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### Ordered Probit Regression

The ordered probit model has the form:

or equivalently,

where

is the inverse of the cumulative standard normal distribution function, often referred to as probit or normit, and denotes the cumulative standard normal distribution function.

#### Ordered Probit Regression with SAS

##### LOGISTIC Procedure

To fit an ordered probit model in PROC LOGISTIC, use the LINK=NORMIT (or PROBIT) option as:

```
proc logistic;

run;

```
Example 22: SAS Ordered Probit Regression in PROC LOGISTIC

Using the data set CHEESE in Example 19, if you use:

```
proc logistic data=cheese;

freq f;

run;

```

you will have:

```
Sample Program: Ordered Probit Regression

The LOGISTIC Procedure

Data Set: WORK.CHEESE

Response Variable: Y

Response Levels: 9

Number of Observations: 28

Frequency Variable: F

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 Equal Slopes Assumption

Chi-Square = 15.0251 with 21 DF (p=0.8217)

Model Fitting Information and Testing Global Null Hypothesis BETA=0

Intercept

Intercept        and

Criterion       Only       Covariates    Chi-Square for Covariates

AIC             875.802       729.391         .

SC              902.502       766.104         .

-2 LOG L        859.802       707.391      152.411 with 3 DF (p=0.0001)

Score              .             .         108.491 with 3 DF (p=0.0001)

Analysis of Maximum Likelihood Estimates

Parameter   Standard      Wald         Pr >      Standardized

Variable   DF    Estimate     Error    Chi-Square   Chi-Square     Estimate

INTERCP1   1      -4.0762     0.2867     202.1202       0.0001              .

INTERCP2   1      -3.5087     0.2496     197.6200       0.0001              .

INTERCP3   1      -2.8628     0.2248     162.2226       0.0001              .

INTERCP4   1      -2.2356     0.2067     117.0124       0.0001              .

INTERCP5   1      -1.4641     0.1858      62.0947       0.0001              .

INTERCP6   1      -0.9155     0.1730      28.0144       0.0001              .

INTERCP7   1      -0.0276     0.1607       0.0296       0.8634              .

INTERCP8   1       0.8779     0.1841      22.7341       0.0001              .

X1         1       0.9643     0.2119      20.7122       0.0001       0.418540

X2         1       2.8618     0.2508     130.2471       0.0001       1.242206

X3         1       1.9408     0.2296      71.4236       0.0001       0.842421

The LOGISTIC Procedure

Association of Predicted Probabilities and Observed Responses

Concordant = 58.4%          Somers' D = 0.512

Discordant =  7.2%          Gamma     = 0.781

Tied       = 34.4%          Tau-a     = 0.443

(18635 pairs)               c         = 0.756

```

This result shows eight fitted regression lines as:

##### PROBIT Procedure

You can use the SAS PROC PROBIT to fit an ordered probit model:

```
proc probit;

class y;

model y = x1 x2;

run;

```
Example 23: SAS Ordered Probit Regression in PROC PROBIT

Using the data set CHEESE2 in Example 21, you can use:

```
proc probit data=cheese2;

class y;

model y = x1-x3;

run;

```

```
Sample Program: Ordered Probit 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 NORMAL -353.6953428

Probit Procedure

Variable  DF   Estimate  Std Err ChiSquare  Pr>Chi Label/Value

INTERCPT   1 -4.0761911   0.2868       202  0.0001 Intercept

X1         1  0.9642503 0.211624    20.761  0.0001

X2         1 2.86184814  0.24956  131.5057  0.0001

X3         1 1.94080682 0.230248  71.05121  0.0001

INTER.2    1 0.56749271 0.166663                              2

INTER.3    1 1.21337982 0.200722                              3

INTER.4    1 1.84054882 0.216171                              4

INTER.5    1 2.61204661 0.229904                              5

INTER.6    1 3.16070252 0.241469                              6

INTER.7    1 4.04855742 0.263792                              7

INTER.8    1 4.95412966 0.298786                              8

```

This result shows eight fitted regression lines as:

which are the same results as we obtained in Example 22.

Example 24: Predicted Probability Computation

Predicted probability computation can be easily obtained using:

```
proc probit data=cheese2;

class y;

model y = x1-x3;

output out=prob2 prob=phat;

run;

proc print data=prob2;

run;

```

As a result, you will have:

```
Sample Program: Ordered Probit Regression

OBS    X1    X2    X3    Y    _LEVEL_      PHAT

1     1     0     0    3       1       0.00093

2     1     0     0    3       2       0.00547

3     1     0     0    3       3       0.02881

4     1     0     0    3       4       0.10179

5     1     0     0    3       5       0.30857

6     1     0     0    3       6       0.51945

7     1     0     0    3       7       0.82552

8     1     0     0    3       8       0.96728

.

.

1657     0     0     0    9       1       0.00002

1658     0     0     0    9       2       0.00023

1659     0     0     0    9       3       0.00210

1660     0     0     0    9       4       0.01269

1661     0     0     0    9       5       0.07158

1662     0     0     0    9       6       0.17997

1663     0     0     0    9       7       0.48898

1664     0     0     0    9       8       0.81001

```

Next: Models for Unordered Multiple Choices
Prev: Ordered Logit Regression
Up: Contents