Conditional Autoregressive Model

Definition

A conditional autoregressive (CAR) model specifies spatial random effects $Z = (Z_1, \ldots, Z_q)'$ for $q$ geographic regions through their full conditional distributions: each $Z_i | Z_{-i}$ follows a normal distribution whose mean is a weighted average of neighboring values. When the implied $q \times q$ precision matrix $B$ is positive definite the joint distribution of $Z$ is proper; when $B$ is only nonnegative definite (singular), the joint distribution is improper — a "partially informative normal" — requiring additional identification from the data for the posterior to be well-defined.

Key Ideas

• Conditional specification (Besag 1974): $f(Z_i | Z_{-i}) = N\!\bigl(\sum_{j \in \Lambda_i} \beta_{ij} Z_j,\; \delta_1/a_i\bigr)$, where $\Lambda_i$ is the set of regions adjacent to $i$. The precision matrix $B$ has diagonal elements $a_i$ and off-diagonal element $-a_i\beta_{ij}$ for $j \in \Lambda_i$.
• Proper joint distribution: If $B$ is symmetric positive definite then $Z \sim N(0, \delta_1 B^{-1})$ — a proper multivariate Normal. Besag (1974) shows this is the unique proper joint consistent with the conditionals.
• Partially informative normal: When $B$ is singular of rank $r < q$, $f(Z) \propto \delta_1^{-q/2}\exp\!\bigl(-\tfrac{1}{2\delta_1}Z'BZ\bigr)$ is improper; the $(q-r)$-dimensional null space of $B$ is completely unidentified from the prior alone.
• Three model variants (Sun-Tsutakawa-Speckman 1999): (i) Model 1A — $B = D - \rho C$, $|\rho| < 1$, where $D$ = diagonal degree matrix and $C$ = adjacency matrix; proper joint. (ii) Intrinsic CAR (Besag-York-Mollié 1991) — $B = D - C$ (i.e., $\rho=1$), all row/column sums zero, rank $q-1$, improper; standard in disease mapping. (iii) Clayton-Kaldor (1987) — $B = I_q - \rho C$; proper iff $\lambda_1^{-1} < \rho < \lambda_q^{-1}$ where $\lambda_1 < 0 < \lambda_q$ are the extreme eigenvalues of $C$.
• Posterior propriety rank condition: For the hierarchical model $V = X_1\theta + X_2 Z + e$ with flat prior on $\theta$ and inverse-gamma priors on variance components, the posterior is proper if and only if $\mathrm{rank}(X_2'R_1X_2 + B) = q$ where $R_1 = I_n - X_1(X_1'X_1)^{-1}X_1'$. This condition is both necessary and sufficient (Theorems 2–3 of Sun-Tsutakawa-Speckman 1999).
• Functional compatibility: The improper intrinsic CAR prior has full conditionals that are "functionally compatible" with the improper joint (Hobert-Casella 1998) — the conditionals are proper and consistent, but the joint they formally imply has infinite mass. Gibbs sampling on such conditionals runs without error but explores an improper distribution if the rank condition fails.
• Auto-models and the large-scale/small-scale split (Cressie-Chan 1989): the CAR conditionals are an instance of Besag's auto-models (Markov random fields) — auto-Gaussian for continuous data (a Gaussian CAR) and auto-Poisson for counts. Cressie-Chan model a regional map as large-scale variation (a spatial trend/mean surface) plus small-scale variation (the variance and CAR spatial dependence in the residuals), fit by resistant (robust) trend estimation followed by auto-Gaussian fitting to the residuals. Their analysis of the North Carolina Sudden-Infant-Death (SIDS) counts, 1974–78 became the canonical benchmark dataset for areal spatial statistics.

How It Works

1. Construct adjacency structure: Build adjacency matrix $C$ (entries $c_{ij}=1$ if regions $i$ and $j$ share a boundary, 0 otherwise) and degree matrix $D = \mathrm{diag}(d_1, \ldots, d_q)$ with $d_i = \#\{j: c_{ij}=1\}$.
2. Choose CAR variant: Intrinsic CAR ($B = D - C$, $\rho=1$) encodes "no overall level" — only local contrasts are identified from the prior. Clayton-Kaldor ($B = I - \rho C$) has a proper joint for $|\rho| < 1/\lambda_q$.
3. Check rank condition: Before fitting, verify $\mathrm{rank}(X_2'R_1X_2 + B) = q$. For intrinsic CAR with grand mean in $X_1$: the rank condition is equivalent to checking that the regression design $X_2$ identifies the null space of $D - C$ (the constant vector $\mathbf{1}_q$).
4. Gibbs sampling: The full conditional $Z_i | Z_{-i}, \theta, \delta_0, \delta_1$ is Normal; the full conditional for $\delta_1 | Z$ is inverse-gamma — both conjugate. The sampler is straightforward to implement but will appear to converge even when the posterior is improper.
5. Generalized Linear Mixed Model (GLMM) extension: Replace the Gaussian observation layer with a Poisson log-linear response ($Y_i \sim \mathrm{Poisson}(m_i e^{V_i})$) for disease mapping. The sufficient condition for a proper posterior requires at least $n$ observations with integrable likelihoods and the rank condition on the design matrices (Theorem 4).

Why It Matters

• Disease mapping: The Besag-York-Mollié intrinsic CAR is the default spatial random effect in WinBUGS, R-INLA, and CARBayes. Its improper joint prior means the posterior's existence depends entirely on the data design satisfying the rank condition — a check that most applied users do not perform.
• Prior choice, ecological bias, and spatial confounding (Wakefield 2007): a critical review of the applied practice. (i) Priors matter — set the variance-component prior on an interpretable scale (the marginal SD / range of relative risks the random effect implies) rather than the reflexive $\mathrm{IG}(\epsilon,\epsilon)$, which is not non-informative. (ii) Ecological bias — an area-level exposure–disease association need not hold at the individual level and can only be repaired with individual data. (iii) Spatial confounding — when the exposure is itself spatially structured, the CAR random effect and the exposure compete for the same variation, distorting the estimated exposure effect; so disease mapping is robust but spatial ecological regression is hazardous. Illustrated on the canonical Scottish male lip-cancer data.
• Geographic mortality: Directly extends Tsutakawa (1988)'s gamma-Poisson model (see Tsutakawa (1988)) to the spatial correlation setting; spatial dependence across counties is now encoded in $B$ rather than treated as independent regional effects.
• Gibbs sampling failure mode: The functionally compatible but improper CAR prior is a canonical example of the "proper conditionals, improper joint" failure mode described in the Gibbs Sampler literature (Casella-George 1992; Hobert-Casella 1996).
• Spatial econometrics: CAR models are the discrete-space analogue of Gaussian random fields; the same conditional-specification idea underlies spatial lag and spatial error models in regional econometrics.
• Spatio-temporal CAR (Mariella-Tarantino 2012): for areal data observed at each location over time, the STCAR model handles spatial dependence between sites and temporal dependence across periods in one space-time autoregressive matrix. Building on the Generalized Multivariate CAR (GMCAR; Jin-Carlin-Banerjee 2005), it reduces the unknowns to a single spatial-association parameter estimated per period. Unlike a VAR (Sims 1980), whose coefficient matrix ignores where the series sit, the STCAR autoregressive matrix encodes the spatial localization of the sampled realizations — so neighbouring areas influence each other's dynamics. Beyond its disease-mapping roots it is applied to bankruptcy risk, reconstructing the space-time distribution of expected SME failures (exploiting the disease-contraction ↔ default-risk analogy).
• Flexible multivariate spatial modelling (Neelon-Gelfand 2014): CAR priors need not sit only on a single Gaussian random-effect layer. Modelling correlated continuous outcomes as a finite mixture of multivariate normals with component-specific spatial random effects, each given a CAR prior, lets the shape of the joint response distribution — not just its mean — vary smoothly across regions. Integrating over the fitted mixture yields region-specific joint, marginal, and conditional inferences, and the hierarchy (individual- and areal-level predictors) is fit by Gibbs sampling with mostly closed-form full conditionals. Applied to North Carolina fourth-grade end-of-grade test scores, it maps achievement disparities (higher in central and coastal counties) — a template for combining CAR smoothing with mixture flexibility beyond count-data disease mapping.

Open Questions

• Identifiability with intrinsic CAR: The intrinsic CAR prior ($\rho=1$) is only identified up to its null space through the data; practitioners typically impose a sum-to-zero or corner constraint on $Z$ to resolve the intercept-level non-identifiability.
• Multivariate CAR: Jointly modeling spatially correlated effects for multiple outcomes (e.g., multi-disease mapping) with a block-structured $B$ raises additional rank conditions and partial-identifiability issues. The foundational proper MCAR construction (Gelfand-Vounatsou 2003) moves from the usual improper univariate CAR to multivariate classes that are genuinely proper — introducing a per-component spatial autoregression parameter $\rho$ and a parametric linear transformation of independent building blocks (extending the Mardia 1988 family, and more general than the Kim et al. 2000 "twofold CAR") — precisely so the object is a legitimate second-stage prior whose posterior does not depend on a design rank condition. Later routes build on it: one (Neelon-Gelfand 2014) places CAR priors on the spatial random effects of each component of a multivariate-normal mixture, letting the response distribution's shape vary across regions; another (Liang 2012) abandons the vector-based block-precision construction for an element-based, graphical one: dependence factors into a fixed spatial graph (from geography) and a random non-spatial graph learned from data via G-Wishart priors and reversible-jump MCMC — so the cross-outcome dependence structure is estimated rather than assumed.
• Hyperprior sensitivity: Even when the rank condition passes, near-improper inverse-gamma hyperpriors on $\delta_1$ (with very small shape parameter $a_1$) can produce near-improper posteriors in finite samples.

Related

• Bayesian Hierarchical Model
• Generalized Linear Mixed Model
• Gibbs Sampler
• Tsutakawa (1988) Mixed Model for Analyzing Geographic Variability in Mortality Rates
• Sun-Tsutakawa-Speckman (1999) Posterior Distribution of Hierarchical Models Using CAR(1) Distributions
• Dongchu Sun
• Robert K. Tsutakawa
• Space-Varying Regression Model
• Gamerman-Moreira-Rue (2003)
• Dani Gamerman
• Wakefield (2007) Disease Mapping and Spatial Regression with Count Data
• Jon Wakefield
• Generalized Linear Mixed Model
• Spatial Autocorrelation
• Mixture Model
• Neelon-Gelfand (2014) A Multivariate Spatial Mixture Model for Areal Data
• Brian Neelon
• Alan E. Gelfand
• Gelfand-Vounatsou (2003) Proper Multivariate Conditional Autoregressive Models for Spatial Data Analysis
• Penelope Vounatsou
• Mariella-Tarantino (2012) Spatial Temporal Conditional Auto-Regressive Model
• Leonardo Mariella
• Marco Tarantino
• Liang (2012) Graph-based Multivariate Conditional Autoregressive Models
• Ye Liang
• Cressie-Chan (1989) Spatial Modeling of Regional Variables
• Noel Cressie
• Spatial Autocorrelation