Modeling Rates and Proportions in SAS – 8

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From Wensui's blog on Sina

<div><b>7. FRACTIONAL LOGIT MODEL</B></DIV>
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<div>Different from all models introduced previously that assume
specific distributional families for the proportional outcomes of
interests, the fractional logit model proposed by Papke and
Wooldridge (1996) is a quasi-likelihood method that does not
specify the full distribution but only requires the conditional
mean to be correctly specified for consistent parameter estimates.
Under the assumption E(Y|X) = G(X`B) = 1 / (1 + EXP(-X`B)), the
fractional logit has the identical likelihood function to the one
for a Bernoulli distribution such that</DIV>
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<div>F(Y) = (G(X`B) ** Y) * (1 &ndash; G(X`B)) ** (1 &ndash; Y) with 1
&gt;= Y &gt;= 0&nbsp;</DIV>
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<div>Based upon the above formulation, parameter estimates are
calculated in the same manner as in the binary logistic regression
by maximizing the log likelihood.&nbsp;</DIV>
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<div>In SAS, the most convenient way to implement the fractional
logit model is with GLIMMIX procedure. In addition, we can also use
NLMIXED procedure by explicitly specifying the likelihood function
as shown above.</DIV>
<div><a href="http://blog.photo.sina.com.cn/showpic.html#url=http://s11.sinaimg.cn/orignal/a28fc28agc01dec8ee4ea" TARGET="_blank"><img SRC="http://s11.sinaimg.cn/middle/a28fc28agc01dec8ee4ea&amp;690" WIDTH="690" HEIGHT="527" NAME="image_operate_74731337134705746" /></A><br />
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