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 k random regressor variables denoted by (), then the distribution of the response variable Y is assumed to have the form
where denotes the distribution of Y with parameters and denote the regression parameters (coefficients). For the effective distribution of Y to be a valid distribution from the same parametric family as , it is necessary for to have a scale parameter. The effective distribution of Y 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 f and the CDF F 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 SCALEMODEL statement. The DFMIXTURE= option in the SCALEMODEL statement specifies the method of computing the CDF estimates that are used to compute the EDF-based statistics of fit.
proc severity data=test_sev3 crit=aicc print=all; loss y; scalemodel x1-x3 / 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.
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 likelihood-based statistics, whereas the gamma model is the best model according to two of the three EDF-based statistics.
Figure 23.15: 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 | * | 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 | * | |||||
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 23.16. 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.
The final parameter estimates of the lognormal model are shown in Figure 23.17. 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 . Let denote the observed value for regressor X
i. If the lognormal distribution is chosen to model Y, 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).
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 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 Y, then the effective value of the scale parameter is .