Example 21.1 Ordered Data Modeling
Cameron and Trivedi (1986) studied Australian Health Survey data. Variable definitions are given in Cameron and Trivedi (1998, p. 68).
The dependent variable, dvisits, has nine ordered values. The following SAS statements estimate the ordinal probit model:
/*-- Ordered Discrete Responses --*/
proc qlim data=docvisit;
model dvisits = sex age agesq income levyplus
freepoor freerepa illness actdays hscore
chcond1 chcond2 / discrete;
run;
The output of the QLIM procedure for ordered data modeling is shown in Output 21.1.1.
Output 21.1.1
Ordered Data Modeling
0 |
4141 |
79.79 |
1 |
782 |
15.07 |
2 |
174 |
3.35 |
3 |
30 |
0.58 |
4 |
24 |
0.46 |
5 |
9 |
0.17 |
6 |
12 |
0.23 |
7 |
12 |
0.23 |
8 |
6 |
0.12 |
1 |
dvisits |
5190 |
-3138 |
0.0003675 |
82 |
Quasi-Newton |
6316 |
6447 |
789.73 |
2 * (LogL - LogL0) |
7065.9 |
- 2 * LogL0 |
0.1321 |
R / (R+N) |
0.1412 |
1 - exp(-R/N) |
0.1898 |
(1-exp(-R/N)) / (1-exp(-U/N)) |
0.149 |
1 - (1-R/U)^(U/N) |
0.1416 |
1 - ((LogL-K)/LogL0)^(-2/N*LogL0) |
0.1118 |
R / U |
0.2291 |
(R * (U+N)) / (U * (R+N)) |
0.2036 |
|
1 |
-1.378705 |
0.147413 |
-9.35 |
<.0001 |
1 |
0.131885 |
0.043785 |
3.01 |
0.0026 |
1 |
-0.534190 |
0.815907 |
-0.65 |
0.5126 |
1 |
0.857308 |
0.898364 |
0.95 |
0.3399 |
1 |
-0.062211 |
0.068017 |
-0.91 |
0.3604 |
1 |
0.137030 |
0.053262 |
2.57 |
0.0101 |
1 |
-0.346045 |
0.129638 |
-2.67 |
0.0076 |
1 |
0.178382 |
0.074348 |
2.40 |
0.0164 |
1 |
0.150485 |
0.015747 |
9.56 |
<.0001 |
1 |
0.100575 |
0.005850 |
17.19 |
<.0001 |
1 |
0.031862 |
0.009201 |
3.46 |
0.0005 |
1 |
0.061601 |
0.049024 |
1.26 |
0.2089 |
1 |
0.135321 |
0.067711 |
2.00 |
0.0457 |
1 |
0.938884 |
0.031219 |
30.07 |
<.0001 |
1 |
1.514288 |
0.049329 |
30.70 |
<.0001 |
1 |
1.711660 |
0.058151 |
29.43 |
<.0001 |
1 |
1.952860 |
0.072014 |
27.12 |
<.0001 |
1 |
2.087422 |
0.081655 |
25.56 |
<.0001 |
1 |
2.333786 |
0.101760 |
22.93 |
<.0001 |
1 |
2.789796 |
0.156189 |
17.86 |
<.0001 |
By default, ordinal probit/logit models are estimated assuming that the first threshold or limit parameter (
) is 0. However, this parameter can also be estimated when the LIMIT1=VARYING option is specified. The probability that
belongs to the
th category is defined as
where
is the logistic or standard normal CDF,
and
. Output 21.1.2 lists ordinal probit estimates computed in the following program. Note that the intercept term is suppressed for model identification when
is estimated.
/*-- Ordered Probit --*/
proc qlim data=docvisit;
model dvisits = sex age agesq income levyplus
freepoor freerepa illness actdays hscore
chcond1 chcond2 / discrete(d=normal) limit1=varying;
run;
Output 21.1.2
Ordinal Probit Parameter Estimates with LIMIT1=VARYING
1 |
0.131885 |
0.043785 |
3.01 |
0.0026 |
1 |
-0.534181 |
0.815915 |
-0.65 |
0.5127 |
1 |
0.857298 |
0.898371 |
0.95 |
0.3399 |
1 |
-0.062211 |
0.068017 |
-0.91 |
0.3604 |
1 |
0.137031 |
0.053262 |
2.57 |
0.0101 |
1 |
-0.346045 |
0.129638 |
-2.67 |
0.0076 |
1 |
0.178382 |
0.074348 |
2.40 |
0.0164 |
1 |
0.150485 |
0.015747 |
9.56 |
<.0001 |
1 |
0.100575 |
0.005850 |
17.19 |
<.0001 |
1 |
0.031862 |
0.009201 |
3.46 |
0.0005 |
1 |
0.061602 |
0.049024 |
1.26 |
0.2089 |
1 |
0.135322 |
0.067711 |
2.00 |
0.0457 |
1 |
1.378706 |
0.147415 |
9.35 |
<.0001 |
1 |
2.317590 |
0.150206 |
15.43 |
<.0001 |
1 |
2.892994 |
0.155198 |
18.64 |
<.0001 |
1 |
3.090367 |
0.158263 |
19.53 |
<.0001 |
1 |
3.331566 |
0.164065 |
20.31 |
<.0001 |
1 |
3.466128 |
0.168799 |
20.53 |
<.0001 |
1 |
3.712493 |
0.179756 |
20.65 |
<.0001 |
1 |
4.168502 |
0.215738 |
19.32 |
<.0001 |
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© 2008 by SAS Institute Inc., Cary, NC, USA. All
rights reserved.