Soledad y aislamiento social

Political and moral decisions are both treated as outcomes from separate K- dimensional spatial where K = 2 for political decision and K = 5 for moral decisions. Each individualshas an ideal pointφs = (φs1, . . . , φsK) in policy space,

and each question q has a policy direction vector λq = (λq1, . . . , λqK), ηq possible

answers, and a set of cutpoints ζq= (ζq1, . . . , ζqηq) that define the breaks between

answers.

I use a simple ordinal logistic model for answers to questions. Individuals map their ideal pointsφsonto the question vectorλq, then add idiosyncratic noiseǫsqto

getx∗_sq, their ideal answer to the question, and finally they choose ysq, the closest available answer to their ideal answer. Since the questions are all ordinal, we can define a set of cutpoints that give the midpoints between successive answers, so for a question with ηq possible answers, we get ηq −1 cutpoints that define ηq

acceptance interval that contains their ideal answer, and they then choose the corresponding available answer.

Formally, this can be written

x∗_sq=λq·φs+ǫsq (4.1) ysq=                        1 , x∗ sq< ζq1 .. . i , ζq,i−1 ≤x∗sq < ζq,i .. . ηq , ζqηq ≤x ∗ sq (4.2)

where φs does not include a constant term.

The arrangement so far is very similar to those used in legislative voting analy- ses like Clinton et al.(2004) except that this version allows for ordinal rather than only binary decisions.

The key difference comes with how ǫsq is parameterized. I allow for het- eroskedasticity across individuals in the idiosyncratic noise terms:

ǫsq∼Logistic(0, σsg(q)) iid over q,s (4.3)

(σ_sp, σm_s )_∼LogNormal(Zγ,Σσ) (4.4) whereg(q) denotes whether questionq is a political or moral question, andσp _and σm are the political and moral heteroskedasticity terms. Z = (z1, ...zD) is a matrix

of demographic predictors where each column has been centered to have mean 0 (and in practice, scaled to have standard deviation 1). Compared to models with question-specific heteroskedasticity, this arrangement greatly reduces the dimen- sionality of the heteroskedasticity term and aids general interpretation of overall consistency rather than question-specific consistency. Note that this arrangement is very similar to the heteroskedastic one-dimensional legislative voting model of Lauderdale (2010) but applied to ordinal data and extended to multiple dimensions

of preferences and heteroskedasticity.

Finally, φs and Σs are influenced in the model by hierarchical covariates:

φs∼MVN(Zβ,Σφ) iid over s (4.5)

Σφ_∼Wishart(I,14) (4.6)

where Σs allows for covariance between the political and moral dimensions, and

the degrees of freedom parameter is twice the total dimensionality of the latent space and provides moderate shrinkage towards the identity matrix.2 _{The value}

of any particular φs is not of much interest to us, as it’s just the estimate for one

specific individual, so they are effectively nuisance parameters in the model. As we will see later, we can differentiate some individuals’ values of φs from others,

but the real interest is in the group-level values of β and Σφ_.

Identification of latent ideal point models depends on fixing some of the latent parameters to define the scaling and rotation of the latent space (Clinton et al., 2004; Jackman, 2001; Rivers, 2003). Legislative ideal point models usually fix the latent space by fixing the ideal points of a few legislators, with the number depending on the policy space’s dimensionality. That is not possible here since the data comes from a random sample of voters and there is no prior information about individual voter preferences.

Further complicating the matter is the fact that the moral questions were de- signed to have a factor structure that can be identified easily through constraints on question parameters but no orthogonality constraints in the latent space. The political questions, on the other hand, have no such predetermined structure and are easiest to interpret with orthogonal dimensions. The sampler can operate with- out strict orthogonality on the political space, but if the two political dimensions become too correlated, the sampler can hit regions of numerical instability. To avoid this, I allow correlation in the political dimensions within the sampler, but

2_{These parameters create a fairly sharp drop-off in the prior likelihood of correlations over 0.5} between dimensions.

I add a penalty term to such correlations (which hurts mixing slightly but does not bias the results), and then run the actual orthogonalizations after sampling is complete. This takes care of orthogonalization, but we still need to pin down the political dimensions’ scale and rotation.

I fix the relative scaling of ζ and φ by standardizing the noise term of φ to mean 0 and identity variance. I also constrain the political questions to have weight 0 on all moral dimensions, and vice versa, and while the directions of moral questions are specifieda priori, those of political questions are not. Instead, I run the sampler on a version of the model that does not identify the political space’s rotation, and then I apply Procrustes rotations to the political φ drawn in each sample to align them to a common space across all samples. Finally, I manually rotate the political dimensions to simplify discussion.

The final step in identification is checking the cutpointsζ. The scaling ofφis fixed to mean 0 and variance 1 on each dimension, so if we leaveλunconstrained, it is sufficient to check that the mean and scale of ǫsq are both fixed. The mean of ǫsq is constrained since it is normally distributed with mean 0, and the scale is

constrained because the log standard deviation ofǫsqhas mean 0 across individuals. Note that the identification restrictions in latent parameters are distinct from assumptions in that it is possible to transform draws taken from the posterior distribution of this model into draws from the posterior of models with other identification restrictions. In other words, the identification restrictions should be thought of as definitions that make it possible to compare points in latent space, but not as informative restrictions on the model.