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### Ordinal and Multinomial Models

William Simpson

Research Computing Services

http://intranet.hbs.edu/dept/research/statistics/

Types of Models

- Models are generalizations of the logit and probit models
- Ordinal logit and probit deal with ordered data (more than 2 categories)
- Multinomial logit deals with unordered data with more than 2 categories
- (Multinomial probit is not commonly used due to computational difficulties)

Outline of Talk

- Review of Binary Models
- Ordinal Models
- Multinomial Logit

Binary Data – View 1 (CDF)

- View 1 – we compute a number that is a linear combination of our predictors, call it y=+ x. We then convert y into a probability p by using a cumulative distribution function (CDF). Our final outcome is 1 with probability p.

Binary Data – View 2 (Latent or Unobserved Variable)

- View 2 – we compute a number that is a linear combination of our predictors and then add an error term, call it

y*= + x+ u

We then get an outcome of 1 if y* >= 0, outcome 0if y* < 0. In this case, the probabilistic element is the error term u, and y* is an unobserved variable.

Comparing CDF and Latent Variable Views

- The two views are equivalent. Each one can be converted into the other, where the cumulative probability function (CDF) in view 1 matches the CDF of the distribution of u in view 2.

Ordinal Outcomes

- 3 or more categorical outcomes, which can be treated as ordered
- Bond ratings (AAA, AA, … B, C, …)
- Likert scales (e.g. responses on a 1-7 scale, from strongly disagree to strongly agree)
- Often analyzed as continuous

SAS and Stata Code

Stata

oprobit outcome x

or

ologit outcome x

SAS

proc logistic;

class outcome;

model outcome = x / link=probit;

or

model outcome = x ;

run;

Sample Output (Stata oprobit)

---------------------------------------------------------

y | Coef. Std. Err. z P>|z|

---------------------------------------------------------

x | 1.074575 .1209108 8.89 0.000

-------------+-------------------------------------------

_cut1 | -2.076242 .1548201 (Ancillary parameters)

_cut2 | -.9736895 .0807119

_cut3 | -.4528313 .073509

_cut4 | 1.106628 .0781733

_cut5 | 2.079342 .0932966

_cut6 | 3.176076 .167065

----------------------------------------------------------

Interpretation of Stata Output

x | 1.074575 .1209108

-------------+-----------------------

_cut1 | -2.076242 .1548201

_cut2 | -.9736895 .0807119

- Outcome will be in the second ordered category or higher (not the first), if 1.07*x+u > -2.08.
- Outcome will be in the third ordered category or higher (not the first or second), if 1.07*x+u > -.97.
- Outcome will be in the second ordered category exactly, if -.97 > 1.07*x+u > -2.08.

Sample Output (SAS PROC LOGISTIC with LINK=PROBIT)

Parameter DF Estimate Std Error

Intercept 71 -3.17580.1666

Intercept 61 -2.07930.0933

Intercept 51 -1.10660.0781

Intercept 410.45280.0734

Intercept 310.97370.0807

Intercept 212.07620.1555

x 11.07460.1208

Interpretation of SAS Output

- Outcome will be in the second ordered category or higher (not the first), if 1.07*x + 2.08 + u > 0.
- Outcome will be in the third ordered category or higher, if 1.07*x + .97 + u > 0.
- Outcome will be in the second ordered category if 1.07*x + 2.08 + u > 0 and 1.07*x + .97 + u < 0.

Intercept 310.97370.0807

Intercept 212.07620.1555

x 11.07460.1208

Interpreting Coefficients

- Multiple cutpoints with no intercept term, or multiple intercept terms
- Probabilities modeled are probabilities for all outcomes >=k, compared with all outcomes < k.
- Interpret the coefficients the same as in the corresponding binary model.

Assumptions of Ordinal Models

- Relationship between probabilities and + x follows the assumed form (normal for probit, logistic for logit).
- Parallel regressions – Coefficient is the same for every hurdle – aka equal slopes, (proportional odds for logistic models)
- If not, use generalized ordered logit

Sample Likert Scalewith Extra Points

2.3 4.2

1 2 3 4 5 6 7

-----------------------------------------------------------

SD D SoD N SoA A SA

MoD VSA

SD=Strongly Disagree, SoD = Somewhat Disagree

D=Disagree, N=Neutral, A=Agree

SA=Strongly Agree, SoA=Somewhat Agree

MoD=Moderately Disagree

VSA = Very Slightly Agree

Sample Likert Scalewith Uneven Points

1 2 3 4 5 6 7

-----------------------------------------------------------

SD D MoD SoD N VSA SA

(1) (2) (2.3) (3) (4) (4.2) (7)

SD=Strongly Disagree, SoD = Somewhat Disagree

MoD=Moderately Disagree

D=Disagree, N=Neutral

VSA = Very Slightly Agree

SA=Strongly Agree

Multinomial Logit

- A generalization of logistic regression
- More than two outcomes
- Outcomes are not ordered
- We are interested in the relative probabilities of outcomes

Examples

- Choice of transportation – bus, taxi, private car
- Choice of product brand
- Occupational choice (considered as unordered) – craft, blue collar, professional, white collar

Sample Results

-----------------------------------------------------

outcome | Coef. Std. Err. z P>|z|

-------------+---------------------------------------

Taxi |

distance | -.0757664 .1305456 -0.58 0.562

income | .319901 .0830162 3.85 0.000

_cons | -6.22562 1.734012 -3.59 0.000

-------------+---------------------------------------

Car |

distance | .4482523 .1129979 3.97 0.000

income | .0447404 .0581754 0.77 0.442

_cons | -2.587764 1.214103 -2.13 0.033

-----------------------------------------------------

(Outcome outcome==Bus is the comparison group)

Sample Results (2)

-----------------------------------------------------

outcome | Coef. Std. Err. z P>|z|

-------------+---------------------------------------

Bus |

distance | .0757664 .1305456 0.58 0.562

income | -.319901 .0830162 -3.85 0.000

_cons | 6.22562 1.734012 3.59 0.000

-------------+---------------------------------------

Car |

distance | .5240187 .1245058 4.21 0.000

income | -.2751607 .080734 -3.41 0.001

_cons | 3.637855 1.705811 2.13 0.033

-----------------------------------------------------

(Outcome outcome==Taxi is the comparison group)

Taxi Bus

Bus Taxi

Bus Car

Taxi Car

.0757664

-.0757664

.4482523

.5240187

Coefficients on DistanceBus Taxi + Taxi Car = Bus Car

-.0757664 + .5240187 = .4482523

Bus Car=Taxi Car – Taxi Bus

Independence from Irrelevant Alternatives (IIA)

- Relative odds of two categories shouldn’t change when a new category is added
- E.g., if choices are car, bus, and Yellow Cab, the relative proportions shouldn’t change if a new choice is added, e.g. Black & White Cab
- Not realistic in this case. Assumption should be examined carefully.

Other Models for Nominal Outcomes

- Conditional Logit
- Attributes of choices can be used as predictors
- Nested Logit
- Treats a set of choices as a hierarchy
- IIA assumption can be relaxed

References

- Long, J. S. (1997). Regression Models for Categorical and Limited Dependent Variables. Thousand Oaks, CA: Sage.
- Hosmer, D. W. and S. Lemeshow. (2000). Applied Logistic Regression (Second ed.). New York: Wiley.
- Allison, P. D. (1999). Logistic Regression Using the SAS System: Theory and Application. Cary, NC: SAS Institute.
- Long, J. S. & Freese, J. (2001). Regression Models for Categorical Dependent Variables using Stata. College Station, TX: Stata Press.

Ordered Logit (SAS)

proclogistic data = work.ordinals

descending;

model y = x;

run;

The LOGISTIC Procedure

Model Information

Data Set WORK.ORDINALS

..............................................

Model cumulative logit

Optimization Technique Fisher's scoring

Response Profile

Ordered Total

Value y Frequency

1 7 6

.............................

7 1 6

Probabilities modeled are cumulated over the lower Ordered Values.

Analysis of Maximum Likelihood Estimates

Standard Wald

Parameter DF Estimate Error Chi-Square Pr > ChiSq

Intercept 7 1 -6.1912 0.4312 206.1863 <.0001

Intercept 6 1 -3.6194 0.1804 402.7389 <.0001

Intercept 5 1 -1.8611 0.1414 173.2883 <.0001

Intercept 4 1 0.7326 0.1275 33.0150 <.0001

Intercept 3 1 1.7093 0.1520 126.4030 <.0001

Intercept 2 1 4.3014 0.4189 105.4418 <.0001

x 1 1.8479 0.2176 72.1016 <.0001

Ordered Probit (SAS)

proclogistic data = work.ordinals

descending;

model y = X / LINK = PROBIT;

run;

The LOGISTIC Procedure

Model Information

Data Set WORK.ORDINALS

...............................................

Model cumulative probit

Response Profile

Ordered Total

Value y Frequency

1 7 6

............................

7 1 6

Probabilities modeled are cumulated over the lower Ordered Values.

Analysis of Maximum Likelihood Estimates

Standard Wald

Parameter DF Estimate Error Chi-Square Pr > ChiSq

Intercept 7 1 -3.1758 0.1666 363.5568 <.0001

Intercept 6 1 -2.0793 0.0933 496.5331 <.0001

Intercept 5 1 -1.1066 0.0781 200.8158 <.0001

Intercept 4 1 0.4528 0.0734 38.0347 <.0001

Intercept 3 1 0.9737 0.0807 145.4615 <.0001

Intercept 2 1 2.0762 0.1555 178.1792 <.0001

x 1 1.0746 0.1208 79.1034 <.0001

Multinomial Logit (SAS)

/* Use Link = GLOGIT in PROC LOGIT */

/* to estimate a multinomial logit */

/* Refer to the response profile to */

/* determine the reference category */

proclogistic data = transport;

class Mode;

model Mode = Distance Income

/link = glogit;

run;

The LOGISTIC Procedure

Model Information

Data Set WORK.TRANSPORT

Response Variable Mode

Number of Response Levels 3

Model generalized logit

Response Profile

Ordered Total

Value Mode Frequency

1 Bus 27

2 Car 42

3 Taxi 31

Logits modeled use Mode='Taxi' as the reference category.

Analysis of Maximum Likelihood Estimates

Standard Wald

Parameter Mode DF Estimate Error Chi-Square Pr > ChiSq

Intercept Bus 1 6.2253 1.7340 12.8897 0.0003

Intercept Car 1 3.6375 1.7057 4.5475 0.0330

Distance Bus 1 0.0757 0.1305 0.3367 0.5617

Distance Car 1 0.5240 0.1245 17.7135 <.0001

Income Bus 1 -0.3199 0.0830 14.8488 0.0001

Income Car 1 -0.2751 0.0807 11.6155 0.0007

Ordered Logit Syntax and Results (SPSS)

PLUM

y WITH x

/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 .

Ordered Probit Syntax and Results (SPSS)

PLUM

y WITH x

/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 .

Multinomial Logit (SPSS)

Analyze

Regression

Multinomial logit...

NOMREG

choice WITH distance income

/CRITERIA = CIN(95) DELTA(0) MXITER(100) MXSTEP(5)

CHKSEP(20) LCONVERGE(0) PCONVERGE(1.0E-6)

SINGULAR(1.0E-8)

/MODEL

/INTERCEPT = INCLUDE

/PRINT = PARAMETER SUMMARY LRT .

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