;
run;

Some of the key results prepared by PROC HPSEVERITY are shown in Figure 9.8 through Figure 9.12. The descriptive statistics of all the variables are shown in Figure 9.8.

Figure 9.8: Summary Results for the Regression Example

The HPSEVERITY Procedure

Input Data Set
Name	WORK.TEST_SEV3
Label	A Lognormal Sample Affected by Regressors

Descriptive Statistics for y
Observations	100
Observations Used for Estimation	100
Minimum	1.17863
Maximum	6.65269
Mean	2.99859
Standard Deviation	1.12845

Descriptive Statistics for Regressors
Variable	N	Minimum	Maximum	Mean	Standard Deviation
x1	100	0.0005115	0.97971	0.51689	0.28206
x2	100	0.01883	0.99937	0.47345	0.28885
x3	100	0.00255	0.97558	0.48301	0.29709

The comparison of the fit statistics of all the models is shown in Figure 9.9. It indicates that the lognormal model is the best model according to each of the likelihood-based statistics, whereas the gamma model is the best model according to two of the three EDF-based statistics.

Figure 9.9: Comparison of Statistics of Fit for the Regression Example

All Fit Statistics
Distribution	-2 Log Likelihood		AIC		AICC		BIC		KS		AD		CvM
Logn	187.49609	*	197.49609	*	198.13439	*	210.52194	*	1.97544		17.24618		1.21665
Burr	190.69154		202.69154		203.59476		218.32256		2.09334		13.93436	*	1.28529
Gamma	188.91483		198.91483		199.55313		211.94069		1.94472	*	15.84787		1.17617	*
Note: The asterisk (*) marks the best model according to each column's criterion.

The distribution information and the convergence results of the lognormal model are shown in Figure 9.10. The iteration history gives you a summary of how the optimizer is traversing the surface of the log-likelihood function in its attempt to reach the optimum. Both the change in the log likelihood and the maximum gradient of the objective function with respect to any of the parameters typically approach 0 if the optimizer converges.

Figure 9.10: Convergence Results for the Lognormal Model with Regressors

The HPSEVERITY Procedure

Logn Distribution

Distribution Information
Name	Logn
Description	Lognormal Distribution
Distribution Parameters	2
Regression Parameters	3

Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.

Optimization Iteration History
Iter	Function Calls	-Log Likelihood	Change	Maximum Gradient
0	2	93.75285		6.16002
1	4	93.74805	-0.0048055	0.11031
2	6	93.74805	-1.5017E-6	0.00003376
3	10	93.74805	-1.421E-13	3.1051E-12

Optimization Summary
Optimization Technique	Trust Region
Iterations	3
Function Calls	10
Log Likelihood	-93.74805

The final parameter estimates of the lognormal model are shown in Figure 9.11. All the estimates are significantly different from 0. The estimate that is reported for the parameter Mu is the base value for the log-transformed scale parameter $\mu$ . Let $x_ i (1 \leq i \leq 3)$ denote the observed value for regressor Xi. If the lognormal distribution is chosen to model Y, then the effective value of the parameter $\mu$ varies with the observed values of regressors as

$\mu = 1.04047 + 0.65221 \, x_1 - 0.91116 \, x_2 + 0.16243 \, x_3$

These estimated coefficients are reasonably close to the population parameters (that is, within one or two standard errors).

Figure 9.11: Parameter Estimates for the Lognormal Model with Regressors

Parameter Estimates
Parameter	Estimate	Standard Error	t Value	Approx Pr > \|t\|
Mu	1.04047	0.07614	13.66	<.0001
Sigma	0.22177	0.01609	13.78	<.0001
x1	0.65221	0.08167	7.99	<.0001
x2	-0.91116	0.07946	-11.47	<.0001
x3	0.16243	0.07782	2.09	0.0395

The estimates of the gamma distribution model, which is the best model according to a majority of the EDF-based statistics, are shown in Figure 9.12. The estimate that is reported for the parameter Theta is the base value for the scale parameter $\theta$ . If the gamma distribution is chosen to model Y, then the effective value of the scale parameter is $\theta = 0.14293 \, \exp (0.64562 \, x_1 - 0.89831 \, x_2 + 0.14901 \, x_3)$ .

Figure 9.12: Parameter Estimates for the Gamma Model with Regressors

Parameter Estimates
Parameter	Estimate	Standard Error	t Value	Approx Pr > \|t\|
Theta	0.14293	0.02329	6.14	<.0001
Alpha	20.37726	2.93277	6.95	<.0001
x1	0.64562	0.08224	7.85	<.0001
x2	-0.89831	0.07962	-11.28	<.0001
x3	0.14901	0.07870	1.89	0.0613