Consider a scenario in which the magnitude of the response variable might be affected by some regressor (exogenous or independent) variables. The SEVERITY 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 random regressor variables denoted by (), then the distribution of the response variable is assumed to have the form

where denotes the distribution of with parameters and denote the regression parameters (coefficients). For the effective distribution of to be a valid distribution from the same parametric family as , it is necessary for to have a scale parameter. The effective distribution of can be written as

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

where denotes a base value of the scale parameter.

Given this form of the model, PROC SEVERITY 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 is a log-transformed scale parameter. This can be verified by replacing with a parameter , which results in the following expressions for the PDF and the CDF in terms of and , respectively, where denotes the CDF of the standard normal distribution:

With this parameterization, the PDF satisfies the condition and the CDF satisfies the condition. This makes a scale parameter. Hence, 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:

 /*----------- 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 SEVERITY step fits the lognormal, Burr, and gamma distribution models to this data. The regressors are specified in the MODEL statement.

 proc severity data=test_sev3 print=all;
     model y = x1-x3 / crit=aicc;

     dist logn;
     dist burr;
     dist gamma;
 run;

Some of the key results prepared by PROC SEVERITY are shown in Figure 22.12 through Figure 22.16. The descriptive statistics of all the variables are shown in Figure 22.12.