The QLIM Procedure

## 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
 Binary Data

The QLIM Procedure

Discrete Response Profile of dvisits
Index Value Frequency Percent
1 0 4141 79.79
2 1 782 15.07
3 2 174 3.35
4 3 30 0.58
5 4 24 0.46
6 5 9 0.17
7 6 12 0.23
8 7 12 0.23
9 8 6 0.12

Model Fit Summary
Number of Endogenous Variables 1
Endogenous Variable dvisits
Number of Observations 5190
Log Likelihood -3138
Maximum Absolute Gradient 0.0003675
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 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
 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

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