Modeling Rates and Proportions in SAS – 7

shiyiming

From Wensui's blog on Sina

<div><b>6. SIMPLEX MODEL (单纯模型：一种不常见的统计模型)</B></DIV>
<div><br /></DIV>
<div>Dispersion models proposed by Jorgensen (1997) can be
considered a more general case of Generalized Linear Models by
McCullagh and Nelder (1989) and include a dispersion parameter
describing the distributional shape. The simplex model developed by
Barndorff-Nielsen and Jorgensen (1991) is a special dispersion
model and is useful to model proportional outcomes. A simplex model
has the density function given by</DIV>
<div>F(Y) = (2 * pi * sigma ^ 2 * (Y * (1 &ndash; Y)) ^ 3) ^ (-0.5) *
EXP((-1 / (2 * sigma ^ 2)) * d(Y; Mu))</DIV>
<div>where d(Y; Mu) = (Y &ndash; Mu) ^ 2 / (Y * (1 &ndash; Y) * Mu ^ 2 * (1 &ndash;
Mu) ^ 2) is a unit deviance function.&nbsp;</DIV>
<div><br /></DIV>
<div>Similar to the Beta model, a simplex model also consists of 2
components. The first is a sub-model to describe the expected mean
Mu. Since 0 &lt; Mu &lt; 1, the logit link
function can be used to specify the relationship between the
expected mean and covariates X such that LOG(Mu / (1 &ndash; Mu)) = X`B.
The second is a sub-model to describe the pattern of dispersion
parameter sigma ^ 2 also by a set of covariates Z such that
LOG(sigma ^ 2) = Z`G. Due to the similar nature of parameterization
between Beta model and Simplex model, model performances of these 2
often have been compared with each other. However, it is still an
open question which model is able to outperform its
competitor.</DIV>
<div><br /></DIV>
<div>Similar to the case of Beta model, there is no out-of-box
procedure in SAS estimating the simplex model. However, following
its density function, we are able to model the simplex model with
NLMIXED procedure as given below.&nbsp;</DIV>
<div><a HREF="http://blog.photo.sina.com.cn/showpic.html#url=http://s6.sinaimg.cn/orignal/a28fc28agbfde4857a905" TARGET="_blank"><img NAME="image_operate_91311336861355426" SRC="http://s6.sinaimg.cn/middle/a28fc28agbfde4857a905&amp;690" WIDTH="690" HEIGHT="411" /></A></DIV>