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A Spatial Econometric Analysis of the Calls to the Portuguese National Health Line ^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Spatial Dependence

#### Moran’s I Statistics

#### 2.2. Spatial Models for Count Data—Hierarchical Bayesian Approach

#### 2.2.1. Hierarchical log-Poisson regression model

#### Besag-York-Mollié Model

#### Leroux, Lei and Breslow Model

#### 2.3. Spatial Autoregressive Bayesian Models for Count Data

#### 2.3.1. Bayesian Spatial Lag Model

#### 2.3.2. The Spatial Lag Poisson Model, Classical Perspective

#### 2.3.3. The Spatial Lag Poisson Model, Bayesian Perspective

#### 2.4. Model Selection

#### 2.4.1. Deviance Information Criterion

#### 2.4.2. Watanabe-Akaike Information Criterion

## 3. Results

#### 3.1. The LS24 Data

#### 3.2. Non-Spatial Modelling, the Log-Poisson Regression Model

- Case 1: The average number of years of schooling, the proportion of elderly residents, the unemployment rate, the rurality index, the number of hospitals and health centres per 1000 inhabitants and the proportion of women in each municipality (AIC: 29530);
- Case 2: The monthly average income, the proportion of children, the unemployment rate, the rurality index, the number of hospital and health centres (both per 1000 inhabitants), and the proportion of women in each municipality (AIC: 36980).

#### 3.3. Spatial Correlation

#### 3.4. Spatial Modelling

#### 3.4.1. Spatial Hierarchical Log-Poisson Regression Model

#### Model A: BYM model

#### Model B: Leroux model

#### 3.4.2. The Spatial Lag Poisson Model

#### Model C: Spatial lag Poisson model

#### 3.5. Comparison of Results

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

LS24 | Portuguese national health line |

slm | spatial autoregressive model |

sem | spatial error model |

MCMC | Markov Chain Monte Carlo |

INLA | Integrated Nested Laplace Approximations |

CAR | Conditional autoregressive |

GMRF | Gaussian Markov Random Field |

BYM | Besag-York-Mollié |

slmINLA | spatial lag model |

splmINLA | spatial lag Poisson model |

AIC | Akaike information criteria |

DIC | Deviance information criteria |

WAIC | Watanabe-Akaike information criteria |

TAE | Triage, counseling and routing |

AT | Therapeutic counseling |

LSP | Assistance in Public Health |

IGS | General Health Information |

SCR | Standard Call Rate |

RRMSE | Relative Root Mean Square Error |

MVN | Multivariate Normal |

## Appendix A. The Integrated Nested Laplace Approximation

- i
- Computation of an approximation to the posterior distribution of the hyperparameters $\pi \left(\mathit{\theta}\right|\mathit{y})$ as in (A4),
- ii
- Use again the Laplace approximation to obtain $\pi \left({u}_{i}\right|\mathit{\theta},\mathit{y})$. For example, rewriting the vector of parameters as $u=({u}_{i},{\mathit{u}}_{-i})$,$$\tilde{\pi}\left({u}_{i}\right|\mathit{\theta},\mathit{y})\propto \frac{\pi (\mathit{u},\mathit{\theta}|\mathit{y})}{\tilde{\pi}\left({\mathit{u}}_{-i}\right|{u}_{i},\mathit{\theta},\mathit{y})}{|}_{{\mathit{u}}_{-i}={\mathit{u}}_{-i}^{*}({u}_{i},\mathit{\theta})}$$
- iii
- Using the previous steps and a numerical integration,$$\tilde{\pi}\left({u}_{i}\right|\mathit{y})=\int \tilde{\pi}\left({u}_{i}\right|\mathit{\theta},\mathit{y})\tilde{\pi}\left(\mathit{\theta}\right|\mathit{y})d\mathit{\theta}$$$$\tilde{\pi}\left({u}_{i}\right|\mathit{y})\approx \sum _{m}\tilde{\pi}\left({u}_{i}\right|{\mathit{\theta}}^{m},\mathit{y})\tilde{\pi}\left({\mathit{\theta}}^{m}\right|\mathit{y}){\mathrm{\Delta}}_{m}$$

## Appendix B. R-Code Used in R-INLA to Implement Spatial Lag Poisson Model

#### slmINLA Model: ## Index for the latent model BD3$idx ## Define adjacency using a row-standardised matrix concelhos_listw<-nb2listw(concelhos_nb) W <- as(as_dgRMatrix_listw(concelhos_listw), "CsparseMatrix") ## Model definition log_n.cham_pop.resi/log.cham.SMR f1 <- log.cham.SMR ~Escolaridade + Perc.idosos+TxDesemp+IndRural+ +Hospitais1000+CentrosSaude1000+ PropMulheres f2 <- log.cham.SMR ~Perc.criancas + TxDesemp+Rendim+IndRural + +Hospitais1000+CentrosSaude1000+ PropMulheres #Covariate matrix### mmatrix1 <- model.matrix(f1,BD3) mmatrix2<- model.matrix(f2,BD3) ## Zero-variance for error term zero.variance = list(prec=list(initial = 25, fixed=TRUE)) ## Compute eigenvalues for slm model, used to obtain rho.min and ## rho.max e = eigenw(concelhos_listw) re.idx = which(abs(Im(e)) < 1e-6) rho.max = 1/max(Re(e[re.idx])) rho.min = 1/min(Re(e[re.idx])) rho = mean(c(rho.min, rho.max)) ## Precision matrix for beta coeffients’ prior betaprec <- .0001 Q.beta = Diagonal(n=ncol(mmatrix), betaprec) ## Priors on the hyperparameters hyper = list( prec = list( prior = "loggamma", param = c(0.01, 0.01)), rho = list( initial=0, prior = "logitbeta", param = c(1,1))) Next, there is the R-code for fitting the spatial lag poisson model: ## slpmINLA Model ## Fit model for SCR case1 slmm1 <- inla( n.chamadas ~ -1 + f(idx, model="slm", args.slm=list( rho.min = rho.min, rho.max = rho.max, W=W, X=mmatrix1, Q.beta=Q.beta), hyper=hyper), data=BD3, family="poisson", control.family = list(hyper=zero.variance), control.compute=list(dic=TRUE, cpo=TRUE, waic=TRUE), offset=log(e_st) ) n <- nrow(BD3) slmm1$summary.random$idx[n+1:ncol(mmatrix1),] ## Fit model for SCR case2 slmm2 <- inla( n.chamadas ~ -1 + f(idx, model="slm", args.slm=list( rho.min = rho.min, rho.max = rho.max, W=W, X=mmatrix2, Q.beta=Q.beta), hyper=hyper), data=BD3, family="poisson", control.family = list(hyper=zero.variance), control.compute=list(dic=TRUE, cpo=TRUE, waic=TRUE), offset=log(e_st) ) n <- nrow(BD3) slmm2$summary.random$idx[n+1:ncol(mmatrix2),]

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**Figure 6.**Posterior marginal distribution of the spatial autocorrelation parameter for Model C, case 1 (

**Left**) and case 2 (

**Right**).

**Table 1.**Covariates and their estimated coefficients for the quasi-Poisson log-regression model, case 1, for the LS24 2014 data.

Variable | Id | Coefficients | p-Values |
---|---|---|---|

Average number of years of schooling | ${x}_{1}$ | $0.322$ | <2 $\times {10}^{-16}$ |

Proportion of elderly residents | ${x}_{2}$ | $4.456$ | 9.52 $\times {10}^{-13}$ |

Unemployment rate | ${x}_{3}$ | $-0.743$ | 0.3156 |

Rurality index | ${x}_{4}$ | $-0.741$ | 4.10 $\times {10}^{-6}$ |

Number of hospitals | ${x}_{5}$ | $-3.822$ | 6.68 $\times {10}^{-6}$ |

Number of health centres | ${x}_{6}$ | $-1.289$ | 0.0437 |

Proportion of women | ${x}_{7}$ | $-5.509$ | 0.0661 |

Intercept | $-0.288$ | 0.8390 |

**Table 2.**Covariates and their estimated coefficients for the quasi-Poisson log-regression model, case 2, for the LS24 2014 data.

Variable | Id | Coefficients | p-Values |
---|---|---|---|

The monthly average income | ${x}_{1}$ | $0.001$ | 5.97 $\times {10}^{-16}$ |

Proportion of children | ${x}_{2}$ | $5.727$ | 0.0004 |

Unemployment rate | ${x}_{3}$ | $-1.679$ | 0.0391 |

Rurality index | ${x}_{4}$ | $-0.212$ | 0.1884 |

Number of hospitals | ${x}_{5}$ | $-0.398$ | 0.6504 |

Number of health centres | ${x}_{6}$ | $-0.902$ | 0.1702 |

Proportion of women | ${x}_{7}$ | $11.810$ | 0.0003 |

Intercept | $-7.930$ | 7.64 $\times {10}^{-6}$ |

**Table 3.**Parameter estimates for the Besag, York and Mollié (BYM) hierarchical log-Poisson model, case 1, for the LS24 2014 data.

Variable | Id | Median | 2.5% | 97.5% |
---|---|---|---|---|

Average number of years of schooling | ${x}_{1}$ | $0.1931$ | $0.0062$ | $3.5386$ |

Proportion of elderly residents | ${x}_{2}$ | $0.4840$ | $-1.6883$ | $2.6921$ |

Unemployment rate | ${x}_{3}$ | $1.8680$ | $-1.5338$ | $4.7276$ |

Rurality index | ${x}_{4}$ | $-0.0930$ | $-0.4980$ | $0.3126$ |

Number of hospitals | ${x}_{5}$ | $-0.2549$ | $-1.9709$ | $1.4916$ |

Number of health centres | ${x}_{6}$ | $0.3777$ | $-1.1932$ | $1.8507$ |

Proportion of women | ${x}_{7}$ | $-1.2479$ | $-10.1867$ | $7.7633$ |

Intercept | $-1.3506$ | $-6.4541$ | $3.5386$ | |

${\tau}^{2}$ | $0.2268$ | $0.1440$ | $0.3494$ | |

${\sigma}^{2}$ | $0.0326$ | $0.0140$ | $0.0592$ |

**Table 4.**Parameter estimates for the BYM hierarchical log-Poisson model, case 2, for the LS24 2014 data.

Variable | Id | Median | 2.5% | 97.5% |
---|---|---|---|---|

The monthly average income | ${x}_{1}$ | $0.001$ | $0.0$ | $0.0021$ |

Proportion of children | ${x}_{2}$ | $1.7348$ | $-2.1687$ | $6.3659$ |

Unemployment rate | ${x}_{3}$ | $1.9042$ | $-1.5689$ | $5.4343$ |

Rurality index | ${x}_{4}$ | $-0.1838$ | $-0.5460$ | $0.2105$ |

Number of hospitals | ${x}_{5}$ | $-0.2282$ | $-2.0447$ | $1.3542$ |

Number of health centres | ${x}_{6}$ | $-0.0613$ | $-1.3820$ | $1.3585$ |

Proportion of women | ${x}_{7}$ | $0.6238$ | $-8.9309$ | $9.4824$ |

Intercept | $-1.9574$ | $-7.2782$ | $3.4761$ | |

${\tau}^{2}$ | $0.2443$ | $0.1594$ | $0.3690$ | |

${\sigma}^{2}$ | $0.0313$ | $0.0149$ | $0.0540$ |

**Table 5.**Parameter estimates for the Leroux hierarchical log-Poisson model, case 1, for the LS24 2014 data.

Variable | Id | Median | 2.5% | 97.5% |
---|---|---|---|---|

Average number of years of schooling | ${x}_{1}$ | $0.1897$ | $0.0141$ | $0.3187$ |

Proportion of elderly residents | ${x}_{2}$ | $0.8586$ | $-1.3888$ | $2.8322$ |

Unemployment rate | ${x}_{3}$ | $2.1413$ | $-0.7070$ | $4.8630$ |

Rurality index | ${x}_{4}$ | $-0.1129$ | $-0.4562$ | $0.2337$ |

Number of hospitals | ${x}_{5}$ | $-0.2606$ | $-1.7421$ | $1.1745$ |

Number of health centres | ${x}_{6}$ | $0.3458$ | $-0.8881$ | $1.6717$ |

Proportion of women | ${x}_{7}$ | $-1.9482$ | $-9.8121$ | $6.2431$ |

Intercept | $-1.1342$ | $-5.1422$ | $3.1621$ | |

${\tau}^{2}$ | $0.3492$ | $0.2829$ | $0.4494$ | |

$\rho $ | $0.9059$ | $0.7008$ | $0.9888$ |

**Table 6.**Parameter estimates for the Leroux hierarchical log-Poisson model, case 2, for the LS24 2014 data.

Variable | Id | Median | 2.5% | 97.5% |
---|---|---|---|---|

The monthly average income | ${x}_{1}$ | $0.001$ | $0.0001$ | $0.0019$ |

Proportion of children | ${x}_{2}$ | $1.8345$ | $-1.9861$ | $5.7947$ |

Unemployment rate | ${x}_{3}$ | $1.6751$ | $-1.3984$ | $4.7253$ |

Rurality index | ${x}_{4}$ | $-0.2145$ | $-0.5562$ | $0.0639$ |

Number of hospitals | ${x}_{5}$ | $-0.3568$ | $-1.8623$ | $1.1085$ |

Number of health centres | ${x}_{6}$ | $-0.0069$ | $-1.2701$ | $1.2484$ |

Proportion of women | ${x}_{7}$ | $0.5789$ | $-7.0199$ | $8.6312$ |

Intercept | $-1.9231$ | $-6.5246$ | $2.4100$ | |

${\tau}^{2}$ | $0.3581$ | $0.2911$ | $0.4508$ | |

$\rho $ | $0.8936$ | $0.6879$ | $0.9855$ |

Variable | Id | Median | 2.5% | 97.5% |
---|---|---|---|---|

Average number of years of schooling | ${x}_{1}$ | $0.179$ | $0.121$ | $0.237$ |

Proportion of elderly residents | ${x}_{2}$ | $0.591$ | $-0.288$ | $1.473$ |

Unemployment rate | ${x}_{3}$ | $0.605$ | $-0.851$ | $2.048$ |

Rurality index | ${x}_{4}$ | $-0.005$ | $-0.209$ | $0.197$ |

Number of hospitals | ${x}_{5}$ | $-0.179$ | $-1.300$ | $0.941$ |

Number of health centres | ${x}_{6}$ | $0.173$ | $-0.316$ | $0.660$ |

Proportion of women | ${x}_{7}$ | $-0.919$ | $-4.702$ | $2.848$ |

Intercept | $-1.071$ | $-3.012$ | $0.868$ | |

${\tau}^{2}$ | $0.070$ | $0.568$ | $0.085$ | |

$\rho $ | $0.897$ | $0.857$ | $0.920$ |

Variable | Id | Median | 2.5% | 97.5% |
---|---|---|---|---|

The monthly average income | ${x}_{1}$ | $0.001$ | $0.000$ | $0.001$ |

Proportion of children | ${x}_{2}$ | $0.972$ | $-1.232$ | $3.190$ |

Unemployment rate | ${x}_{3}$ | $0.065$ | $-1.408$ | $1.520$ |

Rurality index | ${x}_{4}$ | $-0.098$ | $-0.291$ | $0.094$ |

Number of hospitals | ${x}_{5}$ | $0.151$ | $-0.989$ | $1.290$ |

Number of health centres | ${x}_{6}$ | $-0.095$ | $-0.560$ | $0.369$ |

Proportion of women | ${x}_{7}$ | $0.307$ | $-3.570$ | $4.182$ |

Intercept | $-1.015$ | $-3.180$ | $1.141$ | |

${\tau}^{2}$ | $0.074$ | $0.061$ | $0.089$ | |

$\rho $ | $0.897$ | $0.861$ | $0.926$ |

**Table 9.**Deviance Information Criterion (DIC) and Watanabe-Akaike Information Criterion (WAIC) measured for the 3 models fitted for case 1.

Model | DIC | pD | WAIC | pW |
---|---|---|---|---|

Baseline Model MCMC | 2816.9 | 287.2 | 2830.8 | 198.5 |

Model A | 2801.1 | 275.2 | 2773.8 | 179.9 |

Model B | 2788.6 | 267.16 | 2744.5 | 157.2 |

Model C | 2778.63 | 261.96 | 2717.6 | 144.9 |

Model | DIC | pD | WAIC | pW |
---|---|---|---|---|

Baseline Model MCMC | 2816.9 | 287.2 | 2830.8 | 198.5 |

Model A | 2800.0 | 275.6 | 2770.7 | 169.5 |

Model B | 2793.2 | 272.4 | 2743.0 | 156.7 |

Model C | 2777.3 | 262.6 | 2714.1 | 143.8 |

Model | RRMSE |
---|---|

Baseline Model | 0.588 |

Model A | 0.029 |

Model B | 0.028 |

Model C | 0.037 |

Model | RRMSE |
---|---|

Baseline Model | 0.469 |

Model A | 0.025 |

Model B | 0.026 |

Model C | 0.034 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Simões, P.; Carvalho, M.L.; Aleixo, S.; Gomes, S.; Natário, I.
A Spatial Econometric Analysis of the Calls to the Portuguese National Health Line . *Econometrics* **2017**, *5*, 24.
https://doi.org/10.3390/econometrics5020024

**AMA Style**

Simões P, Carvalho ML, Aleixo S, Gomes S, Natário I.
A Spatial Econometric Analysis of the Calls to the Portuguese National Health Line . *Econometrics*. 2017; 5(2):24.
https://doi.org/10.3390/econometrics5020024

**Chicago/Turabian Style**

Simões, Paula, M. Lucília Carvalho, Sandra Aleixo, Sérgio Gomes, and Isabel Natário.
2017. "A Spatial Econometric Analysis of the Calls to the Portuguese National Health Line " *Econometrics* 5, no. 2: 24.
https://doi.org/10.3390/econometrics5020024