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1. HGLM

2. HGLM • It really is little more than the combination of GLM and HLM • Example • Estimating the probability of voting • Data: Cumulative NES file (24 NESs)

3. Data • Micro level variables: • Partisan strength • Education • Age • White • Income

4. Data • Macro level • Random term • Presidential election

5. R command • It is a hybrid of GLM and lmer M1<-lmer(y~x1+x2+x3+x4+x5+(1|year), family=binomial(link="logit"))

6. Results Generalized linear mixed model fit using Laplace Formula: y ~ x1 + x2 + x3 + x4 + x5 + (1 | year) Family: binomial(logit link) AIC BIC logLik deviance 40427 40486 -20206 40413 Random effects: Groups Name Variance Std.Dev. year (Intercept) 0.243 0.493 number of obs: 36752, groups: year, 24 Estimated scale (compare to 1 ) 1 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) 3.582592 0.122241 29.3 < 2e-16 *** x1 -0.363597 0.012498 -29.1 < 2e-16 *** x2 -0.294791 0.008584 -34.3 < 2e-16 *** x3 -0.027922 0.000802 -34.8 < 2e-16 *** x4 -0.206562 0.032863 -6.3 3.3e-10 *** x5 -0.317875 0.012053 -26.4 < 2e-16 ***

7. Add a year level variable • Note the standard deviation of the random term is 0.493

8. Results Generalized linear mixed model fit using Laplace Formula: y ~ x1 + x2 + x3 + x4 + x5 + z + (1 | year) Family: binomial(logit link) AIC BIC logLik deviance 40391 40459 -20187 40375 Random effects: Groups Name Variance Std.Dev. year (Intercept) 0.0451 0.212 number of obs: 36752, groups: year, 24 Estimated scale (compare to 1 ) 1 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) 4.062479 0.096465 42.1 < 2e-16 *** x1 -0.364464 0.012496 -29.2 < 2e-16 *** x2 -0.291768 0.008535 -34.2 < 2e-16 *** x3 -0.027870 0.000802 -34.8 < 2e-16 *** x4 -0.213119 0.032828 -6.5 8.5e-11 *** x5 -0.319210 0.012050 -26.5 < 2e-16 *** z -0.885312 0.090501 -9.8 < 2e-16 ***

9. Results • Adding the indicator of whether or not it is a presidential election year soaks up a lot of the mean level variance • Do we still need the random term? • Remember—this is a nuisance term. It is there to account for what we do not specify in the intercept equation. • Test is a deviance test • Difference in deviance is 183—yes we need it.

10. Add a random slope • We want to see if we need random slopes • Start with a random slope for x1 • M3<-lmer (y~x1+x2+x3+x4+x5+z+(1+x1|year), family=binomial(link="logit"))

11. Results Generalized linear mixed model fit using Laplace Formula: y ~ x1 + x2 + x3 + x4 + x5 + z + (1 + x1 | year) Family: binomial(logit link) AIC BIC logLik deviance 40023 40108 -20002 40003 Random effects: Groups Name Variance Std.Dev. Corr year (Intercept) 0.4595 0.678 x1 0.0543 0.233 -0.951 number of obs: 36752, groups: year, 24 Estimated scale (compare to 1 ) 1 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) 4.16969 0.16339 25.5 < 2e-16 *** x1 -0.38179 0.04930 -7.7 9.7e-15 *** x2 -0.29784 0.00861 -34.6 < 2e-16 *** x3 -0.02845 0.00081 -35.1 < 2e-16 *** x4 -0.21587 0.03314 -6.5 7.3e-11 *** x5 -0.32396 0.01213 -26.7 < 2e-16 *** z -0.89406 0.08948 -10.0 < 2e-16 ***

12. Results • Ok, first that correlation is really high • Why? What is the intercept? • When all the x’s equal zero. • But none of the x’s are ever zero • The data are not centered • So, subtract off the median

13. New Results Generalized linear mixed model fit using Laplace Formula: y ~ x1b + x2b + x3b + x4b + x5b + z + (1 + x1b | year) Family: binomial(logit link) AIC BIC logLik deviance 40023 40108 -20002 40003 Random effects: Groups Name Variance Std.Dev. Corr year (Intercept) 0.0468 0.216 x1b 0.0543 0.233 0.252 number of obs: 36752, groups: year, 24 Estimated scale (compare to 1 ) 1 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.06417 0.07202 -0.9 0.37 x1b -0.38132 0.04929 -7.7 1.0e-14 *** x2b -0.29784 0.00861 -34.6 < 2e-16 *** x3b -0.02844 0.00081 -35.1 < 2e-16 *** x4b -0.21587 0.03314 -6.5 7.4e-11 *** x5b -0.32394 0.01213 -26.7 < 2e-16 *** z -0.89418 0.08939 -10.0 < 2e-16 ***

14. Results • The correlation is moderate • The coefficient on X1 changed slightly, but not much from model without random effect • Deviance test says we need the random slope term • But what if the slope varies as a function of presidential election? • Add the term

15. Generalized linear mixed model fit using Laplace Formula: y ~ x1b + x2b + x3b + x4b + x5b + z + x1b:z + (1 | year) Family: binomial(logit link) AIC BIC logLik deviance 40365 40442 -20174 40347 Random effects: Groups Name Variance Std.Dev. year (Intercept) 0.0443 0.210 number of obs: 36752, groups: year, 24 Estimated scale (compare to 1 ) 1 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.051128 0.071366 -0.7 0.47 x1b -0.299611 0.017549 -17.1 < 2e-16 *** x2b -0.291545 0.008530 -34.2 < 2e-16 *** x3b -0.027875 0.000802 -34.8 < 2e-16 *** x4b -0.212297 0.032868 -6.5 1.1e-10 *** x5b -0.319416 0.012048 -26.5 < 2e-16 *** z -0.916731 0.089988 -10.2 < 2e-16 *** x1b:z -0.128740 0.024592 -5.2 1.7e-07 *** • The interaction term adds to the model now, but what if we add the random slope?

16. Generalized linear mixed model fit using Laplace Formula: y ~ x1b + x2b + x3b + x4b + x5b + z + x1b:z + (1 + x1b | year) Family: binomial(logit link) AIC BIC logLik deviance 40024 40118 -20001 40002 Random effects: Groups Name Variance Std.Dev. Corr year (Intercept) 0.0467 0.216 x1b 0.0513 0.226 0.248 number of obs: 36752, groups: year, 24 • Estimated scale (compare to 1 ) 1 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.05055 0.07299 -0.7 0.49 x1b -0.32210 0.07065 -4.6 5.1e-06 *** x2b -0.29783 0.00861 -34.6 < 2e-16 *** x3b -0.02845 0.00081 -35.1 < 2e-16 *** x4b -0.21610 0.03314 -6.5 7.0e-11 *** x5b -0.32389 0.01213 -26.7 < 2e-16 *** z -0.91938 0.09224 -10.0 < 2e-16 *** x1b:z -0.11019 0.09619 -1.1 0.25 • Now the interaction term is insignificant!

17. The interaction does not add to the model • If we run the paired model comparisons we find: • Including the interaction term is better than omitting it if there is no random slope • Including the random slope is better than omitting it • The model with the random term and the interaction is not superior to the model with only the random slope • It is insignificant • The deviance test tells us to reject including it • So? Best model omits it. We don’t need it for specification (though we might for theory).

18. Other x’s • Long story short, both the random term and the interaction with presidential election improve the model for x2, x4, & x5 • Only the random term improves fit for x3

19. Multiple random slopes If we add a random slope on x2, we improve the model Formula: y ~ x1b + x2b + x3b + x4b + x5b + z + x2b:z + x4b:z + x5b:z + (1 + x1b + x2b | year) Family: binomial(logit link) AIC BIC logLik deviance 39425 39561 -19697 39393 Random effects: Groups Name Variance Std.Dev. Corr year (Intercept) 0.0789 0.281 x1b 0.0709 0.266 0.328 x2b 0.0228 0.151 0.383 0.988 number of obs: 36752, groups: year, 24 Estimated scale (compare to 1 ) 1

20. Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.142094 0.093006 -1.5 0.12657 x1b -0.295615 0.055877 -5.3 1.2e-07 *** x2b -0.246501 0.034072 -7.2 4.7e-13 *** x3b -0.029652 0.000818 -36.3 < 2e-16 *** x4b -0.148631 0.047011 -3.2 0.00157 ** x5b -0.243784 0.016835 -14.5 < 2e-16 *** z -0.462598 0.124808 -3.7 0.00021 *** x2b:z -0.045045 0.023232 -1.9 0.05251 . x4b:z -0.251922 0.065749 -3.8 0.00013 *** x5b:z -0.177566 0.024155 -7.4 2.0e-13 ***

21. The correlation between the random slopes is really high • Adding a random slope for x3 is intractable—you get negative estimates of the standard deviations. • Serious problem

22. Item response • Basic idea • Each person has multiple indicators which tap the underlying concept of interest • Usually, not everyone gets the same indicators • No indicator is used for every person • Indicators differ in their difficulty • So, the dv is the probability that the answer, from the specific person on the specific question is a success (a 1)

23. IRT • Where the αj is person j’s latent ability • and βk is question k’s difficulty • If people get different questions, we need to add a subscript i to denote which response we are speaking of

24. IRT • Examples: • Roll call votes • Supreme Court decisions • Test scores (SAT, GRE) • Democracy • Knowledge • Inherent problem is that we need to estimate a person’s ability at the same time as we estimate the question’s difficulty

25. Identifiability • The other problem is that the model is not identified • Add a constant to all of abilities and all of the difficulties and get the same answer • Just need to constrain the problem somehow—force a question to have a difficulty or a person to have an ability • If 0 & 1 aren’t natural you have a reflexive problem too • Easy

26. IRT • How about we make this a multilevel problem: • Solve identification by fixing one of the means to zero.

27. IRT • We can easily add group level predictors: • What are these? • The X’s in the first equation are things that we think predict the person’s ability (gender, race, party,…) • In the second equation they are whatever information we have about the difficulty of the question that is separate from what the data have to tell us.

28. IRT • Hold on a minute • What are these? • The basic idea that is that everyone has a probability of getting a question right. • That probability is based on two things: • Your ability • How hard the question is • Given these two things, the probability can be defined for each person

29. IRT

30. IRT

31. IRT • The basic prediction is easy: Person j will be a success on question k if his or her ability is greater than the difficulty of the question. • So you want to find the set of difficulty and ability parameters that best fit the data

32. IRT-discrimination • The graph two slides ago (parallel lines) assumed that the questions were equally good at discriminating based on ability • More specifically that the effect of ability on each of the questions was the same—fixed slope • We can allow that to vary: • Gamma defines the ability of the question to discriminate—higher values mean that the question is a better predictor. That also means a sharper curve

33. IRT

34. IRT • We won’t estimate this yet • It is really hard to estimate and needs Bayesian • It is, however, pretty cool. Lots of applications • This is a fundamental measurement issue • Can improve on classic factor analyses or standard scaling techniques

35. Other HGLM models • The R code is basically the same as the logit—put a different link and family • Problem is that the Likelihood based estimation problems get worse. My experience is that the binary choice is the easiest. • More parameters to estimate • Wait until we can do this as Bayesian

36. Next time, MCMC • This gets into detailed probability theory • Gelman and Hill Chapter 18 • Jackman, Simon. 2000. Estimation and Inference via Bayesian Simulation. AJPS 375-404. • Casella and George. 1992. Explaining the Gibbs Sampler. American Statistician. 167-174. • “Markov Chain Monte Carlo in Practice: A Roundtable Discussion.” American Statistician 1998. 93-100.