A generalized angular regression model with a circular random intercept and a scalar random slope

Clustered circular responses arise in repeated orientation experiments, movement ecology, and sensor studies, where both directionality and within-cluster dependence matter.

We propose a parsimonious mixed-effects extension of generalized angular regression in which the mean direction is defined by the orientation of a two-dimensional consensus vector.

The model combines a von Mises circular random intercept with a pre-specified Gaussian scalar random slope acting on one consensus-vector coefficient.

Conditional on the scalar slope, the circular intercept integrates analytically, yielding a one-dimensional marginal likelihood and avoiding the high-dimensional integration required by general random-slope models.

We establish high-level design-conditional identifiability conditions, cluster-asymptotic likelihood theory away from the variance boundary, and a practical framework for deterministic quadrature, diagnostics, and model assessment.

Simulation studies investigate numerical stability and finite-sample performance.

An application to repeated sandhopper orientation data illustrates the proposed methodology and highlights practical considerations for variance-component inference.