2.1.1 Observed data

For each member $l\in \{1,2,\ldots ,L\}$ , we register how that individual voted on proposal $p\in \{1,2,\ldots ,P\}$ : $V_{lp}\in \{0,1\}$ , where $V_{lp}=1$ corresponds to an “Aye” vote, while “Nay” votes map to $V_{lp}=0$ . We also observe each legislator’s count for term $w$ : $T_{lw}\in \{0,1,2,\ldots \}$ . We operationalize these terms as stemmed bigrams.

2.1.2 Vote choice

Our latent space model of voting parallels the development of Ladha (Reference Ladha1991) and Clinton, Jackman, and Rivers (Reference Clinton, Jackman and Rivers2004), where preferences and choices are embedded in a Euclidean space. We denote by $x_{ld}$ the dimension $d\in \{1,2,\ldots ,D\}$ coordinate of legislator $l$ ’s most preferred outcome. Likewise, each policy alternative subject to a vote is itself associated with a $D$ -dimensional location in the same latent space as the legislators’ locations. We denote the coordinate for the $d$ th dimension of the proposal by $z_{pd}^{\text{aye}}$ while we label the dimension $d$ coordinate for the status quo against which it is compared as $z_{pd}^{\text{nay}}$ . When a legislator chooses whether to vote for the proposal, she compares the sum of the squared distances between her most preferred coordinates and those of the proposal with the comparable sum of squared distances for the status quo. We allow some dimensions to be more important than others; $a_{d}\geqslant 0$ denotes the weight placed on dimension $d$ . We join Clinton, Jackman, and Rivers (Reference Clinton, Jackman and Rivers2004) in assuming that the dimension weights are the same for all legislators. Higher values of $a_{d}$ are associated with more important dimensions, while dimensions with a weight of $0$ are irrelevant. This gives us the propensity to vote “Aye”:

(1) $$\begin{eqnarray}\displaystyle & & \displaystyle U_{l}^{\text{vote}}\left(Aye;\{x_{ld}\}_{d=1}^{D},\{z_{pd}^{\text{aye}}\}_{d=1}^{D}\right)-U_{l}^{\text{vote}}\left(Nay;\{x_{ld}\}_{d=1}^{D},\{z_{pd}^{\text{nay}}\}_{d=1}^{D}\right)\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{1}{2}\mathop{\sum }_{d=1}^{D}a_{d}(z_{pd}^{\text{aye}}-x_{ld})^{2}-\left(-\frac{1}{2}\mathop{\sum }_{d=1}^{D}a_{d}(z_{pd}^{\text{nay}}-x_{ld})^{2}\right)-\tilde{\unicode[STIX]{x1D716}}_{lp}^{\text{vote}},\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D716}}_{lp}^{\text{vote}}$ is a standard normal random variable. Clinton, Jackman, and Rivers (Reference Clinton, Jackman and Rivers2004) restrict the nonzero dimension weights to all equal $1$ , whereas we allow more general weights; our model of the decision whether to vote for the proposal is isomorphic with theirs.

Simplifying expression (1) and combining terms gives us the legislator’s latent disposition to cast an “Aye” vote $V_{lp}^{\ast }$ , such that larger values of the latent variable means the member is more likely to favor the proposal.Footnote ¹

(2) $$\begin{eqnarray}V_{lp}^{\ast }=c_{l}^{\text{vote}}+b_{p}^{\text{vote}}+\mathop{\sum }_{d=1}^{D}a_{d}x_{ld}g_{pd}^{\text{vote}}-\unicode[STIX]{x1D716}_{lp}^{\text{vote}},\end{eqnarray}$$

where $c_{l}^{\text{vote}}$ and $b_{p}^{\text{vote}}$ are individual- and proposal-specific effects, $a_{d}$ and $x_{ld}$ are the dimension weights and ideal points described above, and $g_{pd}^{\text{vote}}$ is the signed distance between the “Aye” and “Nay” alternatives. The terms $c_{l}^{\text{vote}}$ and $b_{p}^{\text{vote}}$ are amalgams of structural parameters that serve to model the baseline propensity for a given proposal to receive support and a given member to support any proposal.

2.1.3 Term choice

We now extend the voting model to the choice of terms. We take as the choice variable the emphasis legislator $l$ places on term $w$ : $T_{lw}^{\ast }$ . While we do not observe this emphasis directly, it maps to an observed count for each term, $T_{lw}$ . The member chooses $T_{lw}^{\ast }$ on the basis of its proximity to her ideal point, and various features of its pertinence: the aptness of the term to the issues of the day, $s_{w}$ ; the verbosity of the legislator, $v_{l}$ ; and the degree to which the word has become hackneyed from overuse. Ideological proximity is the distance from her ideal point to the term’s spatial location. If the member only selected terms on the basis of ideology, then she would simply utter her most preferred term ad infinitum, regardless of external circumstance. But members do not choose terms this way. One countervailing concern is the aptness of a term to the debate at hand $(s_{w})$ —some terms are more appropriate in some years than others; for example, we find discussion of mortgage backed securities in 2009 that were not relevant in 1999. The frequency of word use is also a function of a legislator’s baseline verbosity $(v_{l})$ . If ideological proximity and aptness were the only factors at work then in each session each legislator would monotonously repeat the political buzzword of her faction ad nauseam. Actual legislators do not do this, because the value of emphasizing a given term diminishes as it becomes shopworn with overuse. We build this into our model by inducing a negative quadratic term in $T_{lw}^{\ast }$ . Formally, the choice is taken from maximizing:

(3) $$\begin{eqnarray}U_{lw}^{\text{term}}\left(T_{lw}^{\ast };\{x_{ld}\}_{d=1}^{D},\{z_{wd}^{\text{term}}\}_{d=1}^{D}\right)=\underbrace{-\frac{1}{2}T_{lw}^{\ast }\mathop{\sum }_{d=1}^{D}a_{d}(x_{ld}-g_{wd}^{\text{term}})^{2}}_{\text{Ideology}}+\underbrace{T_{lw}^{\ast }\left(s_{w}+v_{l}-\frac{1}{2}T_{lw}^{\ast }-\tilde{\unicode[STIX]{x1D716}}_{lw}^{\text{term}}\right)}_{\text{Pertinence}}.\end{eqnarray}$$

Rearrangement and substitution give an optimal choice of the form:

(4) $$\begin{eqnarray}T_{lw}^{\ast }=c_{l}^{\text{term}}+b_{w}^{\text{term}}+\mathop{\sum }_{d=1}^{D}a_{d}x_{ld}g_{wd}^{\text{term}}-\unicode[STIX]{x1D716}_{lw}^{\text{term}},\end{eqnarray}$$

where $c_{l}^{\text{term}}$ and $b_{w}^{\text{term}}$ are individual- and term-specific effects. The ideal points and dimension weights, $a_{d}$ and $x_{ld}$ , are precisely those from the vote equation.

While expression (4) is similar to the criterion given in expression (2) for choosing whether to vote in favor of or against a bill, the choice of how intensely to emphasize a term depends on the characteristics of that term, whereas the vote choice hinges on the difference between the proposal and the status quo. That is, while it is well known among legislative scholars that someone might vote in favor of a bad bill if it is presented as the alternative to an even worse status quo, politicians are free to emphasize terms that express their most preferred positions, subject to those terms being apt to the discussion at hand, and not having already become hackneyed by overuse.

2.1.4 Placing votes and words in a common space

The lynchpin of the SFA model is the collection of legislator preference parameters $\{\{x_{ld}\}_{d=1}^{D}\}_{l=1}^{L}$ . While it has become standard to use a latent space to model vote choice, see Ladha (Reference Ladha1991), Clinton, Jackman, and Rivers (Reference Clinton, Jackman and Rivers2004) or a latent space for term choice (e.g., Elff Reference Elff2013), SFA integrates both sets of observed outcomes within a common latent space (e.g., Murray et al. Reference Murray, Dunson, Carin and Lucas2013).

2.1.5 Operationalizing the model

We assume the error terms $\unicode[STIX]{x1D716}_{lp}^{\text{vote}}$ and $\unicode[STIX]{x1D716}_{lw}^{\text{term}}$ , are independent and each follow a standard normal distribution. As in Clinton, Jackman, and Rivers (Reference Clinton, Jackman and Rivers2004) positive values of the voting propensity (i.e., $V_{lp}^{\ast }\geqslant 0$ ) result in votes in favor of proposal $p$ : $V_{lp}=1$ , while negative propensities (i.e., $V_{lp}^{\ast }<0$ ) are associated with “Nay” votes: $V_{lp}=0$ . We connect the term use propensities $T_{lw}^{\ast }$ with the observed frequencies $T_{lw}$ through a set of cutpoints; $\{\unicode[STIX]{x1D70F}_{k}\}_{k=-1}^{\infty }$ such that the probability of observing a given term count is the probability of the latent variable falling between two adjacent cutpoints:

(5) $$\begin{eqnarray}\displaystyle & & \displaystyle \Pr (T_{lw}=k|\cdot )=\Pr (\unicode[STIX]{x1D70F}_{k-1}\leqslant T_{lw}^{\ast }<\unicode[STIX]{x1D70F}_{k}|\cdot )\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6F7}\left(\unicode[STIX]{x1D70F}_{k}-c_{l}^{\text{term}}-b_{w}^{\text{term}}-\mathop{\sum }_{d=1}^{D}a_{d}x_{ld}g_{wd}^{\text{term}}\right)-\unicode[STIX]{x1D6F7}\left(\unicode[STIX]{x1D70F}_{k-1}-c_{l}^{\text{term}}-b_{w}^{\text{term}}-\mathop{\sum }_{d=1}^{D}a_{d}x_{ld}g_{wd}^{\text{term}}\right)\end{eqnarray}$$

with the convention that $\unicode[STIX]{x1D70F}_{-1}=-\infty$ and $\unicode[STIX]{x1D6F7}(\cdot )$ denotes the distribution of the standard normal density.

2.2.1 Statistical challenges

The model developed in the last section integrates voting and speech using a common set of preference parameters inside a shared latent space; we must now estimate the connection of this latent space with binary voting choices and, at the same time, with the term counts found in legislators’ speeches. Next, we face the question of how many dimensions in the latent space—to assume an answer is to face the twin dangers of artificially truncating the latent space if one conjectures too few dimensions, or of over fitting “nonsense dimensions” that add noise to the estimator. Finally, term count data is characterized by “zero inflation”: the legislator-term matrix contains a surfeit of zero counts compared with what Poisson models of speech would lead us to expect.

To connect vote and speech data in a common space, we estimate a Gaussian copula factor model for mixed scale data (e.g., Murray et al. Reference Murray, Dunson, Carin and Lucas2013). This entails extending the Bayesian estimation framework of Clinton, Jackman, and Rivers (Reference Clinton, Jackman and Rivers2004) to encompass text data. To address the question of the dimensionality of the latent space, our estimator selects a sparse model of the number of dimensions (Tibshirani Reference Tibshirani1996; Park and Casella Reference Park and Casella2008). We contend with the issue of zero inflation that infests text data by estimating a flexible semiparametric cutpoint that accounts for and adjusts to the zero counts one encounters in text.

For all the parameters except the cutpoints $\{\unicode[STIX]{x1D70F}_{k}\}_{k=-1}^{\infty }$ and the dimension weights, $a_{d}$ , we assume conjugate priors that are normal for mean parameters and inverse gamma for variance parameters. The separate mean-zero normal priors over each of the term and individual specific effects, whose number grows with the sample size, place us in a Bayesian framework where the incidental parameter problem (Neyman and Scott Reference Neyman and Scott1948) does not arise. We likewise place a Jeffreys hyperprior over the variance.

2.2.2 Gaussian copula factor models

Our paper joins the strand of literature on semiparametric Gaussian copulas for discrete data spawned by Hoff (Reference Hoff2007) and applied to factor analysis by Murray et al. (Reference Murray, Dunson, Carin and Lucas2013). Accordingly, the error terms $\unicode[STIX]{x1D716}_{lp}^{\text{vote}}$ and $\unicode[STIX]{x1D716}_{lw}^{\text{term}}$ are assumed independent and identically distributed standard normal variables. The connection from the latent space to the voting data via a probit link is a standard element of the Bayesian item response theory (IRT) model (see Jackman (Reference Jackman2009)). Whereas the probit link (Albert and Chib Reference Albert and Chib1993; Clinton, Jackman, and Rivers Reference Clinton, Jackman and Rivers2004) imposes a cut point of 0 on the choice of whether to vote in favor of a proposal, we estimate a richer set of cutpoints $\{\unicode[STIX]{x1D70F}_{k}\}_{k=-1}^{\infty }$ using a flexible semiparametric model for the marginal distribution for term frequencies. The model is semiparametric in that it uses a nonlinear function of the ranks to generate the cutpoints.

2.2.3 Estimating the number of dimensions

We place a Laplacian (LASSO) prior of Park and Casella (Reference Park and Casella2008) over the dimension weights (for similar work, see Pitt, Chan, and Kohn Reference Pitt, Chan and Kohn2006; Hahn, Carvalho, and Scott Reference Hahn, Carvalho and Scott2012; Murray et al. Reference Murray, Dunson, Carin and Lucas2013):

(6) $$\begin{eqnarray}\Pr (a_{d})\sim \frac{\unicode[STIX]{x1D706}}{2}\exp (-\unicode[STIX]{x1D706}|a_{d}|).\end{eqnarray}$$

This prior provides a principled means of culling erroneous dimensions from the estimated model. As part of our estimation, we naturally recover the maximum likelihood estimate of $a_{d}$ , $\widehat{a}_{d}^{ML}$ . Given $\unicode[STIX]{x1D706}$ , the maximum a posteriori (MAP) estimate for $a_{d}$ is:

(7) $$\begin{eqnarray}\widehat{a}_{d}^{MAP}=\left\{\begin{array}{@{}ll@{}}\widehat{a}_{d}^{ML}-\unicode[STIX]{x1D706}\quad & a_{d}^{ML}>\unicode[STIX]{x1D706},\\ 0\quad & a_{d}^{ML}\leqslant \unicode[STIX]{x1D706}.\end{array}\right.\end{eqnarray}$$

The threshold parameter $\unicode[STIX]{x1D706}$ is estimated within a Gibbs sampler (Park and Casella Reference Park and Casella2008). Our prior over the dimension weights leads to ex post estimates of some weights as zero, as we describe below. For earlier work using a variable selection prior for matrix subspace selection, see, in particular, Mazumder, Hastie, and Tibshirani (Reference Mazumder, Hastie and Tibshirani2010) (especially comparing their equation (9) to our expression (7) above) and Witten, Tibshirani, and Hastie (Reference Witten, Tibshirani and Hastie2009) for extensions. Further technical details on the estimation of SFA appear in the online supplemental materials.

For a discussion of variable selection on latent dimensions in a Gaussian copula framework, see Pitt, Chan, and Kohn (Reference Pitt, Chan and Kohn2006), Hahn, Carvalho, and Scott (Reference Hahn, Carvalho and Scott2012), Murray et al. (Reference Murray, Dunson, Carin and Lucas2013).Footnote ² Correctly estimating the number of dimensions in the latent space is substantively important in its own right. Imposing too few dimensions will attenuate the models explanatory power, while estimating too many dimensions leaves the analyst interpreting chimerical dimensions, over fitting in sample, and predicting poorly out of sample.

An alternative to our procedure would be to fit the model using maximum likelihood while selecting the number of dimensions via some ancillary criterion. Doing so would raise the metaquestion of which of a potpourri of statistics to choose,Footnote ³ each of which may give different results. Given that our model simultaneously provides us with a criterion to select the number of dimensions while at the same time yielding a joint posterior density over dimensions and the other parameters of the model, we believe that researchers will find the modest additional effort to estimate the model worthwhile.

2.2.4 Zero inflation and robust cut point estimation

To cope with zero inflation in text data we must move beyond the Poisson to a more flexible marginal density for term counts. If data are generated by a Poisson density, it should be the case that the ratio of zero term counts to single counts should approximately match the reciprocal of the mean.Footnote ⁴ For instance, using the text data for the Congressional example we consider in Section 3, the ratio of zero counts to single counts is approximately ${\textstyle \frac{3}{2}}$ , fully twenty eight times the reciprocal of the mean which is about ${\textstyle \frac{3}{56}}$ .

The map from the latent space to the marginal density presents another challenge. The link between the emphasis chosen by speaker $l$ for term $w$ and the term frequency $T_{lw}$ given by expression (5) depends on the $\unicode[STIX]{x1D70F}_{k}$ . To avoid having to estimate what are potentially thousands of parameters, we instead model these cutpoints in terms of a much smaller number of parameters. We take the empirical cumulative distribution function (CDF) for a given legislative session as our point of departure:

(8) $$\begin{eqnarray}\widehat{F}(c)=\frac{1}{LW}\mathop{\sum }_{l=1}^{L}\mathop{\sum }_{w=1}^{W}\mathbf{1}(T_{lw}\leqslant c).\end{eqnarray}$$

Except for ties, using the empirical CDF is equivalent to embedding ranks in the interval $(0,1)$ (Hoff Reference Hoff2007; Murray et al. Reference Murray, Dunson, Carin and Lucas2013). We model all cutpoints as depending on a fixed set of three parameters. The $c$ th cut point is:

(9) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{c}|\unicode[STIX]{x1D6FD}_{0},\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}=\unicode[STIX]{x1D6FD}_{0}+\unicode[STIX]{x1D6FD}_{1}\widehat{F}(c-1)^{\unicode[STIX]{x1D6FD}_{2}}\end{eqnarray}$$

where:

(10) $$\begin{eqnarray}\displaystyle & \widehat{F}(-1)=0 & \displaystyle\end{eqnarray}$$

(11) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FD}_{0}=\unicode[STIX]{x1D70F}_{0}=\unicode[STIX]{x1D6F7}^{-1}(\widehat{F}(0)) & \displaystyle\end{eqnarray}$$

(12) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}>0. & \displaystyle\end{eqnarray}$$

Modeling the cutpoints in terms of $\widehat{F}(c)$ instead of $c$ leaves them less sensitive to extreme outliers, as the observed frequencies constrain the extremes. Forcing the intercept, and hence first cut point, to be $\unicode[STIX]{x1D6F7}^{-1}(\widehat{F}(0))$ addresses zero inflation directly: the probability density below the intercept is equal to the proportion of zeros in the data. The quasilinear form of $\unicode[STIX]{x1D6FD}_{1}$ and $\unicode[STIX]{x1D6FD}_{2}$ allows some flexibility in modeling the cutpoints while still ensuring that they are an increasing function of $c$ . We estimate the values of $\unicode[STIX]{x1D6FD}_{1}$ and $\unicode[STIX]{x1D6FD}_{2}$ through a Hamiltonian Monte Carlo sampler that we implemented (Neal Reference Neal, Brooks, Gelman, Jones and Meng2011).Footnote ⁵

2.2.5 Motivation behind cut point model

We note that common practice in text modeling involves a “pre-estimation” stage in which analysts remove sparse and rarely used terms from their data. Different choices at this stage result in a different total proportion of zeros in the term-document matrix, making it an example of what Simmons, Nelson, and Simonsohn (Reference Simmons, Nelson and Simonsohn2011) refer to as a “researcher degree of freedom:” a choice made while collecting and formatting the data that is given only cursory discussion and may have a substantial impact on the results. Rather than quietly adjusting the data cleaning rule to produce a desired result, we believe strongly that consequential data processing decisions should be part of the estimation process and so our cut point density includes a parameter that adapts to the total number of zeros in the term-document matrix.

The count probabilities implied by expression (5) bear a close formal resemblance to an ordered probit model with an infinitude of thresholds, though because our model applies to count data, it possesses no maximal category and so, technically, it is not an ordered probit model (see McKelvey and Zavoina (Reference McKelvey and Zavoina1975) or Greene (Reference Greene2000), pp. 875–879). Because we observe counts from 0 to several thousand, we do not fit a cut point for each value. Instead, we formulate a model for the cutpoints that reflects three attributes common to text data. First, the data is zero-inflated: most members do not use most terms in a given year, and this is problematic for the Poisson model. Second, the data is highly skewed: the observed counts range from 0 to the thousands. Third, the largest values are highly variable from year to year.

Figure 1. Accounting for zero inflation. This figure presents the fitted values for the 38,585 cases of an observed zero count in the legislator-term matrix for a Poisson density over term counts (left) and SFA (right). Of these zero counts, SFA gives a fitted value of below one for 23,532 of them, versus 3037 for the Poisson model. The marginal density used by SFA better fits the zeros in this data than does the Poisson model.

While we have taken a semiparametric approach to the marginal distribution over term frequencies, we note that Poisson densities (Chib and Winkelmann Reference Chib and Winkelmann2001) have also been used as the marginal densities of Gaussian copula models. We prefer $\widehat{F}(c)$ because even compared with a Poisson model that has been adjusted for zero inflation, $\widehat{F}(c)$ better accommodates the empirical term frequencies than would Poisson marginals. To illustrate, we fit both SFA and a latent Poisson model, implemented using the Wordfish algorithm of Slapin and Proksch (Reference Slapin and Proksch2008) and Lo, Proksch, and Slapin (Reference Lo, Proksch and Slapin2014), to the term-document matrix for floor speeches from the 112th Senate. Though we describe the data from our example more fully below, we bring it up now to compare the two methods. Figure 1 presents boxplots of the fitted values for the 38,585 cases of an observed zero count in the legislator-term matrix for a latent Poisson model (left) and SFA (right). Of these zero counts, SFA gives a fitted value of below one for 23,532 of them, versus only 3037 for the Poisson model.Footnote ⁶ The more flexible marginal density allows SFA better to fit the in-truth-zero terms, a key attribute of the data.

2.2.6 Prior sensitivity

The sensitivity of posterior inference to prior parameter specification is a generic concern for Bayesian models. With the exception of parameters about which the existing scaling literature has developed standard priors (e.g., Lauderdale and Clark Reference Lauderdale and Clark2014, Section 3.2) we select noninformative prior densities for our parameters. One component of our model, the Bayesian LASSO, uses two hyperprior parameters set by the researcher. We take as our hyperprior parameters on the Gamma prior those used in the literature (Park and Casella Reference Park and Casella2008; Kyung et al. Reference Kyung, Gill, Ghosh and Casella2010), namely a shape of 1 and rate of 1.78. This prior expresses the expected size of effects we would expect to see, on a $z$ -scale. This prior captures the belief, before seeing the data, that we will see effects of size about $1/1.78\approx 0.57$ . We encourage varying these parameters to assess posterior sensitivity, which our software allows. We illustrate in the example below.

2.2.7 Balancing words and votes

As there are often an order of magnitude more terms than votes, the researcher may fear that the term data is swamping the vote data. We therefore introduce a parameter, $\unicode[STIX]{x1D6FC}$ , that controls the relative information coming from each source.Footnote ⁷

At $\unicode[STIX]{x1D6FC}=0$ , all information on the scaled locations comes from votes; at $\unicode[STIX]{x1D6FC}=1$ , all information on the scaled locations comes from words. Even when using information from only text or only votes, SFA offers additional insight over existing methods. When scaling off only the votes, SFA will place words in the same latent space as the votes, giving an ordering to terms driven by the vote information. This can help the researcher interpret the latent dimension, through finding words that are extreme in the vote dimension. Scaling the two datasets separately does not allow one dataset to inform the other.

We suggest three ways to select $\unicode[STIX]{x1D6FC}$ . The first involves fitting $\unicode[STIX]{x1D6FC}$ at a range of values and present the results, showing how they change along these shifts. This is the strategy we follow in our example below, presenting results for $\unicode[STIX]{x1D6FC}\in \{0,1/2,1\}$ . As a default, we suggest this strategy, unless the researcher has a substantive reason to select $\unicode[STIX]{x1D6FC}$ by one of the data-driven strategies below.

As a second strategy, we suggest selecting $\unicode[STIX]{x1D6FC}$ such that the ideal points are maximally discriminatory.Footnote ⁸ Denote as a function of $\unicode[STIX]{x1D6FC}$ both the dimension weights $a_{d}(\unicode[STIX]{x1D6FC})$ and the most preferred outcomes $x_{ld}(\unicode[STIX]{x1D6FC})$ . Our criterion favors strong dimensions $(\text{large values of }\{a_{d}(\unicode[STIX]{x1D6FC})\}_{d=1}^{D})$ as well as ideal points $(\{x_{ld}\}_{(l,d)=(1,1)}^{(L,D)})$ that provide maximal discrimination among individuals. The criterion we suggest is:

(13) $$\begin{eqnarray}\text{disc}(\unicode[STIX]{x1D6FC})=\mathop{\sum }_{d=1}^{D}\mathop{\sum }_{l=1}^{L}\mathop{\sum }_{l^{\prime }=1}^{L}a_{d}(\unicode[STIX]{x1D6FC})^{2}(x_{ld}(\unicode[STIX]{x1D6FC})-x_{l^{\prime }d}(\unicode[STIX]{x1D6FC}))^{2}\end{eqnarray}$$

with $\unicode[STIX]{x1D6FC}$ chosen to make the left-hand side of equation (13) as large as possible. All the elements needed to calculate $\text{disc}(\unicode[STIX]{x1D6FC})$ are returned from the MCMC output, so our software returns the full posterior density of this statistic, and the optimal value can be selected on the basis of which has the highest mean discrimination.

As a third criterion, we also implement the WAIC statistic, a Bayesian approximation of cross-validated prediction performance (Vehtari, Gelman, and Gabry Reference Vehtari, Gelman and Gabry2017). Though we favor a default of $1/2$ and of presenting results from $\unicode[STIX]{x1D6FC}$ of both 0 and 1, our discrimination and WAIC statistic give a means of a data-driven means of selecting $\unicode[STIX]{x1D6FC}$ . We illustrate the use of the discrimination and WAIC statistics in the online supplemental materials.

2.2.8 Software

We offer two implementations of the software. The first is a full MCMC implementation, which generates samples from the full posterior given the data. This allows the researcher to not only estimate the spatial locations as well as the number of latent dimensions, but also to characterize the uncertainty estimates. The second is an EM implementation. While it only returns point estimates, it is faster than the MCMC implementation and often useful for preliminary results during the process of practical modeling.Footnote ⁹

2.2.9 Additional uses

We have focused on a situation where both votes and term counts are present. There are cases where we observe legislative speech, but we either lack their votes or the votes are so heavily whipped we do not trust them. In this case, as we show in the context of our Congressional data, SFA can leverage words to recover reliable ideal point estimates even in the absence of reliable vote data for nonparty leaders.

Moreover, SFA estimates the spatial location of terms. Existing studies estimate the political affect of terms by attributing left leaning content to those used by Democrats, and rightward import to those spoken by Republicans (Laver, Benoit, and Garry Reference Laver, Benoit and Garry2003; Gentzkow and Shapiro Reference Gentzkow and Shapiro2010), or modelers posit that terms and votes arise from two conditionally independent data generating processes (Gerrish and Blei Reference Gerrish, Blei, Pereira, Burges, Bottou and Weinberger2012; Lauderdale and Clark Reference Lauderdale and Clark2014). SFA scales terms and votes simultaneously, providing natural structural estimates of word affect.

2.3.1 Topic models and related methods

We note that our method differs from the popular topic model approach to text analysis (Blei, Ng, and Jordan Reference Blei, Ng and Jordan2003; Grimmer Reference Grimmer2010; Roberts et al. Reference Roberts, Stewart, Tingley, Lucas, Leder-Luis, Gadarian, Albertson and Rand2014). In brief, when a researcher is looking for an underlying latent structure ordering actors and their choices, and they are comfortable making the model’s assumptions, then SFA should be implemented. In the absence of any structural or behavioral assumptions, topic models will always return a summary of clusters within the data. We emphasize that this is not an either/or distinction; SFA and topic models calibrate disparate features of the data.

Lastly, we situate SFA in the context of several other related methods. Many scholars have combined voting data with a topic model (Gerrish and Blei Reference Gerrish, Blei, Pereira, Burges, Bottou and Weinberger2012; Wang et al. Reference Wang, Salazar, Dunson and Carin2013; Lauderdale and Clark Reference Lauderdale and Clark2014). The methods, particularly when applied to courts, offer great insight: we may think of courts as voting on a wide array of topics, and judicial preference may vary greatly from one to the next. These models differ from SFA in two crucial aspects. First, the topic model methods do not give an ideological position to terms. Terms are not placed on, say, a left–right dimension shared by the actors. The topic models capture what actors are voting about, but not the ideological structure of the topics. Second, the models offer a disjointed relationship between words and votes. Actors are choosing both words and votes, but only vote choice is grounded in a spatial model. SFA differs by offering a tight, and formal, coupling of the ideological preferences that generate terms and votes. SFA is a model for term selection that is fully compatible with the standard and accepted models of vote choice.

2.3.2 Comparison with additional methods

SFA is related to several existing methods for scaling votes, scaling text, and combining multiple outcomes in a single factor analytic model. Our model is an extension of Ladha (Reference Ladha1991) and Clinton, Jackman, and Rivers (Reference Clinton, Jackman and Rivers2004) who model votes in a latent space, which they connect with binary choice using a probit model, and of Slapin and Proksch (Reference Slapin and Proksch2008) and Lo, Proksch, and Slapin (Reference Lo, Proksch and Slapin2014) who model spatial choice in a Poisson framework. Here we connect the latent space with count as well as binary outcomes (Murray et al. Reference Murray, Dunson, Carin and Lucas2013). Our estimation technique relies on probabilistic principal components analysis (Tipping and Bishop Reference Tipping and Bishop1999), whereby singular vectors and latent factors correspond if the errors are assumed independent and identically Gaussian. The method is factor analytic, as opposed to a singular value decomposition, because the scaling takes place in a latent space rather than operating directly on the observed data. We connect the two spaces using a Gaussian copula for mixed data.

Rather than matrix decompositions in a latent space, several works have turned to a Poisson or negative binomial model in order to model term counts. For example, Wordfish of Slapin and Proksch (Reference Slapin and Proksch2008), Lo, Proksch, and Slapin (Reference Lo, Proksch and Slapin2014) is similar to SFA, for terms, but under a Poisson or negative binomial link instead of a probit link. SFA leads to an identical formulation for the latent systematic component of word choice, except the latent component is exponentiated in order to guarantee positivity (see also Elff Reference Elff2013; Bonica Reference Bonica2014). SFA differs in that it places both word and vote choice in the same latent z-space, allowing both types of data to be modeled jointly. Our cut point model allows for a more flexible mapping from the latent to observed data space. We also model the zero inflation directly and are robust to outliers, as described above. SFA also estimates the underlying dimensionality.

The use of the Poisson by Slapin and Proksch (Reference Slapin and Proksch2008) notwithstanding, many analysts eschew both the Poisson and the Negative Binomial in their analysis of text. Examples include: the widely used Wordscores model of Laver, Benoit, and Garry (Reference Laver, Benoit and Garry2003) or the nonparametric content analysis of Hopkins and King (Reference Hopkins and King2010); the text based “slant” measures used by Gentzkow and Shapiro (Reference Gentzkow and Shapiro2010); the LDA formulation of Blei, Ng, and Jordan (Reference Blei, Ng and Jordan2003), Gerrish and Blei (Reference Gerrish and Blei2011) which converts term counts to proportions, thereby admitting a Dirichlet prior; PCA or kernel PCA on the tf-idf matrix Spirling (Reference Spirling2012); and semiparametric copula models for mixed data Hoff (Reference Hoff2007). Like these, SFA is not based on a Poisson model; see Murray et al. (Reference Murray, Dunson, Carin and Lucas2013, esp. 2.1) for a formulation close to SFA’s.

Mixed factor analysis models have turned to copulas to combine data of different types. In these models, mixed data such as counts, binary, and continuous data, are placed on and analyzed on a common underlying latent scale. The Gaussian copula, which places all data on an underlying common $z$ -scale, is a typical choice. For example, Quinn (Reference Quinn2004) converts observed continuous data to a $z$ -scale, and then combines it with ordinal and categorical data on the same scale; for recent extensions, see Hoff (Reference Hoff2007), Murray et al. (Reference Murray, Dunson, Carin and Lucas2013). Our model is closely related to that given in Murray et al. (Reference Murray, Dunson, Carin and Lucas2013, Section 2.1). We differ in that we only have two types of data, text and votes, and hence have to estimate only one set of cutpoints, whereas Murray et al. (Reference Murray, Dunson, Carin and Lucas2013) consider the problem of generic types of data. Murray et al. (Reference Murray, Dunson, Carin and Lucas2013) work with the transformed ranks of all variables, eschewing cutpoints entirely. For that reason, our method is more powerful when modeling votes and text. SFA also introduces a cut point modeled tailored to several common aspects of text data, as we describe above.

Other methods have estimated dimensionality (Heckman and Snyder Reference Heckman and Snyder1997; Hahn, Carvalho, and Scott Reference Hahn, Carvalho and Scott2012). Additionally, Aldrich, Montgomery, and Sparks (Reference Aldrich, Montgomery and Sparks2014) show that sufficiently large cross-party variance can mask important within-party dimensions.Footnote ¹¹