3.3 Latent variable models
3.3.4 Willingness to admit
parameter estimate of Fear and Quiz than we saw in the previous section, and this is likely because Quiz is now forced to absorb some of INR’s demographic variation.
If we look at the Quiz/INR facet of Figure 3.16, Age and Hisp are more positive for INR than for Quiz, and when we compare the Quiz/Fear facet in Figure 3.16 with the Figure 3.21, we see that Age and Hisp were the main changes, and both moved in a positive direction. Whether this movement is a modeling artifact or a more accurate representation of political knowledge cannot be determined from the available data, but the changes are relatively small and so that distinction can be ignored with acceptable risk of misinterpretation.
The other plots of latent variables and their correlations are similar to the previous model, but with slightly more correlation between INR and the other terms that follow from the explicit structural connection. The main change in the plots of individual-specific variations and their correlations is that INR’s noise is no longer correlated with Fear’s or Quiz’s. This is by construction since the individual-specific variation on INR is fixed to be independent of the other terms in Equation 3.9.
Table 3.7: Posterior quantiles of correlations in latent Fear, INR, and Quiz param- eters across draws from posterior in the willingness-to-admit model
Quantile
Variables 2.5% 5% 50% 95% 97.5%
INR, Fear 0.260 0.267 0.303 0.341 0.347 Quiz, Fear 0.217 0.230 0.311 0.390 0.406 Quiz, INR 0.660 0.676 0.722 0.763 0.769
The key results of interest from the willingness-to-admit model are the effects of Fear and Quiz on INR from Equation 3.9. Distribution of the raw variables are shown in Figure 3.26 with quantiles in Table 3.9. The connection to Quiz is apparent, and a change of 1 in Quiz moves INR by nearly .75. The directions of other parameters are fairly uncertain, but we will discuss what they suggest. Fear
Fem
Educ
Age
Ideol
Black
Hisp ORace
Income
NoCh
ChWeek
BornAg
−0.2 0.0 0.2
−0.2 0.0 0.2
Quiz
Fear
Figure 3.21: Fitted parameters of the willingness-to-admit model. The circles are the joint 40% confidence ellipses for each parameter in each pair of latent dimensions. 40% ellipses correspond tot= 1, and they are used to avoid excessive overlap in the visualization. 90% confidence ellipses are about 2.1 times larger, and 95% ellipses are about 2.4 times larger. The ellipses are based on the mean and covariance of each parameter’s draws, which is an approximation of the actual joint distribution of the draws.
INR Quiz
−2 0 2 4
−2 0 2 4
FearINR
−2 0 2 4 −2 0 2 4
Figure 3.22: Mean latent Fear, INR, and Quiz estimates for each individual in the willingness-to-admit model. The dotted red ellipse is a representative 95% joint credible ellipse for an individual at (0,0).
INR Quiz
0 5 10 15
0 5 10 15
FearINR
0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 1.00
Correlation
density
Figure 3.23: Distribution of correlations in latent Fear, INR, and Quiz parameters across draws from posterior in the willingness-to-admit model
INR Quiz
−2
−1 0 1 2
−2
−1 0 1 2
FearINR
−1 0 1 2 −1 0 1 2
Figure 3.24: Mean individual-specific variation in latent Fear, INR, and Quiz terms in the willingness-to-admit model. The dotted red ellipse is a representative 95%
joint credible ellipse for an individual at (0,0).
INR Quiz
0 3 6 9 12
0 3 6 9 12
FearINR
−0.50 −0.25 0.00 0.25 0.50−0.50 −0.25 0.00 0.25 0.50 Correlation
density
Figure 3.25: Distribution of correlations in individual-specific latent Fear, INR, and Quiz variation across draws from posterior in the willingness-to-admit model
Table 3.8: Posterior quantiles of correlations in individual-specific latent Fear, INR, and Quiz variation across draws from posterior in the willingness-to-admit model
Quantile
Variables 2.5% 5% 50% 95% 97.5%
INR, Fear -0.094 -0.086 -0.030 0.026 0.041 Quiz, Fear -0.067 -0.050 0.061 0.170 0.189 Quiz, INR -0.112 -0.102 -0.050 0.003 0.011
appears to have a small positive effect on INR (more fearful people skipping more questions), and this effect appears to be strongest for people with less information (Fear × Quiz likely has a negative effect). There appears to be gender variation in this interaction, which will be clearer below.
Table 3.9: Posterior quantiles of Fear and Quiz effects on INR in the willingness- to-admit model
Quantile
Parameter 2.5% 5% 50% 95% 97.5%
Fear -0.075 -0.038 0.096 0.228 0.249
Fear x Fem -0.140 -0.116 -0.008 0.094 0.119
Quiz 0.622 0.647 0.766 0.911 0.938
Quiz x Fem -0.128 -0.112 -0.027 0.054 0.069 Fear x Quiz -0.184 -0.163 -0.068 0.011 0.025 Fear x Quiz x Fem -0.044 -0.029 0.053 0.149 0.165
Comparisons between males and females are difficult in the graph with raw parameters, so Figure 3.27 and Table 3.10 transform the variables into separate effects for males and females. The impact of Quiz is apparent for both, and Fear’s effect has a similar distribution for males and females, but the interaction between
0 2 4 6
0 2 4 6
0 2 4 6
0 2 4 6
0 2 4 6
0 2 4 6
FearFear x FemQuizQuiz x FemFear x Quiz Fear x Quizx Fem
0.0 0.4 0.8
value
density
Figure 3.26: Distributions of Fear and Quiz effects on INR in the willingness-to- admit model
Fear and Quiz seems to only impact males—the estimate for females in noisy but has mean close to 0.
0 2 4 6
0 2 4 6
0 2 4 6
0 2 4 6
0 2 4 6
0 2 4 6
Male FearFemale FearMale QuizFemale Quiz MaleFear x Quiz FemaleFear x Quiz
0.0 0.5 1.0
value
density
Figure 3.27: Distributions of Fear and Quiz effects on INR in the willingness-to- admit model transformed for easier comparisons between males and females
As mentioned before, only Quiz has a clear enough effect to be conclusive, but it is comforting to see that the interaction that we observed in the traditional model is reasonably well replicated in a model that is extremely different methodologically.
Table 3.10: Posterior quantiles of Fear and Quiz effects on INR in the willingness- to-admit model transformed for easier comparisons between males and females
Quantile
Parameter 2.5% 5% 50% 95% 97.5%
Male Fear -0.075 -0.038 0.096 0.228 0.249 Female Fear -0.143 -0.106 0.087 0.278 0.315 Male Quiz 0.622 0.647 0.766 0.911 0.938 Female Quiz 0.572 0.591 0.740 0.897 0.926 Male Fear x Quiz -0.184 -0.163 -0.068 0.011 0.025 Female Fear x Quiz -0.175 -0.146 -0.011 0.103 0.124