Modeling Rates and Proportions in SAS – 6

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From Wensui's blog on Sina

<span STYLE="font-weight: bold;">5. BETA REGRESSION
(贝塔模型)</SPAN><br />
<br />
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<br />
F(Y) = Gamma(W + T) / (Gamma(W) * Gamma(T)) * Y ^ (W &ndash; 1) * (1 &ndash; Y)
^ (T &ndash; 1)<br />
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 &ndash; Mu), where Mu is the
expected mean and Phi is another parameter governing the variance
such that sigma ^ 2 = Mu * (1 &ndash; Mu) / (1 + Phi).<br />
<br />
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 &ndash; 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.<br />
<br />
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.<br />
<br />
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