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Modeling Rates and Proportions in SAS – 6


Beta regression is a flexible modeling technique based upon the 2-parameter beta distribution and can be employed to model any dependent variable that is continuous and bounded by 2 known endpoints, e.g. 0 and 1 in our context. Assumed that Y follows a standard beta distribution defined in the interval (0, 1) with 2 shape parameters W and T, the density function can be specified as
F(Y) = Gamma(W + T) / (Gamma(W) * Gamma(T)) * Y ^ (W – 1) * (1 – Y) ^ (T – 1)
In the above function, while W is pulling the density toward 0, T is pulling the density toward 1. Without the loss of generality, W and T can be re-parameterized and translated into 2 other parameters, namely location parameter Mu and dispersion parameter Phi such that W = Mu * Phi and T = Phi * (1 – Mu), where Mu is the expected mean and Phi is another parameter governing the variance such that sigma ^ 2 = Mu * (1 – Mu) / (1 + Phi).

Within the framework of generalized linear models (GLM), Mu and Phi can be modeled separately with 2 overlapping or identical sets of covariates X and Z, a location sub-model for Mu and the other dispersion sub-model for Phi. Since the expected mean Mu is bounded by 0 and 1, a natural choice of the link function for location sub-model is logit such that LOG(Mu / (1 – Mu)) = X`B. With the strictly positive nature of Phi, a log function seems appropriate to serve our purpose such that LOG(Phi) = – Z`G, in which the negative sign is only for the purpose of easy interpretation such that the positive G represents a positive impact on the variance.

SAS does not provide the out-of-box procedure to estimate Beta regression. While GLIMMIX procedure is claimed to accommodate Beta modeling, it can only estimate a simple-form of Beta regression without the dispersion sub-model. However, with the density function of Beta distribution, it is extremely easy to model Beta regression with NLMIXED procedure by specifying the log likelihood function. In addition, for the data with a relatively small size, Beta regression estimated with NLMIXED procedure converges very well by setting initial values of parameter estimates equal to parameters from TOBIT model in the previous session.


Written by statcompute

April 8, 2012 at 5:31 pm

Posted in SAS, Statistical Models