### Example 22.1 Ordered Data Modeling

Cameron and Trivedi (1986) studied Australian Health Survey data. Variable definitions are given in Cameron and Trivedi (1998, p. 68).

data docvisit;
input sex age agesq income levyplus freepoor freerepa
illness actdays hscore chcond1 chcond2 dvisits;
y = (dvisits > 0);
if ( dvisits > 8 ) then dvisits = 8;
datalines;
1 0.19 0.0361 0.55  1  0  0  1  4  1  0  0  1
1 0.19 0.0361 0.45  1  0  0  1  2  1  0  0  1

... more lines ...

1 0.37 0.1369 0.25  0  0  1  1  0  1  0  0  0
1 0.52 0.2704 0.65  0  0  0  0  0  0  0  0  0
0 0.72 0.5184 0.25  0  0  1  0  0  0  0  0  0
;


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 22.1.1.

Output 22.1.1: Ordered Data Modeling

 Binary Data

The QLIM Procedure

Discrete Response Profile of dvisits
Index Value Total Frequency
1 0 4141
2 1 782
3 2 174
4 3 30
5 4 24
6 5 9
7 6 12
8 7 12
9 8 6

Model Fit Summary
Number of Endogenous Variables 1
Endogenous Variable dvisits
Number of Observations 5190
Log Likelihood -3138
Number of Iterations 82
Optimization Method Quasi-Newton
AIC 6316
Schwarz Criterion 6447

Goodness-of-Fit Measures
Measure Value Formula
Likelihood Ratio (R) 789.73 2 * (LogL - LogL0)
Upper Bound of R (U) 7065.9 - 2 * LogL0
Aldrich-Nelson 0.1321 R / (R+N)
Cragg-Uhler 1 0.1412 1 - exp(-R/N)
Cragg-Uhler 2 0.1898 (1-exp(-R/N)) / (1-exp(-U/N))
Estrella 0.149 1 - (1-R/U)^(U/N)
Adjusted Estrella 0.1416 1 - ((LogL-K)/LogL0)^(-2/N*LogL0)
McFadden's LRI 0.1118 R / U
Veall-Zimmermann 0.2291 (R * (U+N)) / (U * (R+N))
McKelvey-Zavoina 0.2036

Parameter Estimates
Parameter DF Estimate Standard Error t Value Approx
Pr > |t|
Intercept 1 -1.378705 0.147413 -9.35 <.0001
sex 1 0.131885 0.043785 3.01 0.0026
age 1 -0.534190 0.815907 -0.65 0.5126
agesq 1 0.857308 0.898364 0.95 0.3399
income 1 -0.062211 0.068017 -0.91 0.3604
levyplus 1 0.137030 0.053262 2.57 0.0101
freepoor 1 -0.346045 0.129638 -2.67 0.0076
freerepa 1 0.178382 0.074348 2.40 0.0164
illness 1 0.150485 0.015747 9.56 <.0001
actdays 1 0.100575 0.005850 17.19 <.0001
hscore 1 0.031862 0.009201 3.46 0.0005
chcond1 1 0.061601 0.049024 1.26 0.2089
chcond2 1 0.135321 0.067711 2.00 0.0457
_Limit2 1 0.938884 0.031219 30.07 <.0001
_Limit3 1 1.514288 0.049329 30.70 <.0001
_Limit4 1 1.711660 0.058151 29.43 <.0001
_Limit5 1 1.952860 0.072014 27.12 <.0001
_Limit6 1 2.087422 0.081655 25.56 <.0001
_Limit7 1 2.333786 0.101760 22.93 <.0001
_Limit8 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 22.1.2 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 22.1.2: Ordinal Probit Parameter Estimates with LIMIT1=VARYING

 Binary Data

The QLIM Procedure

Parameter Estimates
Parameter DF Estimate Standard Error t Value Approx
Pr > |t|
sex 1 0.131885 0.043785 3.01 0.0026
age 1 -0.534181 0.815915 -0.65 0.5127
agesq 1 0.857298 0.898371 0.95 0.3399
income 1 -0.062211 0.068017 -0.91 0.3604
levyplus 1 0.137031 0.053262 2.57 0.0101
freepoor 1 -0.346045 0.129638 -2.67 0.0076
freerepa 1 0.178382 0.074348 2.40 0.0164
illness 1 0.150485 0.015747 9.56 <.0001
actdays 1 0.100575 0.005850 17.19 <.0001
hscore 1 0.031862 0.009201 3.46 0.0005
chcond1 1 0.061602 0.049024 1.26 0.2089
chcond2 1 0.135322 0.067711 2.00 0.0457
_Limit1 1 1.378706 0.147415 9.35 <.0001
_Limit2 1 2.317590 0.150206 15.43 <.0001
_Limit3 1 2.892994 0.155198 18.64 <.0001
_Limit4 1 3.090367 0.158263 19.53 <.0001
_Limit5 1 3.331566 0.164065 20.31 <.0001
_Limit6 1 3.466128 0.168799 20.53 <.0001
_Limit7 1 3.712493 0.179756 20.65 <.0001
_Limit8 1 4.168502 0.215738 19.32 <.0001