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HGLM. 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). Data. Micro level variables: Partisan strength Education Age White Income. Data. Macro level Random term

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Hglm
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)


Hglm
Data

  • Micro level variables:

    • Partisan strength

    • Education

    • Age

    • White

    • Income


Hglm
Data

  • Macro level

    • Random term

    • Presidential election


R command
R command

  • It is a hybrid of GLM and lmer

    M1<-lmer(y~x1+x2+x3+x4+x5+(1|year), family=binomial(link="logit"))


Results
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 ***


Add a year level variable
Add a year level variable

  • Note the standard deviation of the random term is 0.493


Results1
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 ***


Results2
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.


Add a random slope
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"))


Results3
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 ***


Results4
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


New results
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 ***


Results5
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


Hglm

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?


Hglm

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!


The interaction does not add to the model
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).


Other x s
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


Multiple random slopes
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


Hglm

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 ***


Hglm


Item response
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)


Hglm
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


Hglm
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


Identifiability
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


Hglm
IRT

  • How about we make this a multilevel problem:

  • Solve identification by fixing one of the means to zero.


Hglm
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.


Hglm
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




Hglm
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


Irt discrimination
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



Hglm
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


Other hglm models
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


Next time mcmc
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.


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