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 logtransformed 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 logtransformed 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 logtransformed 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 x1x3 label='A Lognormal Sample Affected by Regressors'); array x{*} x1x3; 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 SCALEMODEL statement. The DFMIXTURE= option in the SCALEMODEL statement specifies the method of computing the CDF estimates that are used to compute the EDFbased statistics of fit.
proc severity data=test_sev3 crit=aicc print=all; loss y; scalemodel x1x3 / dfmixture=full; dist logn burr gamma; run;
Some of the key results prepared by PROC SEVERITY are shown in Figure 23.14 through Figure 23.18. The descriptive statistics of all the variables are shown in Figure 23.14.
Figure 23.14: Summary Results for the Regression Example
Input Data Set  

Name  WORK.TEST_SEV3 
Label  A Lognormal Sample Affected by Regressors 
Descriptive Statistics for Variable y  

Number of Observations  100 
Number of Observations Used for Estimation  100 
Minimum  1.17863 
Maximum  6.65269 
Mean  2.99859 
Standard Deviation  1.12845 
Descriptive Statistics for the Regressor Variables  

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 23.15. It indicates that the lognormal model is the best model according to each of the likelihoodbased statistics, whereas the gamma model is the best model according to two of the three EDFbased statistics.
Figure 23.15: Comparison of Statistics of Fit for the Regression Example
All Fit Statistics Table  

Distribution  2 Log Likelihood 
AIC  AICC  BIC  KS  AD  CvM  
Logn  187.49609  *  197.49609  *  198.13439  *  210.52194  *  0.68991  *  0.74299  0.11044  
Burr  190.69154  202.69154  203.59476  218.32256  0.72348  0.73064  0.11332  
Gamma  188.91483  198.91483  199.55313  211.94069  0.69101  0.72219  *  0.10546  * 
The distribution information and the convergence results of the lognormal model are shown in Figure 23.16. The iteration history gives you a summary of how the optimizer is traversing the surface of the loglikelihood 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 23.16: Convergence Results for the Lognormal Model with Regressors
Distribution Information  

Name  Logn 
Description  Lognormal Distribution 
Number of Distribution Parameters  2 
Number of Regression Parameters  3 
Convergence Status for Logn Distribution 

Convergence criterion (GCONV=1E8) satisfied. 
Optimization Iteration History for Logn Distribution  

Iter  Number of Function Evaluations 
Log Likelihood  Change in Log Likelihood  Maximum Gradient 
0  2  93.75285  .  6.16002 
1  4  93.74805  0.0048055  0.11031 
2  6  93.74805  1.50188E6  0.0000338 
3  10  93.74805  1.279E13  3.119E12 
Optimization Summary for Logn Distribution  

Optimization Technique  Trust Region 
Number of Iterations  3 
Number of Function Evaluations  10 
Log Likelihood  93.74805 
The final parameter estimates of the lognormal model are shown in Figure 23.17. All the estimates are significantly different from . The estimate that is reported for the parameter Mu is the base value for the logtransformed scale parameter . Let denote the observed value for regressor X
. If the lognormal distribution is chosen to model , then the effective value of the parameter varies with the observed values of regressors as

These estimated coefficients are reasonably close to the population parameters (that is, within one or two standard errors).
Figure 23.17: Parameter Estimates for the Lognormal Model with Regressors
Parameter Estimates for Logn Distribution  

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 EDFbased statistics, are shown in Figure 23.18. The estimate that is reported for the parameter Theta is the base value for the scale parameter . If the gamma distribution is chosen to model , then the effective value of the scale parameter is .
Figure 23.18: Parameter Estimates for the Gamma Model with Regressors
Parameter Estimates for Gamma Distribution  

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 