# Disease Mapping and Regression with Count Data in the Presence of Overdispersion and Spatial Autocorrelation: A Bayesian Model Averaging Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Esophageal Cancer Incidence Data in the Caspian Region of Iran

_{1}, …, X

_{k}. A general form for this type of model for J geographically-defined units (areas) is given by:

_{j}) = logλ

_{j}= X

_{j}β

^{T}+ θ

_{j}+ logE

_{j}j = 1, …, J

_{j}is the count for area j and E

_{j}denotes an “expected” count in area j that is assumed known, X

_{j}= (1, X

_{j}

_{1}, …, X

_{jk}) is a 1 × (k + 1) vector of area-level risk factors, β = (β

_{0}, β

_{1}, …, β

_{k}) is a 1× (k + 1) vector of regression parameters and θ

_{j}represents a residual with no spatial structure (so that θ

_{i}and θ

_{j}are independent for i ≠ j).

**Figure 1.**(

**a**) Geographic boundaries of wards (bold polygons), cities (grey polygons) and rural agglomerations within wards, in Mazandaran and Golestan provinces; (

**b**) Observed spatial pattern; and (

**c**) model adjusted SIR.

#### 2.2. Model & Data Structure

_{j}, and population counts, N

_{j}in region j. The expected count when adjusting for the age structure of an agglomeration, E

_{j}, was obtained by age-standardisation. Then, using the theoretical relationship (SIR = ).

#### 2.3. Distributions for Disease Counts

_{j}ǀ(E

_{j}, λ

_{j}) ~ Poisson(λ

_{j})

_{j}) = V(Y

_{j}) = λ

_{j}.

_{j}, specifically:

_{j}ǀ(ε

_{j}, E

_{j}, λ

_{j}) ~ Poisson(ε

_{j}λ

_{j}), ε

_{j}ǀϑ~ gamma(ϑ, ϑ),

_{j}= 0, 1, 2, 3, …, where ϑ > 0. The resulting probability distribution function marginal to ε

_{j}is:

_{j}= 0, 1, 2, 3, …, with E(Y

_{j}) = λ

_{j}and V(Y

_{j}) = λ

_{j}+ (λ

_{j})

^{2}/ϑ.

_{j}converges to a Poisson random variable.

_{j}= 0, 1, 2, 3, … and has E(Y

_{j}) = λ

_{j}and V(Y

_{j}) = λ

_{j}(1 − ω)

^{−2}. For ω = 0, the generalized Poisson reduces to the Poisson distribution with mean λ

_{j}.

_{m}, m), and the corresponding prior distribution f(γ

_{m}ǀm), where γ

_{m}is a parameter vector under model m and Y is the outcome variable vector. We use the following hierarchical structure on model parameters:

^{k}. The usual choice for the prior on model m is the uniform distribution over the covariate parameter space M = {β

_{1}, …, β

_{k}}. We used this uniform distribution because the prior can be thought of as noninformative in the sense of favouring all candidate models equally within the same probability model class.

#### 2.4. Hierarchical Models for Relative Risks

_{j}}

_{j}

_{ = 1, …,j}follows a spatial model that incorporates assumptions about the spatial relationships between areas. We then extend (1) as:

_{j}) = logλ

_{j}= X

_{j}β

^{T}+ θ

_{j}+ ϕ

_{j}+ log E

_{j}j = 1, …, j

_{j}, represents a residual with spatial structure with ϕ

_{i}and ϕ

_{j}, i ≠ j, modelled to have positive spatial dependence. Two approaches are used for modelling the J-dimensional random variable ϕ: distance-based and neighbourhood-based spatial correlation structures.

_{i}and Σ is a J × J positive definite matrix. If d

_{ij}denotes the distance between centroids of agglomerations i and j, then we specify:

_{ij}= f (d

_{ij}; v,k)

_{ij}; v, k) = exp[(−vd

_{ij})

^{k}]. In this specification ν > 0 controls the rate of decrease of correlation with distance, with large values representing rapid decay, and τ is a scalar parameter representing the overall precision parameter. The parameter κ ϵ (0,2] controls the amount by which spatial variations in the data are smoothed. Large values of κ lead to greater smoothing, with κ = 2 corresponding to the Gaussian correlation function [15]. The distance-based parameters are jointly referred to as .

_{i}~N(0, ) , describing the spatial variation in the heterogeneity component so that geographically close areas tend to present similar risks. One way of expressing this spatial structure is via Markov random fields models where the distribution of each ϕ

_{i}given all the other elements {ϕ

_{1}, …, ϕ

_{i}

_{ – 1,}ϕ

_{i}

_{ + 1}, …, ϕ

_{J}} depends only on its neighbourhood [17]. A commonly used form for the conditional distribution of each ϕ

_{i}is the Gaussian:

_{i}is defined as a weighted average of the other ϕ

_{j}, j ≠ i, and the weights π

_{ij}define the relationship between area i and its neighbours. The precision parameter σ

_{ϕ}controls the amount of variability for the random effect.

_{ij}in Equation (8) are constants and specified as π

_{ij}= 1 if i and j are adjacent and π

_{ij}= 0 otherwise. In that case, the conditional prior mean of ϕ

_{i}is given by the arithmetic average of the spatial effects from its neighbours and the conditional prior variance is proportional to the number of neighbours.

#### 2.5. Specification of Priors

_{θ}, σ

_{ϕ}and δ are defined below.

#### 2.6. Specification of Hyperpriors

_{θ}, σ

_{ϕ}and δ. The estimation of relative risks can be highly dependent on the choice of prior parameters [3] and within a class of Gamma priors, the Gamma(0.5, 0.0005) distribution has been suggested as a sensible choice [2] and was adopted here for the parameters σ

_{θ}and σ

_{ϕ}. For the δ parameters a Gamma(0.001, 0.001) prior was used for τ and uniform distributions Unif(0.05, 1.95) and Unif(0.05, 20) were used for κ and ν respectively.

#### 2.7. Gibbs Variable Selection, GVS

^{κ}, where ψ is a set of binary indicator variables ψ

_{g}(g = 1, …, k), where ψ

_{g}= 1 or 0 represents respectively the presence or absence of covariate g in the model, and α denotes other structural properties of the model. For the generalised linear models in this study, α describes the distribution, link function, variance function and (un)structured terms, and the linear predictor may be written as:

_{ψ}and β

_{\ψ}corresponding to those components of β that are included ψ

_{g}= 1 or not included ψ

_{g}= 0 in the model. Then, the prior f (βǀψ) may be partitioned into a “model” prior f (β

_{ψ}ǀψ) and a “pseudo” prior f (β

_{\ψ}ǀ β

_{ψ}, ψ) [18]. The full posterior distributions for the model parameters are given by:

_{ψ}ǀβ

_{\ψ}, ψ, y)~f (yǀβ,ψ) f (β

_{ψ}ǀψ) f (β

_{\ψ}ǀβ

_{\ψ}, ψ)

_{\ψ}ǀβ

_{\ψ}, ψ, y)~f (β

_{\ψ}ǀβ

_{\ψ}, ψ)

_{ψ}and the inactive parameters β

_{\ψ}are a priori independent given ψ. This assumption implies that f (β

_{ψ}ǀβ

_{\ψ}, ψ, y)~f (yǀβ,ψ) f (β

_{ψ}ǀψ) and f (β

_{\ψ}ǀβ

_{ψ}, ψ, y) ∝ f(β

_{\ψ}ǀψ).

- (1).
- Sample the parameters included in the model from the posterior:f (β
_{ψ}ǀβ_{\ψ}, ψ, y) ∝ (yǀβ,ψ) f (β_{ψ}ǀψ) - (2).
- Sample the parameters excluded from the model from the pseudoprior:f (β
_{\ψ}ǀβ_{ψ}, ψ, y) ∝ f(β_{\ψ}ǀψ) - (3).
- Sample each variable indicator j from a Bernoulli distribution with success probability ; where O
_{g}is given by:

_{\}

_{g}denotes all terms of ψ except ψ

_{g}.

_{g}for each model ψ. Then, each prior for β

_{g}ǀ ψ consists of a mixture of true prior f (β

_{g}ǀψ

_{g}= 1, ψ

_{\}

_{g}) for the parameter and a pseudoprior f (β

_{g}ǀ ψ

_{g}= 0, ψ

_{\}

_{g}) As a result:

_{g}ǀ ψ

_{g}) = ψ

_{g}f (β

_{g}ǀ ψ

_{g}= 1) + (1− ψ

_{g})f (β

_{g}ǀψ

_{g}= 0)

_{g}s resulting in:

_{g}ǀψ

_{g}= 1)~ N(0, ∑

_{g})

_{g}ǀψ

_{g}= 0)~ N(µ

_{G}, S

_{G})

_{G}, S

_{G}are the mean and variance respectively in the corresponding pseudoprior distributions and Ʃ

_{g}is the prior variance when covariate g is included in the model.

_{g}ǀψ

_{g}) = ψ

_{g}N(0, ∑

_{g}) + (1 − ψ

_{g})N(µ

_{G}, S

_{f})

_{g}ǀψ

_{g}= 0) does not affect the posterior distribution of model coefficients.

_{g}is f(ψ

_{g}) = Bernoulli (0.5) [20]. The Gibbs sampler was begun with all ψ

_{g}= 1, which corresponds to starting with the full model.

_{g}. Zellner’s g prior framework was used to define prior variance structure for Ʃ [21]. The choices µ

_{G}= 0 and S

_{g}= with p = 10 were made as they have also been shown to be adequate [18]. The pseudoprior parameters µ

_{G}and S

_{g}are only relevant to the behaviour of the MCMC chain and do not affect the posterior distribution [20].

^{K}competing models are considered M = {m

_{1}, m

_{2}, m

_{3}, …, m

_{2}k}, and the posterior probability of model ma ϵ M is defined as:

_{g}≠ 0ǀy), g = 1, …, k, by summing the posterior model probabilities across those regressors that are included in the model.

#### 2.7.1. Fully Bayesian Estimation

#### 2.7.2. Comparison of Model Performance

^{2}[26], deviance statistic [27], Moran scatter plot [28] and absolute deviance residuals versus fitted values [29] were used for estimating the goodness of fit (GOF) and prediction performance of the competing models. Posterior mean of λ

_{j}were used as the plug-in estimate of to calculate all the goodness of fit measures discussed in this paper.

^{2}is calculated for model comparison and takes values between zero and one. It is based on , however since R

^{2}increases as more parameters are added to a model regardless of their contribution pseudo R

^{2}is defined as Pseudo where d.f. for degrees of freedom equal J minus the effective number of free parameters [26].

## 3. Results

#### 3.1. Automatic Bayesian Model Averaging

**Table 1.**Posterior summaries for Poisson, G-Poisson and Negative Binomial (NB) regression models each with the spatial correlation structures: “IN” independence, “N” neighbourhood-based, “D” distance-based.

Model | Posterior median of regression coefficient β_{1}, (95% credible interval) | Random components | |||||||
---|---|---|---|---|---|---|---|---|---|

Distribution | Spatial structure | income | urbanisation | literacy | unrestricted food choice | restricted food choice | σ_{θ} | σ_{ϕ} | |

Poisson | IN | −0.22, (−0.60, −0.03) | −0.36, (−0.42, −0.15) | −0.16, (−0.26, −0.08) | 0.12, (0.08, 0.16) | −0.32, (−0.41, −0.09) | 0.78 | - | - |

Poisson | IN + N | −0.19, (−0.68, 0.02) | −0.36, (−0.51, −0.16) | −0.15, (−0.22, −0.05) | 0.07, (−0.04, 0.16) | −0.24, (−0.38, −0.06) | 0.35 | 0.73 | - |

Poisson | IN + D | −0.18, (−0.69, 0.07) | −0.35, (−0.51, 0.03) | −0.15, (−0.22, 0.02) | 0.07, (−0.03, 0.16) | −0.23, (−0.38, 0.04) | 0.13 | - | |

G-Poisson | IN | −0.24, (−0.61, −0.09) | −0.38, (−0.51, −0.09) | −0.18, (−0.22, −0.05) | 0.11, (0.09, 0.16) | −0.28, (−0.44, −0.11) | 0.56 | - | - |

G-Poisson | IN + N | −0.19, (−0.69, −0.04) | −0.35, (−0.51, −0.03) | −0.12, (−0.21, −0.03) | 0.07, (−0.02, 0.16) | 0.23, (−0.38, −0.04) | 0.12 | 0.66 | - |

G-Poisson | IN + D | −0.19, (−0.68, 0.01) | −0.36, (−0.51, −0.07) | −0.15, (−0.22, 0.06) | 0.07, (−0.02, 0.16) | −0.24, (−0.39, −0.07) | 0.17 | - | |

NB | IN | −0.23, (−0.59, −0.10) | −0.39, (−0.58, 0.09) | −0.17, (−0.27, −0.7) | 0.17, (0.03, 0.16) | −0.31, (−0.48, −-0.12) | 0.36 | - | - |

NB | IN + N | −0.17, (−0.68−0.06) | −0.35, (−0.51, 0.11) | −0.11, (−0.21, 0.01) | 0.07, (−0.04, 0.16) | −0.23, (−0.38, 0.02) | 0.12 | 0.74 | - |

NB | IN + D | −0.20, (−0.68, 0.10) | −0.35, (−0.51, 0.08) | −0.15, (−0.22, 0.08) | 0.07, (−0.01, 0.16) | −0.24, (−0.38, 0.09) | 0.11 | - |

**Table 2.**The top two candidate covariate models (covariate subsets) based on their posterior probabilities: “IN” stands for independence, “N” stands for neighbourhood-based and “D” stands for distance-based structure.

Model distribution | Spatial structure | Subset | Covariates * | f(m|y) ** |
---|---|---|---|---|

Poisson | IN | 1 | income, restricted food choice | 0.37 |

Poisson | IN | 2 | income, restricted food choice, urbanisation | 0.12 |

Poisson | IN + N | 1 | income, restricted food choice, urbanisation | 0.31 |

Poisson | IN + N | 2 | income, restricted food choice | 0.15 |

Poisson | IN + D | 1 | urbanisation | 0.25 |

Poisson | IN + D | 2 | income | 0.20 |

G-Poisson | IN + D | 2 | income, urbanisation | 0.18 |

G-Poisson | IN | 1 | income, restricted food choice | 0.28 |

G-Poisson | IN | 2 | income, restricted food choice, urbanisation | 0.17 |

G-Poisson | IN + N | 1 | income, urbanisation, restricted food choice | 0.28 |

G-Poisson | IN + N | 2 | urbanisation, restricted food choice | 0.13 |

G-Poisson | IN + D | 1 | restricted food choice | 0.19 |

G-Poisson | IN + D | 2 | income, urbanisation | 0.18 |

NB | IN | 1 | income, restricted food choice | 0.21 |

NB | IN | 2 | restricted food choice, urbanisation | 0.11 |

NB | IN + N | 1 | income | 0.26 |

NB | IN + N | 2 | income, restricted food choice | 0.13 |

NB | IN + D | 1 | income | 0.18 |

NB | IN + D | 2 | restricted food choice | 0.12 |

**Table 3.**Marginal posterior inclusion probability for the top candidate models (covariate subsets): “IN” stands for independence, “N” stands for neighbourhood-based and “D” stands for distance-based structure.

Model | Spatial structure | Subset | Covariates | f(ψ_{g} = 1ǀy) * |
---|---|---|---|---|

distribution | ||||

Poisson | IN | 1 | income | 0.67 |

restricted food choice | 0.42 | |||

Poisson | IN + N | 1 | income | 0.61 |

restricted food choice | 0.48 | |||

urbanisation | 0.37 | |||

Poisson | IN + D | 1 | urbanisation | 0.40 |

G-Poisson | IN | 1 | income | 0.57 |

restricted food choice | 0.33 | |||

G-Poisson | IN + N | 1 | income | 0.59 |

urbanisation | 0.43 | |||

restricted food choice | 0.25 | |||

G-Poisson | IN + D | 1 | restricted food choice | 0.22 |

NB | IN | 1 | income | 0.64 |

restricted food choice | 0.42 | |||

NB | IN + N | 1 | income | 0.47 |

NB | IN + D | 1 | income | 0.55 |

#### 3.2. Prediction Performance

^{2}suggested that approximately one third of the total variation in esophageal cancer counts was explained by each of the subset 1 models with slight improvement for joint independence and spatial models. Figure 2 shows the scatterplot of the observed counts against the model predicted counts; consistent with the pseudo R

^{2}values the scatterplots show better model fit for spatial models.

**Table 4.**Goodness of fit measures: “IN” stands for independence, “N” stands for neighbourhood-based and “D” stands for distance-based structure.

Model | MAD ^{a} | MSPE ^{b} | Pseudo-R^{2} | Deviance index ^{c} | |
---|---|---|---|---|---|

Distribution | Spatial structure | ||||

Poisson | IN | 4.4 | 30.3 | 0.24 | 3.1 |

Poisson | IN + N | 3.7 | 16.6 | 0.32 | 2.8 |

Poisson | IN + D | 2.6 | 13.8 | 0.28 | 2.9 |

G-Poisson | IN | 3.2 | 14.9 | 0.30 | 2.6 |

G-Poisson | IN + N | 2.1 | 10.1 | 0.35 | 1.6 |

G-Poisson | IN + D | 2.3 | 11.6 | 0.33 | 1.7 |

NB | IN | 3.4 | 15.8 | 0.30 | 2.4 |

NB | IN + N | 2.2 | 10.3 | 0.33 | 1.7 |

NB | IN + D | 2.3 | 13.0 | 0.35 | 1.4 |

^{a}Mean absolute deviance;

^{b}Mean-squared predictive error;

^{c}

**Figure 2.**Scatterplots of observed counts (vertical axis) against model predicted counts (horizontal axis) from different models.

#### 3.3. Assessing Overdispersion

#### 3.4. The Moran Scatterplots

**Figure 4.**Moran scatter plot of the residuals from competing models: standardised Pearson residuals against spatially-lagged standardised Pearson residuals.

## 4. Discussion

_{g}~ Bernoulli(0.5) for three reasons: First, our set of covariates was small (k = 5) and it was very unlikely that this choice of prior affects BMA. Second, to minimise any possible tendency towards overparameterised models we implemented a two stage modelling strategy and eliminated covariates with small inclusion probability at the first stage. Third, MCMC computations for fully Bayesian models potentially impose high computational costs. By choosing conventional empirical Bayesian method we aimed to retain useful features of Bayesian variable selection in a pragmatic way.

## 5. Conclusions

## Conflicts of Interest

## Acknowledgements

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Mohebbi, M.; Wolfe, R.; Forbes, A.
Disease Mapping and Regression with Count Data in the Presence of Overdispersion and Spatial Autocorrelation: A Bayesian Model Averaging Approach. *Int. J. Environ. Res. Public Health* **2014**, *11*, 883-902.
https://doi.org/10.3390/ijerph110100883

**AMA Style**

Mohebbi M, Wolfe R, Forbes A.
Disease Mapping and Regression with Count Data in the Presence of Overdispersion and Spatial Autocorrelation: A Bayesian Model Averaging Approach. *International Journal of Environmental Research and Public Health*. 2014; 11(1):883-902.
https://doi.org/10.3390/ijerph110100883

**Chicago/Turabian Style**

Mohebbi, Mohammadreza, Rory Wolfe, and Andrew Forbes.
2014. "Disease Mapping and Regression with Count Data in the Presence of Overdispersion and Spatial Autocorrelation: A Bayesian Model Averaging Approach" *International Journal of Environmental Research and Public Health* 11, no. 1: 883-902.
https://doi.org/10.3390/ijerph110100883