A smoothed ANOVA model for multivariate ecological regression

Abstract

Smoothed analysis of variance (SANOVA) has recently been proposed for carrying out disease mapping. The main advantage of this approach is its conceptual simplicity and ease of interpretation. Moreover, it allows us to fix the combination of diseases of particular interest in advance and to make specific inferences about them. In this paper we propose a reformulation of SANOVA in the context of ecological regression studies. This proposal considers the introduction in a non-parametric way of one (or several) covariate(s) into the model, explaining some pre-specified combinations of the outcome variables. In addition, random effects are also incorporated in order to model geographical variation in the combinations of outcome variables not explained by the covariate. Lastly, the model permits the decomposition of the variance in the set of outcome variables into different orthogonal components, quantifying the contribution of every one of them. The proposed model is applied to the geographical analysis of mortality due to malignant stomach neoplasm among women resident in the city of Barcelona (Spain). The available outcome variables are deaths grouped into two time periods, and a socioeconomic deprivation index is included as a covariate. The model has been implemented through INLA, a novel inference tool for Bayesian statistics.

Appendix

In what follows we explain briefly the general ideas involved in the Integrated nested Laplace approximations (INLA) inference method, the approach used in this paper to approximate the marginal posterior density of the parameters in our multivariate ecological regression model.

Consider a three-stage Bayesian hierarchical model based on an observation model $\pi \left( {\varvec{y}|\varvec{x}} \right) = \mathop \prod_{i} \pi \left( {y_{i} |x_{i} } \right)$, a parameter model $\pi \left( {\varvec{x}|\varvec{\theta}} \right)$, and a hyperprior $\pi \left(\varvec{\theta}\right)$. Here $y = \left( {y_{1} , \ldots ,y_{n} } \right)$ denotes the observed data, x are unknown parameters which typically follow a Gaussian Markov random field (GMRF) and θ are unknown hyperparameters. The marginal posterior density of x _i can be written as

$$ \pi \left( {x_{i} |\varvec{y}} \right) = \mathop \int \limits_{\varvec{\theta}}^{{}} \pi \left( {x_{i} |\varvec{\theta},\varvec{y}} \right)\pi \left( {\varvec{\theta}|\varvec{y}} \right)d\varvec{\theta} $$

and can be approximated by the finite sum

$$ \tilde{\pi }\left( {x_{i} |{\mathbf{y}}} \right) = \mathop \sum \limits_{k} \tilde{\pi }\left( {x_{i} |\varvec{\theta}_{k} ,y} \right)\tilde{\pi }\left( {\varvec{\theta}_{k} |{\mathbf{y}}} \right)\Updelta_{k} $$

Integrated nested Laplace approximations (INLA) approximates these marginal posterior densities using an approximation $\tilde{\pi }\left( {x_{i} |\varvec{\theta},\varvec{y}} \right)$ of $\pi \left( {x_{i} |\varvec{\theta},\varvec{y}} \right)$ and an additional approximation $\tilde{\pi }\left( {\varvec{\theta}|\varvec{y}} \right)$ of the marginal posterior density of the hyperparameters $\pi \left( {\varvec{\theta}|\varvec{y}} \right)$. $\Updelta_{k}$ are chosen appropriately.

To approximate $\pi \left( {\varvec{\theta}|\varvec{y}} \right)$ we consider:

$$ \pi \left( {\user2{x},\varvec{\theta},\user2{y}} \right) =\varvec{\pi} \left( {\user2{x}|\varvec{\theta},\user2{y}} \right) \times \pi \left( {\varvec{\theta}|\user2{y}} \right) \times \pi \left( \user2{y} \right),$$

therefore for any $x$

$$ \pi \left( {\varvec{\theta}|\varvec{y}} \right) \propto \frac{{\pi \left( {\varvec{x},\varvec{\theta},\varvec{y}} \right)}}{{\pi \left( {\varvec{x}|\varvec{\theta},\varvec{y}} \right)}} $$

INLA approximates $\pi \left( {\varvec{\theta}|\varvec{y}} \right)$ using a Laplace approximation (Tierney and Kadane 1986):

$$ \tilde{\pi }\left( {\varvec{\theta}|\varvec{y}} \right)\,\propto\,\frac{{\tilde{\pi }\left( {\varvec{x},\varvec{\theta},\varvec{y}} \right)}}{{\tilde{\pi }_{G} \left( {\varvec{x}|\varvec{\theta},\varvec{y}} \right)}}|_{{\varvec{x} = \varvec{x}^{*} \left(\varvec{\theta}\right)}} $$

where the denominator $\tilde{\pi }_{G} \left( {\varvec{x}|\varvec{\theta},\varvec{y}} \right)$ is the Gaussian approximation of $\pi \left( {\varvec{x}|\varvec{\theta},\varvec{y}} \right)$ and $\varvec{x}^{ * } \left(\varvec{\theta}\right)$ is the mode of the full conditional $\pi \left( {\varvec{x}|\varvec{\theta},\varvec{y}} \right)$ (Rue and Held 2005; Rue et al. 2009). Posterior marginals for the hyperparameters $\tilde{\pi }\left( {\varvec{\theta}|\varvec{y}} \right)$ could be obtained using two strategies. The first, more accurate but also time consuming, consists of defining a grid of points covering the area where most of the mass of $\tilde{\pi }\left( {\varvec{\theta}|\varvec{y}} \right)$ is located (GRID strategy). The second one, named central composite design (CCD strategy), consists in laying out a small amount of “points” in a m-dimensional space in order to estimate the curvature of $\tilde{\pi }\left( {\varvec{\theta}|\varvec{y}} \right)$ (Rue et al. 2009). In this paper the CCD strategy has been used as it needs much less computational time and the differences between the CCD and GRID strategies are expected to be minor. Moreover, the CCD strategy is recommended for problems with high dimensionality of the hyperparameter vector θ.

Secondly, in order to approximate $\tilde{\pi }\left( {x_{i} |\varvec{\theta},\varvec{y}} \right)$, a Gaussian approximation $\tilde{\pi }_{G} \left( {x_{i} |\varvec{\theta},\varvec{y}} \right) = N\left( {x_{i} ;\mu_{i} \left(\varvec{\theta}\right)\sigma_{i}^{2} \left(\varvec{\theta}\right)} \right)$ could be used. However, the approximation can be improved using Laplace approximation or a simplified Laplace approximation based on the skew-normal distribution (Rue et al. 2009). In this paper the Laplace approximation has been used (Tierney and Kadane 1986). This is the most time consuming approximation but on the other hand is the most accurate.