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. The ISCENS variable indicates whether the loss exceeds the policy limit. ISCENS=1 means that the actual value of Y is greater than the recorded value; that is, Y is right-censored. If ISCENS has any other value, then the recorded value of Y is the actual value of the loss.

 /*----- Lognormal Model with left-truncation and censoring -----*/
 data test_sev2(keep=y iscens threshold 
         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);
 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 
         print=all plots(markcensored marktruncated)=pp;
     model y(lt=threshold rc=iscens(1)) / crit=aicc;

     dist logn;
     dist burr;
     dist gamma;
     dist weibull;
 run;

The MODEL statement specifies the left-truncation and right-censoring indicator variables. You need to specify that the value of 1 for the ISCENS variable indicates right-censoring, because the default indicator value is 0. Each candidate distribution needs to be specified by using a separate DIST statement. 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 P-P plots for each candidate distribution be prepared in addition to the default plots. It also instructs the procedure to mark the left-truncated and right-censored observations in the CDF plot.

Some of the key results prepared by PROC SEVERITY are shown in Figure 22.6 through Figure 22.11. The descriptive statistics of Y are shown in the second table of Figure 22.6. In addition to the estimates of the range, mean, and standard deviation of Y, the table also indicates the number of observations that are right-censored, left-truncated, and both right-censored and left-truncated. The "Model Selection Table" in Figure 22.6 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.