Ordered Logit Regression
Ordered logit model has the form:
This model is known as the proportional-odds model because the odds ratio of the event 
is independent of the category j. The odds ratio is assumed to be constant for all categories.
LOGISTIC Procedure
SAS PROC LOGISTIC is a procedure you can use for an ordered multiple outcome model as well as for a binary model. All previous discussions about the binary logistic regression estimation in PROC LOGISTIC are also valid for ordered logit model. To fit an ordered logit model, you can use:
proc logistic; model y=x1 x2; run;
where Y is the ordinally scaled multiple response variable, and X1 and X2 are two regressors of interest.
The following data are from McCullagh and Nelder (McCullagh and Nelder, 1989, Generalized Linear Models, London, Chapman Hall, p. 175) and used in a SAS manual (SAS, 1996, SAS/STAT Software Changes and Enhancements through Release 6.11, pp. 435-438). Consider a study of the effects on taste of various cheese additives. Researchers tested four cheese additives and obtained 52 response ratings for each additive. Each response was measured on a scale of nine values ranging from strong dislike (1) to excellent taste (9). The data set CHEESE has five variables Y, X1, X2, X3, and F. The variable Y contains the response rating and the variables X1, X2, and X3 are dummy variables, representing the first, second, and third additive, respectively; for the fourth additive, X1=X2=X3=0. F gives the frequency of occurrence of the observation. The following DATA step creates the data set CHEESE:
data cheese; input x1 x2 x3 y f; cards; 1 0 0 1 0 1 0 0 2 0 1 0 0 3 1 1 0 0 4 7 1 0 0 5 8 1 0 0 6 8 1 0 0 7 19 1 0 0 8 8 1 0 0 9 1 0 1 0 1 6 0 1 0 2 9 0 1 0 3 12 0 1 0 4 11 0 1 0 5 7 0 1 0 6 6 0 1 0 7 1 0 1 0 8 0 0 1 0 9 0 0 0 1 1 1 0 0 1 2 1 0 0 1 3 6 0 0 1 4 8 0 0 1 5 23 0 0 1 6 7 0 0 1 7 5 0 0 1 8 1 0 0 1 9 0 0 0 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 1 0 0 0 5 3 0 0 0 6 7 0 0 0 7 14 0 0 0 8 16 0 0 0 9 11 ;
Because the response variable Y is ordinally scaled, you can estimate an ordered logit model. You can use:
proc logistic data=cheese; freq f; model y=x1-x3; run;
You will have the following SAS output:
Sample Program: Ordered Logit Regression
The LOGISTIC Procedure
Data Set: WORK.CHEESE
Response Variable: Y
Response Levels: 9
Number of Observations: 28
Frequency Variable: F
Link Function: Logit
Response Profile
Ordered
Value Y Count
1 1 7
2 2 10
3 3 19
4 4 27
5 5 41
6 6 28
7 7 39
8 8 25
9 9 12
NOTE: 8 observation(s) having zero frequencies or weights were excluded since
they do not contribute to the analysis.
Score Test for the Proportional Odds Assumption
Chi-Square = 17.2868 with 21 DF (p=0.6936)
Model Fitting Information and Testing Global Null Hypothesis BETA=0
Intercept
Intercept and
Criterion Only Covariates Chi-Square for Covariates
AIC 875.802 733.348 .
SC 902.502 770.061 .
-2 LOG L 859.802 711.348 148.454 with 3 DF (p=0.0001)
Score . . 111.267 with 3 DF (p=0.0001)
Analysis of Maximum Likelihood Estimates
Parameter Standard Wald Pr > Standardized Odds
Variable DF Estimate Error Chi-Square Chi-Square Estimate Ratio
INTERCP1 1 -7.0802 0.5624 158.4865 0.0001 . .
INTERCP2 1 -6.0250 0.4755 160.5507 0.0001 . .
INTERCP3 1 -4.9254 0.4272 132.9477 0.0001 . .
INTERCP4 1 -3.8568 0.3902 97.7086 0.0001 . .
INTERCP5 1 -2.5206 0.3431 53.9713 0.0001 . .
INTERCP6 1 -1.5685 0.3086 25.8379 0.0001 . .
INTERCP7 1 -0.0669 0.2658 0.0633 0.8013 . .
INTERCP8 1 1.4930 0.3310 20.3443 0.0001 . .
X1 1 1.6128 0.3778 18.2258 0.0001 0.385954 5.017
X2 1 4.9646 0.4741 109.6453 0.0001 1.188080 143.257
X3 1 3.3227 0.4251 61.0936 0.0001 0.795146 27.735
The LOGISTIC Procedure
Association of Predicted Probabilities and Observed Responses
Concordant = 67.6% Somers' D = 0.578
Discordant = 9.8% Gamma = 0.746
Tied = 22.6% Tau-a = 0.500
(18635 pairs) c = 0.789
This result shows eight fitted regression lines as follows:
where p1 is the probability of being strongly disliked, i.e, the probability of Y=1, and so on. Positive coefficients of X1, X2 and X3 indicate that adding those additives is associated with increased probability of the cheese being disliked. The estimated odds are reported 5.017, 143.257 and 27.735 for X1, X2 and X3 respectively. Each odd is constant for all categories.
and so on. However, this computation can be easily obtained for each combination of additives by using:
proc logistic data=cheese; freq f; model y=x1-x3; output out=prob predicted=phat; run; proc print data=prob; run;
You will have the following additional output:
Sample Program: Ordered Logit Regression
OBS X1 X2 X3 Y F _LEVEL_ PHAT
1 1 0 0 1 0 1 0.00420
2 1 0 0 1 0 2 0.01198
3 1 0 0 1 0 3 0.03514
4 1 0 0 1 0 4 0.09587
5 1 0 0 1 0 5 0.28746
6 1 0 0 1 0 6 0.51106
7 1 0 0 1 0 7 0.82432
8 1 0 0 1 0 8 0.95713
.
.
281 0 0 0 9 11 1 0.00084
282 0 0 0 9 11 2 0.00241
283 0 0 0 9 11 3 0.00721
284 0 0 0 9 11 4 0.02070
285 0 0 0 9 11 5 0.07443
286 0 0 0 9 11 6 0.17242
287 0 0 0 9 11 7 0.48329
288 0 0 0 9 11 8 0.81652
In this output you have 8 observations for each additive-response combination. The observation with _LEVEL_=1 shows the predicted probability of Y=1, the observation with _LEVEL_=2 shows the predicted probability of Y=1 or 2, and so on.
PROBIT Procedure
To use the SAS PROC PROBIT to fit an ordered logit model, use the syntax:
proc probit; class y; model y = x1 x2 / d=logistic; run;
where Y is the ordinally scaled multiple response variable, and X1 and X2 are two regressors of interest.
In this example, you are using the same data set as in Example 19. However, SAS PROC PROBIT does not accept a data set in a frequency format. You need to have the same data set in an individual data format; i.e., you need to have:
X1 X2 X3 Y
1 0 0 3
1 0 0 4
1 0 0 4
.
.
1 0 0 4 (7 rows of the same data)
1 0 0 5
1 0 0 5
.
.
1 0 0 5 (8 rows of the same data)
.
.
.
With this new data set, CHEESE2, you can use:
proc probit data=cheese2; class y; model y = x1-x3 / d=logistic; run;
The resulting SAS output will be:
Sample Program: Ordered Logit Regression
Probit Procedure
Class Level Information
Class Levels Values
Y 9 1 2 3 4 5 6 7 8 9
Number of observations used = 208
Probit Procedure
Data Set =WORK.CHEESE2
Dependent Variable=Y
Weighted Frequency Counts for the Ordered Response Categories
Level Count
1 7
2 10
3 19
4 27
5 41
6 28
7 39
8 25
9 12
Log Likelihood for LOGISTIC -355.6739524
Probit Procedure
Variable DF Estimate Std Err ChiSquare Pr>Chi Label/Value
INTERCPT 1 -7.0801649 0.56401 157.5844 0.0001 Intercept
X1 1 1.6127909 0.380544 17.96169 0.0001
X2 1 4.96463991 0.476721 108.4546 0.0001
X3 1 3.32268278 0.42183 62.04439 0.0001
INTER.2 1 1.0551848 0.324654 2
INTER.3 1 2.15474934 0.387165 3
INTER.4 1 3.22336352 0.420573 4
INTER.5 1 4.55961327 0.454216 5
INTER.6 1 5.5116267 0.479248 6
INTER.7 1 7.01328969 0.520899 7
INTER.8 1 8.57313924 0.587685 8
Notice that intercept parameter estimates are computed as:
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