Why Separate Interaction From Main Effects?
A spatial pattern that persists over time and a temporal pattern shared across locations can both produce apparent space-time structure. An interaction is different: it represents latent variation that cannot be described by adding a spatial effect to a temporal effect. Separating these components makes the model’s assumptions about dependence explicit.
For example, a persistent difference between two regions belongs in a spatial main effect; a seasonal change shared by all regions belongs in a temporal main effect. If one region has an additional time-specific departure, that variation belongs in the interaction component. The question is not merely whether observations are correlated, but which component is allowed to carry that correlation.
Let \(i=1,\ldots,S\) index locations and \(t=1,\ldots,T\) index times. For a GLMM, write the conditional mean model as
Here \(g\) is the link function, \(\mathbf{x}_{it}^{\mathsf T}\boldsymbol{\beta}\) is the fixed-effects contribution, \(a_i\) is a spatial main effect, \(b_t\) is a temporal main effect, and \(w_{it}\) is the space-time interaction. The usual GLMM sampling assumption is that observations are conditionally independent given the latent effects and covariates, with a specified response distribution and link. The latent components are commonly modeled as Gaussian and, for a simple decomposition, mutually independent.
Separating the Components
In a balanced \(S\)-location by \(T\)-time array, a useful centering convention is \(\sum_i a_i=0\), \(\sum_t b_t=0\), and interaction constraints \(\sum_t w_{it}=0\) for each location and \(\sum_i w_{it}=0\) for each time. These constraints make the interaction have no location-specific or time-specific average pattern that could be reassigned to a main effect. They are constraints on the latent components, not claims about the observed responses.
The interaction constraints require that the interaction be centered across both dimensions. For instance, a constant value added to every \(w_{it}\) is not a uniquely identifiable interaction: it could instead be represented by an intercept. Likewise, a pattern that is constant over time within each location is spatial main-effect variation, not a purely centered interaction.
Worked Example: Decomposing a Small Latent Pattern
Suppose a hypothetical \(2\times2\) array of latent values is
These are illustrative latent values, not observed outcomes. The grand mean is \((2+0+1+3)/4=1.5\). The row means are \(1\) and \(2\), so the centered spatial effects are \(-0.5\) and \(0.5\). Both column means are \(1.5\), so the centered temporal effects are \(0\) and \(0\).
The interaction is the cell value minus its row mean minus its column mean plus the grand mean. Thus
For example, \(W_{11}=2-1-1.5+1.5=1\), and \(W_{12}=0-1-1.5+1.5=-1\). Each row and column of \(W\) sums to zero. The array therefore contains a nonadditive pattern: the departure at one location-time combination is balanced by departures elsewhere rather than being explained by a location-only or time-only shift.
Four Common Interaction Structures
A standard classification distinguishes interaction models by whether their interaction effects are spatially and/or temporally correlated. To state the models without ambiguity, let \(R_S\) be a spatial correlation matrix and \(R_T\) a temporal correlation matrix, each with unit diagonal. Suppose the interaction has marginal variance \(\sigma_w^2\). Its covariance can be written by cell indices as \(\operatorname{Cov}(w_{it},w_{js})\).
| Structure | Interaction covariance | Dependence represented |
|---|---|---|
| Independent interaction | \(\sigma_w^2\,1(i=j)1(t=s)\) | Neither spatial nor temporal correlation across distinct cells |
| Temporally structured interaction | \(\sigma_w^2\,1(i=j)R_T[t,s]\) | Temporal correlation within a location, but no cross-location interaction correlation |
| Spatially structured interaction | \(\sigma_w^2 R_S[i,j]1(t=s)\) | Spatial correlation within a time, but no cross-time interaction correlation |
| Separable space-time interaction | \(\sigma_w^2 R_S[i,j]R_T[t,s]\) | Both kinds of correlation, combined multiplicatively |
These are often called Types I–IV, respectively, in the common interaction classification: Type I is independent, Type II is temporally structured, Type III is spatially structured, and Type IV is structured in both dimensions. The labels describe the interaction component, not the entire model. A model can also contain correlated spatial and temporal main effects.
For a proper Gaussian interaction, each covariance matrix must be positive semidefinite. If a structure is instead specified through a precision matrix \(Q\), the corresponding proper Gaussian covariance is \(Q^{-1}\). Intrinsic spatial or temporal precision matrices are singular, so a generalized inverse and appropriate constraints are required; their implied covariance is not an ordinary unconstrained covariance matrix. The interaction constraints must be compatible with the chosen precision structure.
Worked Example: Comparing the Four Structures
Consider two locations with spatial correlation \(R_S[1,2]=0.6\), and two times with temporal correlation \(R_T[1,2]=0.5\). Let the interaction variance be \(\sigma_w^2=4\). The diagonal correlations in both matrices are \(1\).
For a temporally structured interaction, two times at the same location have covariance \(4(0.5)=2\), while two locations at the same time have interaction covariance \(0\). For a spatially structured interaction, two locations at the same time have covariance \(4(0.6)=2.4\), while distinct times at the same location have interaction covariance \(0\).
For the separable interaction, the covariance between different locations at the same time is \(4(0.6)(1)=2.4\), and between the same location at different times it is \(4(1)(0.5)=2\). Between different locations and different times it is \(4(0.6)(0.5)=1.2\). Since the marginal interaction variance is \(4\), the corresponding interaction correlations are \(0.6\), \(0.5\), and \(0.3\). In contrast, the independent structure gives zero covariance for every pair of distinct cells.
The calculation shows what separability assumes: the correlation for a pair differing in both location and time is the product of the spatial and temporal correlations. This is a substantive modeling restriction, not a consequence of having measurements arranged on a space-time grid.
Covariance of the Full Latent Predictor
Suppose \(a_i\), \(b_t\), and \(w_{it}\) are mutually independent, centered Gaussian components with covariances
The random part of the linear predictor is \(z_{it}=a_i+b_t+w_{it}\). Independence makes cross-component covariances zero, so the covariance between two cells is the sum of the three component covariances:
This equation also explains why an interaction structure should not be confused with the covariance of the whole latent predictor. For example, a spatial main effect induces covariance between different locations at every pair of times, even if the interaction itself has no spatial dependence.
Worked Example: Adding Main Effects and Interaction
Use the correlations from the previous example and suppose \(\sigma_a^2=1.5\), \(\sigma_b^2=1\), and \(\sigma_w^2=4\). Consider the latent predictor at two different locations at the same time. Its covariance is
At a single cell, the variance is \(1.5+1+4=6.5\), because all three correlation diagonals equal \(1\). Thus the latent-predictor correlation for those two cells is \(4.3/6.5=0.6615\). The interaction contribution alone is \(2.4\), not \(4.3\); the rest comes from the shared spatial and temporal main effects.
For the same location at adjacent times, the covariance is
giving correlation \(4.0/6.5=0.6154\). These are correlations of the latent linear predictor. They are not automatically the correlations of the binary or count responses, whose relationship to latent correlation depends on the GLMM response distribution and link.
Justification: Why the Separable Formula Has a Product
A separable interaction assumes that spatial and temporal dependence act independently on the interaction covariance. One direct construction begins with independent standardized Gaussian variables \(e_{kl}\) arranged in an \(S\times T\) matrix \(E\). Let \(A\) and \(B\) satisfy \(AA^{\mathsf T}=R_S\) and \(BB^{\mathsf T}=R_T\), and set \(W=AEB^{\mathsf T}\). For the cell \(W_{it}\), expand \(W_{it}=\sum_{k,l}A_{ik}E_{kl}B_{tl}\). Because the \(E_{kl}\) are independent with variance \(1\), covariance between two cells is
Multiplying \(W\) by \(\sigma_w\) gives the stated interaction covariance. This derivation establishes the product form under the construction; it does not establish that real space-time dependence must be separable. With column-wise vectorization of \(W\), the covariance matrix is \(\sigma_w^2(R_T\otimes R_S)\); reversing the vectorization order reverses the Kronecker factor order.
Model Choice, Diagnostics, and Limitations
Begin with the scientific pattern and sampling layout. An independent interaction is a useful benchmark when unexplained cell-level variation is plausible but no cross-cell interaction dependence is justified. A temporally structured interaction is appropriate when departures appear to persist over time within locations but there is little support for dependence between locations. A spatially structured interaction makes the opposite restriction. A separable interaction permits both, while imposing the product relationship.
These interaction components differ from an additive model with only \(a_i+b_t\): an additive model cannot represent a location’s departure that changes over time. They also differ from a nonseparable interaction, which allows the spatial dependence pattern to vary with temporal separation or vice versa. A separable model is more parsimonious, but may be too restrictive when dependence changes across space-time lags.
Useful checks are structural. Examine whether the fitted model leaves systematic location-by-time patterns unexplained, whether estimated interaction variance is near zero, and whether conclusions change substantially under plausible alternative interaction structures. A variance estimate near zero can indicate that an interaction is unnecessary, but can also arise when the data provide little information to distinguish it from the main effects. Sparse observations at some location-time combinations, short time series, or few locations can make that distinction difficult.
Common warning signs include unstable variance estimates, correlations pushed toward boundary values, and sensitivity to centering constraints or covariance choices. These indicate that the assumed interaction may not be supported by the design or may be competing with main effects. They do not by themselves identify which alternative is correct. Comparisons should account for the fact that a variance component equal to zero lies on the boundary of its parameter space; a routine likelihood-ratio reference distribution may then be inappropriate without further justification.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- In \(g\{E(Y_{it}\mid\mathbf{u})\}=\mathbf{x}_{it}^{\mathsf T}\boldsymbol{\beta}+a_i+b_t+w_{it}\), which term represents a location-specific departure that changes over time?
- A Type II interaction has temporal correlation \(0.4\) and variance \(9\). What is its covariance between two distinct times at the same location? What is its interaction covariance between different locations?
- For a separable interaction, \(R_S[i,j]=0.5\), \(R_T[t,s]=0.2\), and \(\sigma_w^2=3\). Find the covariance between the two distinct cells.
- Why can a spatial main effect create covariance between observations taken at different times?
- What extra restriction does separability impose beyond having both spatial and temporal dependence?
Solutions. (1) \(w_{it}\), because it is indexed jointly by location and time and captures departures beyond the additive main effects. (2) The same-location covariance is \(9(0.4)=3.6\); the different-location interaction covariance is \(0\) under Type II. (3) \(3(0.5)(0.2)=0.300\). (4) The same \(a_i\) contributes at every time, so its covariance contribution is \(\sigma_a^2R_S[i,j]\) regardless of the time indices. (5) It requires the space-time interaction correlation to factor into the product of spatial and temporal correlations.