The SEVERITY Procedure

PROC SEVERITY enables you to specify that the response variable values are left-truncated or right-censored. The following DATA step expands the data set of the previous example to simulate a scenario that is typically encountered by an automobile insurance company. The values of the variable Y represent the loss values on claims that are reported to an auto insurance company. The variable THRESHOLD records the deductible on the insurance policy. If the actual value of Y is less than or equal to the deductible, then it is unobservable and does not get recorded. In other words, THRESHOLD specifies the left-truncation of Y. LIMIT records the policy limit. If the value of Y is equal to or greater than the recorded value, then the observation is right-censored.

/*----- Lognormal Model with left-truncation and censoring -----*/
data test_sev2(keep=y threshold limit
        label='A Lognormal Sample With Censoring and Truncation');
   set test_sev1;
   label y='Censored & Truncated Response';
   if _n_ = 1 then call streaminit(45679);

   /* make about 20% of the observations left-truncated */
   if (rand('UNIFORM') < 0.2) then
      threshold = y * (1 - rand('UNIFORM'));
   else
      threshold = .;
   /* make about 15% of the observations right-censored */
   iscens = (rand('UNIFORM') < 0.15);
   if (iscens) then
      limit = y;
   else
      limit = .;
run;

The following statements use the AICC criterion to analyze which of the four predefined distributions (lognormal, Burr, gamma, and Weibull) has the best fit for the data:

proc severity data=test_sev2 crit=aicc
       print=all plots=(cdfperdist pp qq);
   loss y / lt=threshold rc=limit;

   dist logn burr gamma weibull;
run;

The LOSS statement specifies the left-truncation and right-censoring variables. The DIST statement specifies the candidate distributions. The PRINT= option in the PROC SEVERITY statement requests that all the displayed output be prepared. The PLOTS= option in the PROC SEVERITY statement requests that the CDF plot, P-P plot, and Q-Q plot be prepared for each candidate distribution in addition to the default plots.

Some of the key results prepared by PROC SEVERITY are shown in Figure 23.6 through Figure 23.13. In addition to the estimates of the range, mean, and standard deviation of Y, the "Descriptive Statistics for y" table shown in Figure 23.6 also indicates the number of observations that are left-truncated or right-censored. The "Model Selection" table in Figure 23.6 shows that models with all the candidate distributions have converged and that the Logn (lognormal) model has the best fit for the data according to the AICC criterion.

PROC SEVERITY also prepares a table that shows all the fit statistics for all the candidate models. It is useful to see which model would be the best fit according to each of the criteria. The "All Fit Statistics" table prepared for this example is shown in Figure 23.7. It indicates that the lognormal model is chosen by all the criteria.

The plot that compares EDF and CDF estimates is shown in Figure 23.8. When you specify left-truncation, both the EDF and CDF estimates are conditional on the response variable being greater than the smallest left-truncation threshold in the sample.

Figure 23.8: EDF and CDF Estimates for the Truncated and Censored Data

When you specify the PLOTS=CDFPERDIST option, PROC SEVERITY prepares a plot that compares the nonparametric EDF estimates with the parametric CDF estimates for each distribution. These plots for lognormal and Weibull distributions are shown in Figure 23.9. These plots also contain the lower and upper confidence limits of EDF for the specified confidence level. Because no confidence level is specified in the EDFALPHA= option in the PROC SEVERITY statement, a default confidence level of 95% is used, which is equivalent to specifying EDFALPHA=0.05. If the CDF estimates lie entirely within the EDF confidence interval, then you can be 95% confident that the parametric and nonparametric estimates are in agreement.

There are two additional ways to compare nonparametric (empirical) and parametric estimates for each model that has not failed to converge:

/*------ Specifying initial values using INIT= option -------*/
proc severity data=test_sev2 crit=aicc print=all plots=none;
   loss y / lt=threshold rc=limit;

   dist burr(init=(theta=4.62348 alpha=1.15706 gamma=6.41227));
run;