This paper develops an extension of Poole’s (Polit Anal 8(3):211–237, 2000) Optimal Classification (OC) scaling procedure to the analysis of polytomous or ordered choice data. This type of data is regularly encountered in public opinion and expert surveys, legislative and judicial bodies where abstention is relevant, and measures of policy that are coded along ordinal scales. OC is nonparametric and requires only minimal assumptions about voters’ utility functions and the error term. As such, Ordered Optimal Classification (OOC) provides a flexible modeling strategy to estimate latent ideological spaces from ordinal choice data. OOC is also easily estimated in multidimensional space without identifying restrictions. After describing the OOC procedure, we perform a series of Monte Carlo experiments and apply the method to analyze survey data from the 2015 Cooperative Congressional Election Study. We then conclude with a discussion of how scholars can utilize OOC in future work involving multidimensional spatial models of choice.
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This includes, for instance, the Chapel Hill Expert Survey (Bakker et al. 2015), the Congressional Election Study (Stone and Simas 2010), the Convention Delegate Study (Layman et al. 2010), the United Nations (Bailey et al. 2017), the IMF (Thacker 1999), human rights policies (Fariss and Schnakenberg 2014), and legal systems (Rosenthal and Voeten 2007).
OOC can also be used to analyze the strategic and sincere components of roll call voting on a series of related dichotomous votes (particularly amendment voting) by coding legislators’ voting patterns (e.g., YY, NY, NN) categorically (Silberman and Durden 1976; Nunez and Rosenthal 2004; Ladha 1991). As discussed in the next section, OOC uses a constrained normal vector to model the dichotomous components of a single ordinal scale. This means that all of the votes that comprise a given voting pattern will have an identical orientation in the recovered ideological space. We thank Howard Rosenthal for raising this point.
However, an underappreciated aspect of measuring mass ideology concerns the researcher’s reliance on which issues and policy alternatives are and are not included in public opinion surveys. This is a subtle—though still consequential—form of agenda control. We thank Bob Erickson for this observation.
Although OC is not guaranteed to find the global maximum, it regularly does so or gets very close to it. Poole (2000) reports the results of extensive Monte Carlo tests in one to ten dimensions which show that, at worst, only about 43 misclassifications per 50,000 total choices occur. Such a figure indicates that OC is very closely approximating the global classification maximum.
The starting values for the ideal points are obtained from an eigenvalue–eigenvector decomposition of the double-centered voter agreement score matrix.
As Adam Bonica has pointed out, OC and SVMs are closely related in their pursuit of a separation hyperplane that optimally divides two classes of data. The major differences between the two methods concerns their loss functions (OC uses overall correct classification rates while SVMs assess classification performance using both the correct classification rate and the robustness of the derived hyperplane), constraints on the separating hyperplane (SVMs allow for nonlinear separating hyperplanes, while OC uses strictly linear separating hyperplanes), and their treatment of the predictor variables X (standard SVMs requires the predictor variables to be observed, while in OC the predictor variables [i.e., the ideal point coordinates of the observations] are treated as latent variables to be estimated) (Hastie et al. 2009; Bonica 2018).
The cost parameter C controls the width of the margin—i.e., the number of observations that are allowed to violate the margin and constitute the support vectors.
We use a linear kernel to simplify analysis, although one of the attractive properties of SVMs is that alternative kernel functions can be used to estimate nonlinear decision boundaries and prediction functions. We note that the adaptation of such kernels provides one possible avenue for future development of the OOC procedure.
Implemented in the krls package in R (Ferwerda et al. 2017).
We follow Hainmueller and Hazlett (2014) and use the Gaussian kernel function.
We have found that support vector regression (SVR) tends to slightly outperform its competitors in terms of classification performance, and does so with reasonable computation efficiency. Hence, we set SVR as the default method for the normal vector routine and use it in the analyses presented in this paper.
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An earlier version of this paper was presented at the Keith T. Poole Career Retrospective Conference, May 2017, Athens, GA. Thanks to Keith Poole for his patient and invaluable guidance throughout this project and to Bob Erickson, Howard Rosenthal, and an anonymous review for their insightful comments and suggestions for improvement. Thanks also to the University of Georgia, the Albert B. Saye Fund, and the School of Public and International Affairs for their generous financial support for conference travel. An R package and accompanying code to estimate Ordered Optimal Classification is available at http://github.com/tzuliu/ooc.
Below we perform two additional sets of Monte Carlo experiments on the statistical properties of the Ordered Optimal Classification estimator. The first assesses OOC’s recovery of the true ideal points of ideologically moderate and ideologically extreme respondents. The second provides an informal test of the consistency of the OOC estimator, replicating the analysis in the main text (specifically, Figs. 3, 4) across an increasing number (25, 50, and 100) of issue questions.
Figure 7 shows the correlations between the true and recovered ideal points separately for ideologically moderate and extreme respondents in two dimensions. Ideologically moderate respondents are defined as those with ideal points in the interquartile range on both dimensions, while ideologically extreme respondents are those with ideal points outside of the interquartile range on both dimensions. OOC clearly performs better in its recovery of extremists’ ideal points, though some of this is an artifact of the wider (more polarized) range of ideal point values for ideologically extreme respondents relative to moderate respondents.
Figures 8 and 9 replicate the analaysis in Figs. 3 and 4 while increasing the number of simulated issue scales from 25 to 50 to 100. If OOC is a consistent estimator, the correlations between true and estimated parameters should increase alongside the number of issues for a given level of error and dimensionality. This is precisely what we observe, with the improvements most apparent in the three-dimensional case in Fig. 9.
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Hare, C., Liu, T. & Lupton, R.N. What Ordered Optimal Classification reveals about ideological structure, cleavages, and polarization in the American mass public. Public Choice 176, 57–78 (2018). https://doi.org/10.1007/s11127-018-0540-6
- Ideal point estimation
- Public opinion
- Optimal Classification