Cameron and Trivedi (1986, 1998) studied the number of doctor visits from the Australian Health Survey 1977-78. In the following data set, the dependent variable, DVISITS, contains the number of doctor visits in the past 2 weeks (0, 1, or more than 2). The explanatory variables are: SEX indicates if the patient is female; AGE is the age in years divided by 100; INCOME is the annual income ($10,000); LEVYPLUS indicates if the patient has private health insurance; FREEPOOR indicates free government health insurance due to low income; FREEREPA indicates free government health insurance for other reasons; ILLNESS is the number of illnesses in the past 2 weeks; ACTDAYS is the number of days the illness caused reduced activity; HSCORE is a questionnaire score; CHCOND1 indicates a chronic condition that does not limit activity; and CHCOND2 indicates a chronic condition that limits activity.
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.
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 | |
N = # of observations, K = # of regressors |
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 jth 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 |
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 |