5. Ordered Logit/Probit Regression Models
Suppose we have an ordinal dependent variable such as the degree of illegal parking (0=none, 1=sometimes, and 2=often). The ordered logit and probit models have the parallel regression assumption, which is violated from time to time.
5.1 Ordered Logit/Probit in STATA (.ologit and .oprobit)
STATA has the .ologit and .oprobit commands to estimate the ordered logit and probit models, respectively.
. ologit parking income age male
Iteration
0: log likelihood = -103.78713
Iteration 1: log likelihood = -92.739147
Iteration 2: log likelihood = -90.036393
Iteration 3: log likelihood = -89.861679
Iteration 4: log likelihood = -89.860105
Iteration 5: log likelihood = -89.860105
Ordered logistic regression Number of obs = 437
LR chi2(3) = 27.85
Prob > chi2 = 0.0000
Log likelihood = -89.860105 Pseudo R2 = 0.1342
------------------------------------------------------------------------------
parking | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
income | -.5140709 1.283192 -0.40 0.689 -3.029082 2.00094
age | -.7362588 .1894339 -3.89 0.000 -1.107542 -.3649752
male | -1.227092 .4705859 -2.61 0.009 -2.149423 -.3047605
-------------+----------------------------------------------------------------
/cut1 | -12.74479 3.787616 -20.16839 -5.321203
/cut2 | -10.83295 3.801685 -18.28412 -3.381786
------------------------------------------------------------------------------
Iteration 1: log likelihood = -92.739147
Iteration 2: log likelihood = -90.036393
Iteration 3: log likelihood = -89.861679
Iteration 4: log likelihood = -89.860105
Iteration 5: log likelihood = -89.860105
Ordered logistic regression Number of obs = 437
LR chi2(3) = 27.85
Prob > chi2 = 0.0000
Log likelihood = -89.860105 Pseudo R2 = 0.1342
------------------------------------------------------------------------------
parking | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
income | -.5140709 1.283192 -0.40 0.689 -3.029082 2.00094
age | -.7362588 .1894339 -3.89 0.000 -1.107542 -.3649752
male | -1.227092 .4705859 -2.61 0.009 -2.149423 -.3047605
-------------+----------------------------------------------------------------
/cut1 | -12.74479 3.787616 -20.16839 -5.321203
/cut2 | -10.83295 3.801685 -18.28412 -3.381786
------------------------------------------------------------------------------
STATA estimates taus, /cut1 and /cut2, assuming beta0=0 (Long and Freese 2003). This parameterization is different from that of SAS and LIMDEP, which assume tau0=0.
. oprobit parking income age male
Iteration
0: log likelihood = -103.78713
Iteration 1: log likelihood = -90.990455
Iteration 2: log likelihood = -89.496288
Iteration 3: log likelihood = -89.430915
Iteration 4: log likelihood = -89.430754
Ordered probit regression Number of obs = 437
LR chi2(3) = 28.71
Prob > chi2 = 0.0000
Log likelihood = -89.430754 Pseudo R2 = 0.1383
------------------------------------------------------------------------------
parking | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
income | -.1869839 .6116037 -0.31 0.760 -1.385705 1.011737
age | -.3594853 .0924817 -3.89 0.000 -.540746 -.1782246
male | -.5867871 .2205253 -2.66 0.008 -1.019009 -.1545655
-------------+----------------------------------------------------------------
/cut1 | -6.000986 1.869046 -9.664248 -2.337724
/cut2 | -5.118676 1.862909 -8.769911 -1.467442
------------------------------------------------------------------------------
Iteration 1: log likelihood = -90.990455
Iteration 2: log likelihood = -89.496288
Iteration 3: log likelihood = -89.430915
Iteration 4: log likelihood = -89.430754
Ordered probit regression Number of obs = 437
LR chi2(3) = 28.71
Prob > chi2 = 0.0000
Log likelihood = -89.430754 Pseudo R2 = 0.1383
------------------------------------------------------------------------------
parking | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
income | -.1869839 .6116037 -0.31 0.760 -1.385705 1.011737
age | -.3594853 .0924817 -3.89 0.000 -.540746 -.1782246
male | -.5867871 .2205253 -2.66 0.008 -1.019009 -.1545655
-------------+----------------------------------------------------------------
/cut1 | -6.000986 1.869046 -9.664248 -2.337724
/cut2 | -5.118676 1.862909 -8.769911 -1.467442
------------------------------------------------------------------------------
5.2 The Parallel Assumption and the Generalized Ordered Logit Model
The .brant command of SPost is valid only in the .ologit command. This command tests the parallel regression assumption of the ordinal regression model. The outputs here are skipped.
. quietly ologit parking income male
. brant
The parallel regression assumption is often violated. If this is the case, you may use the multinomial regression model or estimate the generalized ordered logit model (GOLM) using either the .gologit command written by Fu (1998) or the .gologit2 command by Williams (2005).
. gologit2 parking income age male, autofit
------------------------------------------------------------------------------
Testing parallel lines assumption using the .05 level of significance...
Step 1: male meets the pl assumption (P Value = 0.9901)
Step 2: income meets the pl assumption (P Value = 0.8958)
Step 3: age meets the pl assumption (P Value = 0.7964)
Step 4: All explanatory variables meet the pl assumption
Wald test of parallel lines assumption for the final model:
( 1) [0]male - [1]male = 0
( 2) [0]income - [1]income = 0
( 3) [0]age - [1]age = 0
chi2( 3) = 0.04
Prob > chi2 = 0.9982
An insignificant test statistic indicates that the final model
does not violate the proportional odds/ parallel lines assumption
If you re-estimate this exact same model with gologit2, instead
of autofit you can save time by using the parameter
pl(male income age)
------------------------------------------------------------------------------
Generalized Ordered Logit Estimates Number of obs = 437
Wald chi2(3) = 21.74
Prob > chi2 = 0.0001
Log likelihood = -89.860105 Pseudo R2 = 0.1342
( 1) [0]male - [1]male = 0
( 2) [0]income - [1]income = 0
( 3) [0]age - [1]age = 0
------------------------------------------------------------------------------
parking | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
0 |
income | -.5140709 1.283192 -0.40 0.689 -3.029082 2.00094
age | -.7362588 .1894339 -3.89 0.000 -1.107543 -.3649752
male | -1.227092 .4705859 -2.61 0.009 -2.149423 -.3047605
_cons | 12.74479 3.787616 3.36 0.001 5.321202 20.16839
-------------+----------------------------------------------------------------
1 |
income | -.5140709 1.283192 -0.40 0.689 -3.029082 2.00094
age | -.7362588 .1894339 -3.89 0.000 -1.107543 -.3649752
male | -1.227092 .4705859 -2.61 0.009 -2.149423 -.3047605
_cons | 10.83295 3.801686 2.85 0.004 3.381785 18.28412
------------------------------------------------------------------------------
Testing parallel lines assumption using the .05 level of significance...
Step 1: male meets the pl assumption (P Value = 0.9901)
Step 2: income meets the pl assumption (P Value = 0.8958)
Step 3: age meets the pl assumption (P Value = 0.7964)
Step 4: All explanatory variables meet the pl assumption
Wald test of parallel lines assumption for the final model:
( 1) [0]male - [1]male = 0
( 2) [0]income - [1]income = 0
( 3) [0]age - [1]age = 0
chi2( 3) = 0.04
Prob > chi2 = 0.9982
An insignificant test statistic indicates that the final model
does not violate the proportional odds/ parallel lines assumption
If you re-estimate this exact same model with gologit2, instead
of autofit you can save time by using the parameter
pl(male income age)
------------------------------------------------------------------------------
Generalized Ordered Logit Estimates Number of obs = 437
Wald chi2(3) = 21.74
Prob > chi2 = 0.0001
Log likelihood = -89.860105 Pseudo R2 = 0.1342
( 1) [0]male - [1]male = 0
( 2) [0]income - [1]income = 0
( 3) [0]age - [1]age = 0
------------------------------------------------------------------------------
parking | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
0 |
income | -.5140709 1.283192 -0.40 0.689 -3.029082 2.00094
age | -.7362588 .1894339 -3.89 0.000 -1.107543 -.3649752
male | -1.227092 .4705859 -2.61 0.009 -2.149423 -.3047605
_cons | 12.74479 3.787616 3.36 0.001 5.321202 20.16839
-------------+----------------------------------------------------------------
1 |
income | -.5140709 1.283192 -0.40 0.689 -3.029082 2.00094
age | -.7362588 .1894339 -3.89 0.000 -1.107543 -.3649752
male | -1.227092 .4705859 -2.61 0.009 -2.149423 -.3047605
_cons | 10.83295 3.801686 2.85 0.004 3.381785 18.28412
------------------------------------------------------------------------------
The QLIM, LOGISTIC, and PROBIT procedures estimate ordered logit and probit models. As shown in Tables 3 and 4, the QLIM procedure is most recommended. Note that the DIST=LOGISTIC indicates the logit model to be estimated.
PROC QLIM DATA=masil.students;
MODEL parking = income age male /DISCRETE (DIST=LOGISTIC);
RUN;
The QLIM Procedure
Discrete Response Profile of parking
Index Value Frequency Percent
1 0 413 94.51
2 1 20 4.58
3 2 4 0.92
Model Fit Summary
Number of Endogenous Variables 1
Endogenous Variable parking
Number of Observations 437
Log Likelihood -89.86011
Maximum Absolute Gradient 8.14046E-7
Number of Iterations 23
AIC 189.72021
Schwarz Criterion 210.11988
Goodness-of-Fit Measures
Measure Value Formula
Likelihood Ratio (R) 27.854 2 * (LogL - LogL0)
Upper Bound of R (U) 207.57 - 2 * LogL0
Aldrich-Nelson 0.0599 R / (R+N)
Cragg-Uhler 1 0.0618 1 - exp(-R/N)
Cragg-Uhler 2 0.1633 (1-exp(-R/N)) / (1-exp(-U/N))
Estrella 0.0662 1 - (1-R/U)^(U/N)
Adjusted Estrella 0.0418 1 - ((LogL-K)/LogL0)^(-2/N*LogL0)
McFadden's LRI 0.1342 R / U
Veall-Zimmermann 0.1861 (R * (U+N)) / (U * (R+N))
McKelvey-Zavoina 0.6462
N = # of observations, K = # of regressors
Algorithm converged.
Parameter Estimates
Standard Approx
Parameter Estimate Error t Value Pr > |t|
Intercept 12.744794 3.787615 3.36 0.0008
income -0.514071 1.283192 -0.40 0.6887
age -0.736259 0.189434 -3.89 0.0001
male -1.227092 0.470586 -2.61 0.0091
_Limit2 1.911842 0.468050 4.08 <.0001
Discrete Response Profile of parking
Index Value Frequency Percent
1 0 413 94.51
2 1 20 4.58
3 2 4 0.92
Model Fit Summary
Number of Endogenous Variables 1
Endogenous Variable parking
Number of Observations 437
Log Likelihood -89.86011
Maximum Absolute Gradient 8.14046E-7
Number of Iterations 23
AIC 189.72021
Schwarz Criterion 210.11988
Goodness-of-Fit Measures
Measure Value Formula
Likelihood Ratio (R) 27.854 2 * (LogL - LogL0)
Upper Bound of R (U) 207.57 - 2 * LogL0
Aldrich-Nelson 0.0599 R / (R+N)
Cragg-Uhler 1 0.0618 1 - exp(-R/N)
Cragg-Uhler 2 0.1633 (1-exp(-R/N)) / (1-exp(-U/N))
Estrella 0.0662 1 - (1-R/U)^(U/N)
Adjusted Estrella 0.0418 1 - ((LogL-K)/LogL0)^(-2/N*LogL0)
McFadden's LRI 0.1342 R / U
Veall-Zimmermann 0.1861 (R * (U+N)) / (U * (R+N))
McKelvey-Zavoina 0.6462
N = # of observations, K = # of regressors
Algorithm converged.
Parameter Estimates
Standard Approx
Parameter Estimate Error t Value Pr > |t|
Intercept 12.744794 3.787615 3.36 0.0008
income -0.514071 1.283192 -0.40 0.6887
age -0.736259 0.189434 -3.89 0.0001
male -1.227092 0.470586 -2.61 0.0091
_Limit2 1.911842 0.468050 4.08 <.0001
The SAS QLIM procedure estimates the intercept and , assuming . The estimated intercept of SAS is equivalent to (0-/cut1) in STATA. The _Limit2 of SAS is the difference between cut points of STATA, 1.91184=-10.83295-(-12.74479).
The SAS LOGISTIC and PROBIT procedures are also used to estimate the ordered logit and probit models. These procedures recognize binary or ordinal response models by examining the dependent variable.
PROC LOGISTIC DATA = masil.students DESC;
MODEL parking = income age male /LINK=LOGIT;
RUN;
Like the STATA .ologit command, The LOGISTIC procedure fits the model, assuming the intercept is zero. The parameter estimates and standard errors are slightly different from those of the QLIM procedure and the .ologit command. Other parts of the output are skipped.
Analysis of Maximum Likelihood Estimates
Standard Wald
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 2 1 10.8324 3.8112 8.0784 0.0045
Intercept 1 1 12.7444 3.8021 11.2354 0.0008
income 1 -0.5142 1.2908 0.1587 0.6904
age 1 -0.7362 0.1900 15.0221 0.0001
male 1 -1.2271 0.4709 6.7902 0.0092
Standard Wald
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 2 1 10.8324 3.8112 8.0784 0.0045
Intercept 1 1 12.7444 3.8021 11.2354 0.0008
income 1 -0.5142 1.2908 0.1587 0.6904
age 1 -0.7362 0.1900 15.0221 0.0001
male 1 -1.2271 0.4709 6.7902 0.0092
PROC PROBIT DATA = masil.students;
CLASS parking;
MODEL parking = income age male /DIST=LOGISTIC;
RUN;
The PROBIT procedure returns almost the same results as the QLIM procedure except for the signs of the estimates. Other parts of the output are skipped.
Analysis of Parameter Estimates
Standard 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 -12.7448 3.7876 -20.1684 -5.3212 11.32 0.0008
Intercept2 1 1.9118 0.4680 0.9945 2.8292 16.68 <.0001
income 1 0.5141 1.2832 -2.0009 3.0291 0.16 0.6887
age 1 0.7363 0.1894 0.3650 1.1075 15.11 0.0001
male 1 1.2271 0.4706 0.3048 2.1494 6.80 0.0091
Standard 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 -12.7448 3.7876 -20.1684 -5.3212 11.32 0.0008
Intercept2 1 1.9118 0.4680 0.9945 2.8292 16.68 <.0001
income 1 0.5141 1.2832 -2.0009 3.0291 0.16 0.6887
age 1 0.7363 0.1894 0.3650 1.1075 15.11 0.0001
male 1 1.2271 0.4706 0.3048 2.1494 6.80 0.0091
The QLIM procedure by default estimates a probit model. The DIST=NORMAL, the default option, may be omitted.
PROC QLIM DATA=masil.students;
MODEL parking = income age male /DISCRETE (DIST=NORMAL);
RUN;
The QLIM Procedure
Discrete Response Profile of parking
Index Value Frequency Percent
1 0 413 94.51
2 1 20 4.58
3 2 4 0.92
Model Fit Summary
Number of Endogenous Variables 1
Endogenous Variable parking
Number of Observations 437
Log Likelihood -89.43075
Maximum Absolute Gradient 4.69307E-6
Number of Iterations 17
AIC 188.86151
Schwarz Criterion 209.26117
Goodness-of-Fit Measures
Measure Value Formula
Likelihood Ratio (R) 28.713 2 * (LogL - LogL0)
Upper Bound of R (U) 207.57 - 2 * LogL0
Aldrich-Nelson 0.0617 R / (R+N)
Cragg-Uhler 1 0.0636 1 - exp(-R/N)
Cragg-Uhler 2 0.1682 (1-exp(-R/N)) / (1-exp(-U/N))
Estrella 0.0683 1 - (1-R/U)^(U/N)
Adjusted Estrella 0.0439 1 - ((LogL-K)/LogL0)^(-2/N*LogL0)
McFadden's LRI 0.1383 R / U
Veall-Zimmermann 0.1915 (R * (U+N)) / (U * (R+N))
McKelvey-Zavoina 0.3011
N = # of observations, K = # of regressors
Algorithm converged.
Parameter Estimates
Standard Approx
Parameter Estimate Error t Value Pr > |t|
Intercept 6.000986 1.869053 3.21 0.0013
income -0.186984 0.611605 -0.31 0.7598
age -0.359485 0.092482 -3.89 0.0001
male -0.586787 0.220526 -2.66 0.0078
_Limit2 0.882310 0.196555 4.49 <.0001
Discrete Response Profile of parking
Index Value Frequency Percent
1 0 413 94.51
2 1 20 4.58
3 2 4 0.92
Model Fit Summary
Number of Endogenous Variables 1
Endogenous Variable parking
Number of Observations 437
Log Likelihood -89.43075
Maximum Absolute Gradient 4.69307E-6
Number of Iterations 17
AIC 188.86151
Schwarz Criterion 209.26117
Goodness-of-Fit Measures
Measure Value Formula
Likelihood Ratio (R) 28.713 2 * (LogL - LogL0)
Upper Bound of R (U) 207.57 - 2 * LogL0
Aldrich-Nelson 0.0617 R / (R+N)
Cragg-Uhler 1 0.0636 1 - exp(-R/N)
Cragg-Uhler 2 0.1682 (1-exp(-R/N)) / (1-exp(-U/N))
Estrella 0.0683 1 - (1-R/U)^(U/N)
Adjusted Estrella 0.0439 1 - ((LogL-K)/LogL0)^(-2/N*LogL0)
McFadden's LRI 0.1383 R / U
Veall-Zimmermann 0.1915 (R * (U+N)) / (U * (R+N))
McKelvey-Zavoina 0.3011
N = # of observations, K = # of regressors
Algorithm converged.
Parameter Estimates
Standard Approx
Parameter Estimate Error t Value Pr > |t|
Intercept 6.000986 1.869053 3.21 0.0013
income -0.186984 0.611605 -0.31 0.7598
age -0.359485 0.092482 -3.89 0.0001
male -0.586787 0.220526 -2.66 0.0078
_Limit2 0.882310 0.196555 4.49 <.0001
The QLIM procedure and .oprobit command produce almost the same result except for the tau2 estimate. The _Limit2 of SAS is the difference of the cut points of STATA, .88231=-5.118676-(-6.000986).
The PROBIT and LOGISTIC procedures also estimate the ordered probit model. Keep in mind that the signs of the coefficients are reversed in the PROBIT procedure.
PROC LOGISTIC DATA = masil.students DESC;
MODEL parking = income age male /LINK=PROBIT;
RUN;
Analysis of Maximum Likelihood Estimates
Standard Wald
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 2 1 5.1181 1.8373 7.7601 0.0053
Intercept 1 1 6.0004 1.8441 10.5872 0.0011
income 1 -0.1869 0.6160 0.0921 0.7615
age 1 -0.3595 0.0908 15.6767 <.0001
male 1 -0.5868 0.2203 7.0941 0.0077
Standard Wald
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 2 1 5.1181 1.8373 7.7601 0.0053
Intercept 1 1 6.0004 1.8441 10.5872 0.0011
income 1 -0.1869 0.6160 0.0921 0.7615
age 1 -0.3595 0.0908 15.6767 <.0001
male 1 -0.5868 0.2203 7.0941 0.0077
PROC PROBIT DATA = masil.students;
CLASS parking;
MODEL parking = income age male /DIST=NORMAL;
RUN;
Analysis of Parameter Estimates
Standard 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 -6.0010 1.8691 -9.6643 -2.3377 10.31 0.0013
Intercept2 1 0.8823 0.1966 0.4971 1.2675 20.15 <.0001
income 1 0.1870 0.6116 -1.0117 1.3857 0.09 0.7598
age 1 0.3595 0.0925 0.1782 0.5407 15.11 0.0001
male 1 0.5868 0.2205 0.1546 1.0190 7.08 0.0078
Standard 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 -6.0010 1.8691 -9.6643 -2.3377 10.31 0.0013
Intercept2 1 0.8823 0.1966 0.4971 1.2675 20.15 <.0001
income 1 0.1870 0.6116 -1.0117 1.3857 0.09 0.7598
age 1 0.3595 0.0925 0.1782 0.5407 15.11 0.0001
male 1 0.5868 0.2205 0.1546 1.0190 7.08 0.0078
5.5 Ordered Logit/Probit in LIMDEP (Ordered$)
The LIMDEP Ordered$ command estimates ordered logit and probit models. The Logit$ subcommand runs the ordered logit model.
ORDERED;
Lhs=parking;
Rhs=ONE,income,age,male;
Logit$
Normal
exit from iterations. Exit status=0.
+---------------------------------------------+
| Ordered Probability Model |
| Maximum Likelihood Estimates |
| Model estimated: Sep 18, 2005 at 05:53:44PM.|
| Dependent variable PARKING |
| Weighting variable None |
| Number of observations 437 |
| Iterations completed 13 |
| Log likelihood function -89.86011 |
| Restricted log likelihood -103.7871 |
| Chi squared 27.85404 |
| Degrees of freedom 3 |
| Prob[ChiSqd > value] = .3896741E-05 |
| Underlying probabilities based on Logistic |
| Cell frequencies for outcomes |
| Y Count Freq Y Count Freq Y Count Freq |
| 0 413 .945 1 20 .045 2 4 .009 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Index function for probability
Constant 12.74479424 3.7876161 3.365 .0008
INCOME -.5140708643 1.2831923 -.401 .6887 .61683982
AGE -.7362588281 .18943391 -3.887 .0001 20.691076
MALE -1.227091964 .47058590 -2.608 .0091 .57208238
Threshold parameters for index
Mu(1) 1.911841923 .46804996 4.085 .0000
+---------------------------------------------------------------------------+
| Cross tabulation of predictions. Row is actual, column is predicted. |
| Model = Logistic . Prediction is number of the most probable cell. |
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| Actual|Row Sum| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| 0| 413| 413| 0| 0|
| 1| 20| 20| 0| 0|
| 2| 4| 4| 0| 0|
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
|Col Sum| 437| 437| 0| 0| 0| 0| 0| 0| 0| 0| 0|
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
+---------------------------------------------+
| Ordered Probability Model |
| Maximum Likelihood Estimates |
| Model estimated: Sep 18, 2005 at 05:53:44PM.|
| Dependent variable PARKING |
| Weighting variable None |
| Number of observations 437 |
| Iterations completed 13 |
| Log likelihood function -89.86011 |
| Restricted log likelihood -103.7871 |
| Chi squared 27.85404 |
| Degrees of freedom 3 |
| Prob[ChiSqd > value] = .3896741E-05 |
| Underlying probabilities based on Logistic |
| Cell frequencies for outcomes |
| Y Count Freq Y Count Freq Y Count Freq |
| 0 413 .945 1 20 .045 2 4 .009 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Index function for probability
Constant 12.74479424 3.7876161 3.365 .0008
INCOME -.5140708643 1.2831923 -.401 .6887 .61683982
AGE -.7362588281 .18943391 -3.887 .0001 20.691076
MALE -1.227091964 .47058590 -2.608 .0091 .57208238
Threshold parameters for index
Mu(1) 1.911841923 .46804996 4.085 .0000
+---------------------------------------------------------------------------+
| Cross tabulation of predictions. Row is actual, column is predicted. |
| Model = Logistic . Prediction is number of the most probable cell. |
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| Actual|Row Sum| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| 0| 413| 413| 0| 0|
| 1| 20| 20| 0| 0|
| 2| 4| 4| 0| 0|
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
|Col Sum| 437| 437| 0| 0| 0| 0| 0| 0| 0| 0| 0|
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
LIMDEP and SAS QLIM produce the same results for the ordered logit model. Note that _Limit2 in SAS is equivalent to Mu(1), the threshold parameter, in LIMDEP.
The ordered probit model is estimated by the Ordered$ command without the Logit$ subcommand. The command by default fits the ordered logit model. The output is comparable to that of the QLIM procedure.
ORDERED;
Lhs=parking;
Rhs=ONE,income,age,male;
Normal
exit from iterations. Exit status=0.
+---------------------------------------------+
| Ordered Probability Model |
| Maximum Likelihood Estimates |
| Model estimated: Sep 18, 2005 at 05:55:42PM.|
| Dependent variable PARKING |
| Weighting variable None |
| Number of observations 437 |
| Iterations completed 11 |
| Log likelihood function -89.43075 |
| Restricted log likelihood -103.7871 |
| Chi squared 28.71275 |
| Degrees of freedom 3 |
| Prob[ChiSqd > value] = .2572557E-05 |
| Underlying probabilities based on Normal |
| Cell frequencies for outcomes |
| Y Count Freq Y Count Freq Y Count Freq |
| 0 413 .945 1 20 .045 2 4 .009 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Index function for probability
Constant 6.000985035 1.8690536 3.211 .0013
INCOME -.1869836008 .61160494 -.306 .7598 .61683982
AGE -.3594852294 .92482090E-01 -3.887 .0001 20.691076
MALE -.5867870572 .22052578 -2.661 .0078 .57208238
Threshold parameters for index
Mu(1) .8823095981 .19655461 4.489 .0000
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
+---------------------------------------------------------------------------+
| Cross tabulation of predictions. Row is actual, column is predicted. |
| Model = Probit . Prediction is number of the most probable cell. |
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| Actual|Row Sum| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| 0| 413| 413| 0| 0|
| 1| 20| 20| 0| 0|
| 2| 4| 4| 0| 0|
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
|Col Sum| 874| 437| 0| 0| 0| 0| 0| 0| 0| 0| 0|
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
+---------------------------------------------+
| Ordered Probability Model |
| Maximum Likelihood Estimates |
| Model estimated: Sep 18, 2005 at 05:55:42PM.|
| Dependent variable PARKING |
| Weighting variable None |
| Number of observations 437 |
| Iterations completed 11 |
| Log likelihood function -89.43075 |
| Restricted log likelihood -103.7871 |
| Chi squared 28.71275 |
| Degrees of freedom 3 |
| Prob[ChiSqd > value] = .2572557E-05 |
| Underlying probabilities based on Normal |
| Cell frequencies for outcomes |
| Y Count Freq Y Count Freq Y Count Freq |
| 0 413 .945 1 20 .045 2 4 .009 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Index function for probability
Constant 6.000985035 1.8690536 3.211 .0013
INCOME -.1869836008 .61160494 -.306 .7598 .61683982
AGE -.3594852294 .92482090E-01 -3.887 .0001 20.691076
MALE -.5867870572 .22052578 -2.661 .0078 .57208238
Threshold parameters for index
Mu(1) .8823095981 .19655461 4.489 .0000
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
+---------------------------------------------------------------------------+
| Cross tabulation of predictions. Row is actual, column is predicted. |
| Model = Probit . Prediction is number of the most probable cell. |
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| Actual|Row Sum| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| 0| 413| 413| 0| 0|
| 1| 20| 20| 0| 0|
| 2| 4| 4| 0| 0|
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
|Col Sum| 874| 437| 0| 0| 0| 0| 0| 0| 0| 0| 0|
+-------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
5.6 Ordered Logit/Probit in SPSS
The Plum command estimates the ordered logit and probit models in SPSS. The Threshold points in SPSS are equivalent to the cut points in STATA.
PLUM parking WITH income age male
/CRITERIA = CIN(95) DELTA(0) LCONVERGE(0) MXITER(100) MXSTEP(5)
PCONVERGE(1.0E-6) SINGULAR(1.0E-8)
/LINK = LOGIT /PRINT = FIT PARAMETER SUMMARY .
PLUM parking WITH income age male
/CRITERIA = CIN(95) DELTA(0) LCONVERGE(0) MXITER(100) MXSTEP(5)
PCONVERGE(1.0E-6) SINGULAR(1.0E-8)
/LINK = PROBIT /PRINT = FIT PARAMETER SUMMARY .
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