## Tzu-Ping Liu, Gento Kato and Samuel Fuller (University of California, Davis)

**Abstract**: Bridging ideological estimates of various groups and polities is an important, but relatively troubled branch of the ideological scaling literature. Most previous research utilizes joint scaling, which simply pools (combines) the separate groups into one single dataset, and then estimates a common ideological scale for the pooled data. However, joint scaling requires an assumption of homoskedasticity and is affected by trivial factors such as the relative sizes of groups (Jessee 2016). To date, there are two alternatives to joint scaling: model-based prediction and linear mapping. The former trains an ideological scaling model on only one group and then uses the trained model to predict ideology of other groups (Ibid.). In contrast, the latter first estimates two separate scales for each group and then merges them using a linear transformation, usually through OLS regression (Shor 2010). Both alternatives suffer from significant limitations. When using non-parametric scaling methods, model-based prediction is simply unusable, because non-parametric scaling methods do not generate parameters for new-data prediction. Furthermore, the most common linear mapping methods use parametric regression, which enforces a rigidity to the mapping that is non-conducive to non-parametric scaling techniques. Finally, these methods require a set of common individuals across groups and have no formal generalization to address multi-dimensional ideological spaces. Improving upon current bridging methodologies, we propose a novel linear mapping method for use with non-parametric scaling techniques. Importantly, our method also addresses a major limitation of previous methods: the absence of common individuals and multi-dimensionality of ideological space. Suppose that there are two non-overlapped groups of individuals, A and B. Our method works by: First, randomly sampling a small subset of individuals from A, called “bridging actors;” Second, merging bridging actors from A with B; Third, estimating separate ideological spaces for A and B; Fourth, estimating a transformation matrix using the bridging actors to map B’s ideological space onto A’s space; and finally, merging the ideological spaces of A and B using the estimated transformation matrix. The primary application of this method is in bridging the ideological spaces of politicians and voters. Given that non-parametric ideological scaling makes less stringent assumptions regarding the data generating process than their parametric counterparts, they are particularly appropriate in estimating voters’ ideology, whose data generating process we know little about. Specifically, we use Optimal Classification (OC; Poole 2005) and Ordered Optimal Classification (OOC; Hare, Liu, & Lupton 2018) to estimate ideological spaces. For the non-parametric estimation of the transformation matrix, we use Procrustes transformation, the geometric transformation method that changes the size and positioning, but not the relative distances of points/shape of a geometric object. We demonstrate the utility of our methodology by comparing the performance of our bridging methodology with that of joint scaling, using two sets of voter and politician data. First, in parallel with Jessee (2016), we use the Senate Representation Survey and roll-call votes of senators in 2004 and 2005 to bridge the OC estimates of voters’ and senators’ ideologies in the United States. Second, we utilize the UTokyo-Asahi Survey (UTAS) fielded in Japan during their House of Representatives elections in 2009 and 2012. In each election, UTAS contains two sets of surveys—of voters and candidates—that share an identical set of policy questions. This congruence between the two sets of surveys provides us a great amount of leverage to use our method to bridge the ideological space between voters and candidates (namely, there are no issues of question similarity). Specifically, we use OOC to estimate ideology here, since UTAS offers ordinal response categories for policy questions. The results suggest that our non-parametric linear transformation method can generate bridged ideology estimates comparable to those generated from joint scaling with fewer limitations, less stringent assumptions, and the ability to generalize to multidimensional spaces.