3. Circular GLMMs

Repeated observations can reveal whether each animal keeps a consistent heading and whether different animals agree. A generalized linear mixed model (GLMM), also called a hierarchical or multilevel model, extends the GLM with animal- or group-specific departures from population effects.

Add individual effects to the regression coefficients

Let \(i\) identify an animal and \(n\) an observation from that animal. The coefficient vectors are

\[ \boldsymbol{\beta}_i=\boldsymbol{\beta}+\mathbf{b}_i, \qquad \boldsymbol{\alpha}_i=\boldsymbol{\alpha}+\mathbf{a}_i. \]

\(\boldsymbol{\beta}\) and \(\boldsymbol{\alpha}\) are population-level coefficients. \(\mathbf{b}_i\) and \(\mathbf{a}_i\) are animal-specific offsets for mean direction and concentration. An offset can change a baseline intercept, a treatment response, or a continuous slope. The observation model is

\[ \begin{aligned} \eta_{\mu,in}&=\mathbf{x}_{in}^\top(\boldsymbol{\beta}+\mathbf{b}_i),\\ \eta_{\kappa,in}&=\mathbf{x}_{in}^\top(\boldsymbol{\alpha}+\mathbf{a}_i),\\ \mu_{in}&=g_\mu^{-1}(\eta_{\mu,in}),\\ \kappa_{in}&=g_\kappa^{-1}(\eta_{\kappa,in}),\\ y_{in}\mid\mathbf{x}_{in},\boldsymbol{\beta}_i,\boldsymbol{\alpha}_i &\sim\mathcal{M}(\mu_{in},\kappa_{in}). \end{aligned} \]

The inverse links are the same circular-modulo and softplus functions used in the GLM. The sampling unit can also be a trial, site, or social group; choose the grouping and varying effects to match the experiment.

Partial pooling on the predictor scale

For offsets on an unconstrained scale before the links, a common construction is

\[ \mathbf{b}_i\sim\mathcal{N}(\mathbf{0},\Sigma_\beta), \qquad \mathbf{a}_i\sim\mathcal{N}(\mathbf{0},\Sigma_\alpha). \]

The covariance matrices \(\Sigma_\beta\) and \(\Sigma_\alpha\) describe the spread of animal effects and any modelled association among their intercepts and slopes. Their unknown scales receive hyperpriors, priors on the parameters of these shared distributions. Learning the spread from all animals induces partial pooling: individual estimates share information while retaining differences supported by the observations.

These multivariate Normal priors are one predictor-scale construction. An angular offset can instead use the circular construction below.

Circular offsets: between-animal alignment

For a genuinely angular effect, such as an animal's baseline directional offset, a circular alternative is

\[ b_{i,0}\sim\mathcal{M}(0,\kappa_\mu), \qquad \mu_i(\mathbf{x}=\mathbf{x}_0) =g_\mu^{-1}(\beta_0+b_{i,0}), \]

where \(\mathbf{x}_0=(1,0,\ldots,0)^\top\) is the baseline design vector. This replaces the Normal prior for that angular offset. The remaining offsets can still use appropriate predictor-scale distributions.

\(\kappa_\mu\) measures how tightly animal directional offsets cluster around the population direction. Give it a hyperprior when estimating this alignment from the data. It is different from the observation concentration \(\kappa_{in}\):

Quantity	Biological interpretation
\(\kappa_{in}\)	Consistency of headings within an animal at the specified predictors
\(\kappa_\mu\)	Alignment of animal mean-direction offsets around the population value

Thus, animals can each hold a steady heading (high observation concentration) while pointing in different directions (low between-animal alignment). Angular treatment offsets can be modelled similarly when supported by the experimental design.

Bambi's standard group-effect formulas, such as (1 + cue | animal), specify group effects on the predictor scale. They do not by themselves replace an offset prior with a von Mises distribution. Circular offset constructions can be specified explicitly in PyMC to match the intended model.

Choose the level of prediction

The same fitted model supports different biological summaries:

Target	Calculation
An observed animal	Use its population coefficients plus estimated offsets
The population relationship at zero offsets	Set \(\mathbf{a}_i=\mathbf{b}_i=\mathbf{0}\) before applying the links
A new animal from the population	Draw new offsets from the fitted population distributions and propagate their uncertainty

Setting offsets to zero describes a reference animal. It generally differs from the prediction averaged over animal variation, because the inverse links are nonlinear. In particular, observation concentration and between-animal dispersion both shape the distribution of headings pooled across animals. Posterior predictive checks at both levels help distinguish these patterns.

Next: Extend the observation model with mixtures.