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Ordinal and Multinomial Models

Ordinal and Multinomial Models

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Ordinal and Multinomial Models

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  1. Ordinal and Multinomial Models William Simpson Research Computing Services http://intranet.hbs.edu/dept/research/statistics/

  2. 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)

  3. Outline of Talk • Review of Binary Models • Ordinal Models • Multinomial Logit

  4. 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.

  5. Another CDF View

  6. 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.

  7. Binary Data – Unobserved Variable View PDF of Y*

  8. What Happens When Standard Deviation of u Changes y*=  +  x+ v std(v) > std(u)

  9. 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.

  10. Combining the Two Views

  11. Combining the Two Views

  12. 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

  13. Ordinal Outcomes (Latent Variable View)

  14. Ordinal Outcomes (CDF and Latent Variable View)

  15. Ordinal Outcomes (CDF and Latent Variable View)

  16. Ordinal Outcomes (CDF and Latent Variable View)

  17. Ordinal Outcomes (CDF and Latent Variable View)

  18. 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;

  19. 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 ----------------------------------------------------------

  20. 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.

  21. 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

  22. 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

  23. 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.

  24. Interpreting Coefficients(Ordinal Probit)

  25. Interpreting Coefficients(Ordinal Logit)

  26. 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

  27. Parallel Regressions

  28. Proportional Odds

  29. Interpreting Cutpoints

  30. 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

  31. Probability of Responses

  32. 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

  33. Probabilities with Uneven Scale

  34. Ordinal Outcomes (Latent Variable View)

  35. Multinomial Logit • A generalization of logistic regression • More than two outcomes • Outcomes are not ordered • We are interested in the relative probabilities of outcomes

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

  37. Example Data

  38. Using a Reference Level

  39. 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)

  40. 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)

  41. Taxi  Bus Bus  Taxi Bus  Car Taxi  Car .0757664 -.0757664 .4482523 .5240187 Coefficients on Distance Bus  Taxi + Taxi  Car = Bus  Car -.0757664 + .5240187 = .4482523 Bus  Car=Taxi  Car – Taxi  Bus

  42. Probability Change Plot

  43. Odds Ratio Plot

  44. 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.

  45. 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

  46. 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.

  47. Appendix Programming Examples By James Zeitler

  48. 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

  49. 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

  50. 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