Modeling Rates and Proportions in SAS - 5

shiyiming

From Wensui's blog on Sina

<p STYLE="text-align:justify;text-justify:inter-ideograph; line-height:150%">
<b STYLE="mso-bidi-font-weight:normal"><span STYLE="font-family:&quot;Arial&quot;,&quot;sans-serif&quot;">
4. TOBIT MODEL</SPAN></B></P>
<span STYLE="font-size: 11pt; line-height: 115%; font-family: &quot;Arial&quot;,&quot;sans-serif&quot;;">Tobit
model can be considered a special case of na&iuml;ve OLS regression with
the dependent variable censored and therefore observable only in a
certain interval, which is (0, 1) for rates and proportions in my
study. Specifically, this class of models assumes that there is a
latent variable Y* such that<br />
<br />
&nbsp;&nbsp;&nbsp; Y* = X`B +
e, where e ~ N(0, sigma ^
2)&nbsp;&nbsp;&nbsp;
&amp;&nbsp;&nbsp;&nbsp;
Y = Min(1, Max(0, Y*))<br />
<br />
As a result, the representation of rates and proportions, Y,
bounded by (0, 1) can be considered the observable part of a
normally distributed variable Y* ~ N(X`B, sigma ^ 2) on the real
line. However, a fundamental argument against this censoring
assumption is that the reason that values outside the boundary of
[0, 1] are not observable is not because they are censored but
because they are not defined. Hence, the censored normal
distribution might not be an appropriate assumption for percentages
and proportions.<br />
<br />
In SAS, the most convenient way to model Tobit model is QLIM
procedure in SAS / ETS module. However, in order to clearly
illustrate the log likelihood function of Tobit model, we’d like
stick to NLMIXED procedure and estimate the Tobit model with
maximum likelihood estimation. The maximum likelihood estimator for
a Tobit model assumes that the errors are normal and homoscedastic
and would be otherwise inconsistent. In the previous section, it is
shown that the heteroscedasticity presents due to the nature of
rate and proportion outcomes. As a result, the simultaneous
estimation of a variance model is also necessary to account for
heteroscedasticity by the function VAR(e) = sigma ^ 2 * (1 +
EXP(Z`G)). Therefore, there are 2 components in our Tobit model
specification, a mean sub-model and a variance sub-model. As shown
in the output below, a couple independent variables, e.g. X1 and
X2, are significant in both mean and variance models, confirming
that the conditional variance is not independent of the mean in
proportion outcomes.<br />
<br /></SPAN><a HREF="http://blog.photo.sina.com.cn/showpic.html#url=http://s2.sinaimg.cn/orignal/a28fc28agbc0433b05071" TARGET="_blank"><img SRC="http://s2.sinaimg.cn/middle/a28fc28agbc0433b05071&amp;690" STYLE="margin: 0pt auto;display:block" NAME="image_operate_40201332626180343" HEIGHT="481" WIDTH="690" /></A><br />
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