Concept Comprehension: Unordered outcomes

Here are estimates from the 1996 General Social Survey, which included a question asking respondents what they thought was the best way to decide which couples interested in adopting a child should get to do so: a social service agency [decides], go to the highest bidders, by lottery, or birth mother decides.

The explanatory variables are sex (male/female) and education (in years). The estimates are exponentiated coefficients. I have reformatted the output so that each equation in the model is its own row.

Model 1
go to the highest bidders by lottery birth mothers should decide
(Intercept) 0.045*** 0.033*** 1.170
(0.041) (0.018) (0.342)
Female 0.797 1.003 1.301*
(0.289) (0.212) (0.155)
Education (yrs) 1.003 1.108** 0.944**
(0.064) (0.041) (0.020)
Num.Obs. 1368

  1. In the rightmost column of the model, the exponentiated coefficient for \(\texttt{female}\) is 1.301. In one sentence, interpret this coefficient. [1]

  2. In the middle column of the model, the exponentiated coefficient for \(\texttt{educ}\) is 1.108. In one sentence, interpret this coefficient. [1]

  3. The results above are exponentiated coefficients. What would be the unexponentiated coefficient for \(\texttt{female}\) we would have obtained for “lottery” if the base category of the model was “birth mother should decide”? Show or explain how you calculate this. [1]

Here is a table of the female-male difference in predicted probability of each outcome averaged over all the observations:

Category Change
social service agency ???
go to highest bidder -.0073
by lottery -.0059
birth mothers should decide .0605

  1. What is the value of ??? (compute to at least 3 decimal places)? How do you know? [1]

  2. In the context of the conditional logit model, we described it as a way of being able to incorporate alternative-specific variables in one’s model. Say there was a local school board race with five candidates and you were hired to conduct a poll. What would be an example of an alternative-specific variable you might collect from respondents? (It should be clear from your answer that you understand what an alternative-specific variable is, so elaborate if worried that it might not be clear.) [1]

Here are the unexponentiated coefficients from a conditional logit model in which the outcome is winning an Oscar. The explanatory variables correspond to whether one is nominated for Best Picture and the interaction of this variable and whether the nomination is for a main role or supporting role (e.g., Best Actress vs. Best Supporting Actress).

Model matrix is rank deficient. Parameters `rolesupporting` were not
  estimable.
beta (se)
Nominated for best pic 1.430 (0.196)***
Nominated for best pic x supporting -0.944 (0.264)***
Num.Obs. 1848

  1. The term for the interaction is negative (and statistically significant). Substantively, what does this mean? [1]

  2. Being nominated for best picture increases a nominee’s chances of winning Best Actor/Actress. In terms of a percent change in the odds, by how much? [1]

  3. Even though we specified an interaction term, there is no main effect term in these results for “supporting role.” If I had provided the summary() output from the clogit() function (as below), it would have said the coefficient for “supporting role” was “NA”. Why is there not a coefficient for supporting role? [1]

summary(model)
Call:
coxph(formula = Surv(rep(1, 1848L), winner) ~ nom_best_pic * 
    role + strata(groupid), data = oscars, method = "exact")

  n= 1848, number of events= 374 

                                   coef exp(coef) se(coef)      z Pr(>|z|)    
nom_best_picTRUE                 1.4299    4.1784   0.1956  7.310 2.68e-13 ***
rolesupporting                       NA        NA   0.0000     NA       NA    
nom_best_picTRUE:rolesupporting -0.9442    0.3890   0.2638 -3.579 0.000345 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

                                exp(coef) exp(-coef) lower .95 upper .95
nom_best_picTRUE                    4.178     0.2393    2.8477    6.1309
rolesupporting                         NA         NA        NA        NA
nom_best_picTRUE:rolesupporting     0.389     2.5707    0.2319    0.6524

Concordance= 0.6  (se = 0.015 )
Likelihood ratio test= 68.24  on 2 df,   p=2e-15
Wald test            = 60.96  on 2 df,   p=6e-14
Score (logrank) test = 67.24  on 2 df,   p=3e-15