1. 1 Topic 4 : Ordered Logit Analysis

2. 2 Often we deal with data where the responses are ordered – e.g. : (i) Eyesight tests – bad; average; good (ii) Voting – rank the candidates (iii) Bond ratings – A+++, A++, A+, A, B+++ (iv) We could set this up as a multinomial logit model, but this would ignore an important piece of information in the data  the ordering of the values. Of course, ordinary least squares would have a problem in the opposite direction – the numbers would be used as if the values meant something.

3. 3 To see how we can deal with such data sensibly, let’s reconsider the (binary) logit set-up and motivation. One way to proceed is to assume there is a “latent” (unobservable) variable, y*, and we observe code where  is a threshold value.

4. 4 Then write y* = x  +  if the underlying distribution is symmetric

5. 5 Setting  = 0, we have : and then choose the cdf of logit as the link function.

6. 6 Now let’s use this approach with our ordered data. We have a latent variable, y*, where The U j ’s are unknown parameters to be estimated with the  ’s.

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8. 8 Of course, for all these probabilities to be positive, we require that

9. 9 Now to get the  ’s and the  ’s, note the following. For the i th observation, the log- likelihood is whereI(E)= 1if event E occurs = 0otherwise Then sum over all “n” observations to get the full log likelihood function (assuming independence). As usual, there is a unique maximum.

10. 10 As for the signs of the coefficients, we need to look carefully at the marginal effects – Recall that :

11. 11 So as for the marginal effects –

12. 12 So, sign of P 0 marginal effect is opposite to the coefficient sign; sign of P J marginal effect is the same as the coefficient sign; other signs of ambiguous. One has to be very careful when interpreting the coefficients in this model.

13. 13 Example : Consider the data set in the cast example in which the dependent variable was the response to the question, “If you found a wallet on the street, would you (1) keep the wallet and the money (2) keep the money and return the wallet (3) return both the wallet and the money”. There is an obvious ordering in the responses : 1 is the most unethical response, 3 is the most ethical; 2 is in the middle.

14. 14 Intercept 1 =  0 = – 3.2691 Intercept 2 =  0 = – 1.4913 Likelihood ratio : H 0 =  MALE =  BUSINESS =  PUNISH =  EXPLAIN = 0 H 1 = otherwise Score test for the proportional odds assumptions H 0 =  model1 =  model2 H 1 = otherwise

15. 15 DATA WALLET; INFILE 'D:\TEACHING\MS4225\WALLET.TXT'; INPUT WALLET MALE BUSINESS PUNISH EXPLAIN; PROC LOGISTIC DATA=WALLET; MODEL WALLET=MALE BUSINESS PUNISH EXPLAIN; RUN;

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18. 18 DATA WALLET; INFILE 'D:\TEACHING\MS4225\WALLET.TXT'; INPUT WALLET MALE BUSINESS PUNISH EXPLAIN; DATA A; SET WALLET; IF WALLET=3 THEN WALLET=2; RUN; PROC LOGISTIC DATA=A; MODEL WALLET = MALE BUSINESS PUNISH EXPLAIN; RUN;

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20. 20 DATA WALLET; INFILE 'D:\TEACHING\MS4225\WALLET.TXT'; INPUT WALLET MALE BUSINESS PUNISH EXPLAIN; DATA A; SET WALLET; IF WALLET=1 THEN WALLET=2; RUN; PROC LOGISTIC DATA=A; MODEL WALLET = MALE BUSINESS PUNISH EXPLAIN; RUN;

21. 21 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.3189 0.5465 5.8243 0.0158 MALE 1 1.1845 0.3408 12.0824 0.0005 BUSINESS 1 0.6357 0.3812 2.7808 0.0954 PUNISH 1 0.5071 0.2474 4.2030 0.0404 EXPLAIN 1 -1.0200 0.3662 7.7606 0.0053