Quasi-Binomial Model in SAS
Similar to quasi-Poisson regressions, quasi-binomial regressions try to address the excessive variance by the inclusion of a dispersion parameter. In addition to addressing the over-dispersion, quasi-binomial regressions also demonstrate extra values in other areas, such as LGD model development in credit risk modeling, due to its flexible distributional assumption.
Measuring the ratio between NCO and GCO, LGD could take any value in the range [0, 1] with no unanimous consensus on the distributional assumption currently in the industry. An advantage of quasi-binomial regression is that it makes no assumption of a specific distribution but merely specifies the conditional mean for a given model response. As a result, the trade-off is the lack of likelihood-based measures such as AIC and BIC.
Below is a demonstration on how to estimate a quasi-binomial model with GLIMMIX procedure in SAS.
proc glimmix data = _last_; model y = age number start / link = logit solution; _variance_ = _mu_ * (1-_mu_); random _residual_; run; /* Model Information Data Set WORK.KYPHOSIS Response Variable y Response Distribution Unknown Link Function Logit Variance Function _mu_ * (1-_mu_) Variance Matrix Diagonal Estimation Technique Quasi-Likelihood Degrees of Freedom Method Residual Parameter Estimates Standard Effect Estimate Error DF t Value Pr > |t| Intercept -2.0369 1.3853 77 -1.47 0.1455 age 0.01093 0.006160 77 1.77 0.0800 number 0.4106 0.2149 77 1.91 0.0598 start -0.2065 0.06470 77 -3.19 0.0020 Residual 0.9132 . . . . */
For the comparison purpose, the same model is also estimated with R glm() function, showing identical outputs.
summary(glm(data = kyphosis, Kyphosis ~ ., family = quasibinomial)) #Coefficients: # Estimate Std. Error t value Pr(>|t|) #(Intercept) -2.03693 1.38527 -1.470 0.14552 #Age 0.01093 0.00616 1.774 0.07996 . #Number 0.41060 0.21489 1.911 0.05975 . #Start -0.20651 0.06470 -3.192 0.00205 ** #--- #(Dispersion parameter for quasibinomial family taken to be 0.913249)