The Geometry of Belief Change: Diagnosing and Estimating Graded Asymmetries in Two-Wave Repeated Measures

Observed changes in survey responses appear to be shaped, in part, by the structural characteristics of the instrument designed to measure them. In two-wave panel surveys, the distribution of responses in the second wave is conditioned on participants' initial responses. Deviations from the scale midpoint at baseline impose asymmetric constraints on subsequent responses. The reduced proximity to one boundary and increased proximity to the other bias the response space, concentrating probability mass toward the farther extreme. Even after normalizing each observed change by the maximum possible displacement from the baseline response, consistent asymmetries in both the direction and magnitude of change persist across three separate two-wave panel datasets (n = 3,427). Responses tend to move away from the nearest boundary, with displacement magnitude increasing as the baseline response approaches a scale endpoint. These asymmetries persist beyond what can be accounted for by random error, regression to the mean, or sensitivity to between-wave evidence. We term this systematic tendency open-movement bias: A pattern in which follow-up responses disproportionately shift toward the side of the scale with greater residual support—that is, the direction offering a greater range for movement from the respondent's initial position on a bounded, one-dimensional scale. This observation motivates the core theory proposed in this thesis: that respondents' psychological tendencies interact with the structural affordances of the bounded response scale. The correlation between baseline responses and evidence-discrepancy terms remains directionally invariant, irrespective of between-wave evidence placement, due to their shared dependence on the baseline measure. This invariance reveals structurally embedded asymmetries in potential response change imposed by the bounded scale, and empirical data suggest that respondents are sensitive to these asymmetries. This thesis shifts the analytical focus from what survey responses reveal about latent states to how the geometry of the measurement instrument shapes those responses. In doing so, it advances a methodological framework at the intersection of behavioral science, measurement theory, and survey methodology.