Spatial Confounding Beyond Generalized Linear Mixed Models

---
class: title-page

<div>
 <h3 style="text-align: justify; padding-top:50px">Spatial Confounding Beyond Generalized Linear Mixed Models</h3>
 <h4 style="text-align: right;">Extension to Shared Components and Spatial Frailty Models</h4>
<div>

<div style="padding-top:130px; color:#858585">
 <div style="float:left">
 <h5>
 Douglas R. Mesquita Azevedo 
 Advisor <- Marcos Oliveira Prates 
 Co-advisor <- Dipankar Bandyopadhyay
 
 </h5>
 </div>
 <div style="float:right">
 <img src="img/logo_ufmg.png" alt="ufmg" height="100"/>
 <img src="img/logo_vcu.png" alt="ufmg" height="100"/>
 <img src="img/logo_capes.jpeg" alt="capes" height="100"/>
 </div>
</div>

<div style="align:center; padding-top:135px; color:#858585">
 <h5 style="text-align:center"> 
 September 2021
 
 24º SINAPE
 </h5>
</div>

---

## Summary

+ Motivation
+ Spatial confounding: A brief summary
+ Shared component model
    + Method
    + Simulation
+ Spatial frailty model
    + Method
    + Simulation
    + Application
+ RASCO: An R package to Alleviate the Spatial Confounding
+ Conclusion

---

## Motivation

The spatial confounding happens when .enfase[fixed] and .enfase[latent effects] carry .enfase[similar information] to the model

Example: Time until death by .enfase[lung and bronchus cancer] in California

---

<div class="my-section">
 <h2>
 Spatial Confounding
 </h2>
</div>

---
class: slide-page

## ICAR model

+ Undirected graph

+ Let `$\psi$` be a vector of random variables

`$$\displaystyle \small{\psi_i|\psi_{-i} \sim \text{Normal}\Bigg(\sum_{j \sim i}\frac{\psi_j}{w_{i+}}, \frac{\sigma^2}{w_{i+}}\Bigg) ~ \forall ~ i \in \{1, \ldots, n\}}$$`

+ `$j\sim i$`: regions `$j$` and `$i$` are neighboors
+ `$w_{i+}$`: number of neighboors of region `$i$`

---
class: slide-page

## Reich, Hodges and Zadnik (2006)

+ Reich et al. (2006) showed that

.mathbox[
  	`$$E(\beta|\tau_{\epsilon}, \tau_{\psi}, y) = \displaystyle (X'X)^{-1}X'(y - \hat{\psi}) = \beta_{ols} - (X'X)^{-1}X'\hat{\psi}$$`
  	`$$Var^{-1}(\beta|\tau_{\epsilon}, \tau_{\psi}, y) = \displaystyle \tau_{\epsilon}(X'X) - X' Var(\psi|\beta, \tau_{\epsilon}, \tau_{\psi}, y)X$$`
]

where

+ `$\hat{\psi} = E(\psi|\tau_{\epsilon}, \tau_{\psi}, y)$`

We can notice that
+ `$E(\beta|\tau_{\epsilon}, \tau_{\psi}, y)$` is the OLS estimator minus a `$\psi$`-related factor 
+ `$Var(\beta|\tau_{\epsilon}, \tau_{\psi}, y)$` always increase

---
class: slide-page

## Reich, Hodges and Zadnik (2006)

+ Original model

.mathbox[
  	`$$\displaystyle y|\beta, \psi, \tau_{\epsilon} \sim N(X\beta + P\psi + P^c\psi, \tau_{\epsilon}I_n)$$`
]

+ Restricted model

.mathbox[
  	`$$\displaystyle y|\beta, \psi, \tau_{\epsilon} \sim N(X\beta + P^c\psi, \tau_{\epsilon}I_n)$$`
]

+ `$P = X(X'X)^{-1}X'$` is the .enfase[projection matrix] onto space of `$X$`
+ `$P^c = (I-X(X'X)^{-1}X')$` is the .enfase[projection matrix] onto orthogonal space of `$X$`
+ `$P\psi + P^c\psi = P\psi + (I - P)\psi = \psi$`

---
class: slide-page

<h2> Prates, Assunção, Rodrigues (2019)</h2>

+ .enfase[SP]atial .enfase[O]rthogonal .enfase[C]entroid .enfase["K"]orrection - SPOCK (Prates et al. 2018)

+ Alleviating spatial confounding for areal data problems by displacing the geographical centroids

---
class: slide-page

<h2> Prates, Assunção, Rodrigues (2019)</h2>

+ Let `$c_i = [c_{1i}, c_{2i}] ~ \forall i \in [1, ..., n]$` 
+ We want a .enfase[new set of centroids] `$c^{\ast} = P^c\times[c_1, c_2]$`
+ We want to find `$W^{\ast}$`: an adjancency matrix .enfase["free" of spatial confounding]

To create `$W^{\ast}$` we can use, for example, the .enfase[k-nearest neighbors algorithm]

---
class: slide-page

## Comments

+ There are other approaches for spatial confounding:
 + Moran operator: Hughes and Haran (2013)
 + Lasso regression: Hefley et al., 2017,
 + Structural equation models: Thaden and Kneib, 2018
 + Causal inference: Osama, Zachariah and Schön, 2019
+ Most works are focused on .enfase[GLMM models]
+ There is .error[no software available] to unify approaches

---

<div class="my-section">
 <h2>
 Contributions
 </h2>
</div>

---
class: slide-page

## Our contributions

.mathbox[
<center>
 <h4 style="color:#4a4545; margin-top:0px; margin-bottom:0px">What if we have .enfase[more than one spatial effect] or .enfase[more sample units than areas]?</h4>
</center>
]

+ Restricted shared component model
 + Alleviating the spatial confounding in the presence of multiple spatial effects
 + The correction is made before the analysis is performed
+ Restricted spatial frailty model
 + Alleviating the spatial confounding when the support of fixed and latent effecs differ
 + Samples from restricted and unrestricted models concurrently
 + Efficient method to have sample from both models by using the reduction operator
+ _RASCO_: An R Package to Alleviate Spatial Confounding
 + GLMM, shared component and spatial frailty models
 + RHZ, HH and SPOCK besides our contributions

---

<div class="my-section">
 <h2>
 Restricted shared component model
 </h2>
</div>

---
class: slide-page

## Shared component model

+ Knorr‐Held, L., & Best, N. G. (2001)
+ Sometimes we have interest in .enfase[more than one outcome]
+ Univariate models may .enfase[not] be .enfase[realistic]

.mathbox[
  `$$Y_{id} \sim \text{Poisson}(E_{id}\theta_{id}) ~ \forall ~ i \in \{1, \ldots, n\}, ~ d \in \{1, 2\}$$`
  `$$\log(\theta_{i1}) = X_{i1}\beta_1 + \delta\psi_i + \phi_{i1}$$`
  `$$\log(\theta_{i2}) = X_{i2}\beta_2 + \frac{1}{\delta}\psi_i + \phi_{i2}$$`
  `$$\psi_i \sim \text{ICAR}(W, \tau_{\psi}Q); ~ \phi_1 \sim \text{ICAR}(W, \tau_{\phi_1}Q); ~ \phi_2 \sim \text{ICAR}(W, \tau_{\phi_2}Q)$$`
]

where `$\delta$` is a scale parameter to allow different levels of dependence on the shared component

---
class: slide-page

## Restricted shared component model

+ For the shared component effect:
.mathbox[
    + `$X_{\psi_k}$`: union of `$X_d$` columns for diseases `$d = 1, \ldots, D$` related to shared component `$k, k = 1, \ldots, K$`
    + `$P_{\psi_k}$`: `$X_{\psi_k}(X'_{\psi_k}X_{\psi_k})^{-1}X'_{\psi_k}$`
    + `$c_{\psi_k}$`: `$P^c_{\psi_k}c$`, where `$c$` is the original set of centroids
    + `$W_{\psi_k}$`: neighborhood structure based on `$c_{\psi_k}$` (KNN)
]
+ For the specific spatial effects:
.mathbox[
    + `$X_{\phi_d}$`: `$X_d$` covariates for diseases `$d = 1, \ldots, D$` 
    + `$P_{\phi_d}$`: `$X_{\phi_d}(X'_{\phi_d}X_{\phi_d})^{-1}X'_{\phi_d}$`
    + `$c_{\phi_d}$`: `$P^c_{\phi_d}c$`, where `$c$` is the original set of centroids
    + `$W_{\phi_d}$`: neighborhood structure based on `$c_{\phi_d}$` (KNN)
]

---
class: slide-page

## RSCM - simulation

+ Case `$D = 2$`, `$X_1$` is a two column matrix as well as `$X_2$`. `$X_{12}$` and `$X_{22}$` are:
 + S1: Random;
 + S2: `$X_{12}$` is set of latitudes; S3: `$X_{22}$` is set of latitudes
 + S4: `$X_{12}$` and `$X_{22}$` are the set of latitudes
+ Parameters
 + `$\beta_1 = [0.5, -0.5, -0.2]$`, `$\beta_2 = [0.1, -0.8, -0.4]$`
 + `$\tau_{\psi} = 1$`, `$\tau_{\phi_1} = 10$`, `$\tau_{\phi_2} = 10$`
 + `$\delta = \{1.00, 1.50, 1.75\}$`
+ Priors
 + `$\beta_{dj} \sim \text{Normal}(0, 0.001), ~ d = 1, 2; ~ j = 0, 1, 2,$`
 + `$\log(\gamma) \sim \text{Normal}(0, 0.1) ~~ (\delta = \sqrt{\gamma})$`
 + `$\tau_{\psi} \sim \Gamma(0.5, 0.05), \tau_{\phi_1} \sim \Gamma(0.5, 0.05), \tau_{\phi_2} \sim \Gamma(0.5, 0.05)$`

---
class: slide-page

## RSCM - simulation

---

<div class="my-section">
 <h2>
 Restricted spatial frailty model
 </h2>
</div>

---
class: slide-page

## Proportional hazards models

+ Interest: time until an event
 + Exponential, log-normal, Weibull, ...
+ `$f_{\theta}(t) = h_{\theta}(t)S_{\theta}(t)$`
 + `$f_{\theta}(t)$`: probability density function
 + `$h_{\theta}(t)$`: hazard function
 + `$S_{\theta}(t)$`: survival function
+ Censoring schemes
 + Right censoring, left censoring, interval censoring
+ We want to identify .enfase[risk factors] (covariates)
+ Proportional hazards models:

`$$h_{\theta}(t_i) = h^{\ast}_{\theta}(t_i)\exp\{X_i\beta\}$$`

---
class: slide-page

## Spatial frailty model

+ `$h_{\theta}(t_{ij}) = h^{\ast}_{\theta}(t_{ij})\gamma_i\exp\{X_{ij}\beta\}$`
    + `$\gamma_i$` must be positive (Gamma distribution)
+ Banerjee, Wall e Carlin (2003) proposed the following structure:

.mathbox[
$$
`\begin{aligned}
  h_{\theta}(t_{ij}) & = h_{\theta}^{\ast}(t_{ij}) \times \gamma_i \times  e^{X_{ij} \beta} \\
                     & = h_{\theta}^{\ast}(t_{ij}) \times e^{X_{ij}\beta + log(\gamma_i)} \\
                     & = h_{\theta}^{\ast}(t_{ij}) \times e^{X_{ij}\beta + \psi_i}.
\end{aligned}`
$$
]

We can model `$\psi$` with an ICAR model, for example.

---
class: slide-page

## Restricted spatial frailty model

+ We have .enfase[more subjects] than .enfase[areas]
    + `$\displaystyle P_{N\times N}$`, `$N = \sum_{i = 1}^{n_i}n_i$`
    + `$n_i$` is the number of individuals in area `$i$`, `$i = 1, \ldots, n$`
    + `$\psi$` is a `$n\times 1$` column vector
    + `$c$` is a `$n\times 2$` vector (centroids)
+ Therefore, it is .error[not possible] to make the projections because:

RHZ:
`$$P^c_{N\times N} ~~ \text{and} ~~ \psi_{n\times 1}$$`

SPOCK:
`$$P^c_{N\times N} ~~ \text{and} ~~ c_{n\times 2}$$`

---
class: slide-page

## Restricted spatial frailty model

+ Solution: to create a vector `$\Psi = [\psi_1\times 1_{n_1}, \ldots, \psi_n\times 1_{n_n}]'$` 
    + `$1_{m}$` is a length `$m$` line vector of ones

.mathbox[
$$
`\begin{equation}
    \psi_{n\times 1} =
    \begin{bmatrix}
        \psi_1     \\
        \psi_2     \\
        \vdots  \\
        \psi_n
    \end{bmatrix};
    ~~~~
    \Psi_{N\times 1} =
    \begin{array}{c@{\!\!\!}l}
        \left[ 
            \begin{array}[c]{ccccc}
                \psi_1     \\
                \vdots  \\
                \psi_1     \\
                \vdots  \\
                \psi_n     \\
                \vdots  \\
                \psi_n
            \end{array}  
        \right]
        &
        \begin{array}[c]{@{}l@{\,}l}
            \left. 
                \begin{array}{c} 
                    \vphantom{0}      \\
                    \vphantom{\vdots} \\ 
                    \vphantom{0} 
                \end{array} 
            \right\} 
            & 
            \text{$n_1$ times} \\
            \vphantom{\vdots}  \\
            \left. 
                \begin{array}{c} 
                    \vphantom{0}      \\ 
                    \vphantom{\vdots} \\ 
                    \vphantom{0}  
                \end{array} 
            \right\} 
            & 
            \text{$n_n$ times}
        \end{array}
    \end{array}.
\end{equation}`
$$
]

Now `$P^c_{N\times N}\Psi_{N\times 1}$` .success[is possible]

---
class: slide-page

## Restricted spatial frailty model

To have sample from .enfase[restricted] and .enfase[unrestricted] models .enfase[concurrently] we found the .enfase[equivalence between models]:

.mathbox[
$$
`\begin{align}
    h_{\theta}(t) & = h_{\theta}^0(t)\exp\{X\beta_{rsf} + \Psi_{rsf} + \epsilon_{rsf}\} \\
                  & = h_{\theta}^0(t)\exp\{X\beta_{rsf} + \Psi_{rsf} + \tilde{\psi} + \epsilon_{sf}\} \\
                  & = h_{\theta}^0(t)\exp\{X\beta_{rsf} + P^c\Psi_{sf} + \epsilon_{sf}\} \\
                  & = h_{\theta}^0(t)\exp\{X(\beta_{rsf} - (X'X)^{-1}X'\Psi_{sf}) + \Psi_{sf} + \epsilon_{sf}\} \\
                  & = h_{\theta}^0(t)\exp\{X\beta_{sf} + \Psi_{sf} + \epsilon_{sf}\},
\end{align}`
$$
]

where "rsf" means _restricted spatial frailty_

+ `$\Psi_{rsf} = [\psi_{rsf_1}\times 1_{n_1}, \ldots, \psi_{rsf_n}\times 1_{n_n}]'$` 
    + `$\psi_{rsf_n}$` is the mean by area of `$P^c\Psi_{sf}$`
+ `$\tilde{\psi} = P^c\Psi_{sf} - \Psi_{rsf}$`

---
class: slide-page

## Reduction operator

+ `$X_{N\times p}$`: matrix with entries `$X_{ijk}$` for an index `$i$`, an element `$j$` and column `$k$`, 
+ `$G_{N\times 1}$`: vector of indexes for each line of `$X_{N\times p}$` 1 until `$n$`, `$n \ll N$`.

Then the reduction operator `$\circledR$` is defined by:
`$$X_{N\times p}~\circledR~G = x_{n\times p},$$`

in which `$\displaystyle x_{ik} = \sum_{j = 1}^{n_i}X_{ijk}$`, and `$n_i$` is the number of elements related with index `$i$`.

Then:

.mathbox[
`$$\beta_{rsf} = \beta_{sf} + (X'X)^{-1}X'\Psi_{sf} = \beta_{sf} + (X'X)^{-1}(X~\circledR~G)'\psi_{sf}$$`
`$$\psi_{rsf} = (N^{-1}P^c\Psi_{sf})~\circledR~G$$`
]

---
class: slide-page

## RSFM - simulation

Reduction operator improvement:

---
class: slide-page

## RSFM - simulation

+ `$X$` is a two column matrix. `$X_{1}$` and `$X_{2}$` are:
 + Case 1: random variables
 + Case 2: `$X_{1}$` is random and `$X_{2}$` corresponds to the set of latitudes (spatially correlated)
+ Weibull proportional hazards model
+ Censure in `$\{0\%, 25\%, 50\%, 75\%\}$`
+ Parameters
 + `$\alpha = 1.2$`, `$\beta_1 = −0.3$`, `$\beta_2 = 0.3$`, `$\tau_{\psi} = 0.75$`
+ Priors
 + `$\alpha \sim \text{Gamma}(0.001, 0.001)$`
 + `$\beta_j \sim \text{Normal}(0, 0.001), ~ j = 0, 1, 2$`
 + `$\tau_{\psi} \sim \Gamma(0.5, 0.0005)$`
 
---
class: slide-page

## RSFM - simulation

---
class: slide-page

## RSFM - application

+ .enfase[Months until death] by .enfase[lung and bronchus cancer] in California
+ Same priors as in the simulation study

---
class: slide-page

## RSFM - application

+ Model results

---

## RASCO

+ Available at: https://github.com/DouglasMesquita/RASCO

```r
mod <- rsglmm(data = data, area = "reg",
 formula = Y ~ X1 + X2,
 family = "poisson", neigh = neigh_RJ, 
 model = "restricted_besag", 
 proj = "rhz", nsamp = 1000)

mod <- rscm(data = data, area = "reg",
 formula1 = Y1 ~ X11 + X12, 
 formula2 = Y2 ~ X21 + X12,
 family = c("poisson", "poisson"), neigh = neigh_RJ, 
 proj = "spock", nsamp = 1000)

mod <- rsfm(data = data, area = "reg",
 formula = surv(time = L, event = status) ~ X1 + X2,
 family = "weibull", neigh = neigh_RJ, 
 model = "restricted_besag", 
 proj = "rhz", nsamp = 1000)
```

---

## Conclusion

+ Spatial confouding
 + This is an important limitation and should be checked for models beyond the GLMM family.
+ Shared component model
 + We proposed a model changing the multiple spatial structures present in the model
 + It is interesting since the user can correct the structure before running analysis
+ Spatial frailty model
 + We proposed a restricted model increasing the spatial effects dimension
 + The reduction operator brought computational efficiency to our model
+ RSCM and RSFM
 + The simulation showed the effectiveness of both approaches
 + In the application we showed that one important covariate was confounded and our methods alleviated such confounding in both cases

---
class: slide-page

## Conclusion

+ RASCO
 + Our package is available at: https://github.com/DouglasMesquita/RASCO
 + We have solutions for 
 + GLMM
 + SCM
 + SFM
 + The available approaches are: 
 + RHZ
 + HH
 + SPOCK
 + RSCM
 + RSFM

---
class: slide-page

## Future work

+ To employ the reduction operator for HH and SPOCK in survival models
    + HH: `$((N^{-1}P^c~\circledR~G')~\circledR~G))W((N^{-1}P^c~\circledR~G')~\circledR~G)$`
    + SPOCK: `$c^{\ast} = ((N^{-1}P^c~\circledR~G')~\circledR~G)c$`
+ Study of the spatial and temporal confounding effects in spatio-temporal models
+ To add more methods to RASCO package
    + HH for SCM and SFM
    + MCMC versions

---

## Papers

+ .enfase[Azevedo, D. R., Bandyopadhyay, D., Prates, M. O., Abdel‐Salam, A. S. G., & Garcia, D. (2020)]. **Assessing spatial confounding in cancer disease mapping using R**. _Cancer Reports, 3(4), e1263_.

+ .enfase[Azevedo, D. R., Prates, M. O., & Bandyopadhyay, D. (2021)]. **MSPOCK: Alleviating Spatial Confounding in Multivariate Disease Mapping Models**. _Journal of Agricultural, Biological and Environmental Statistics, 26, 464–491_.

+ .enfase[Azevedo, D. R. M., Prates, M. O., & Bandyopadhyay, D. (2020)]. **Alleviating spatial confounding in spatial frailty models**. _Biometrics (To be submitted)_.

+ .enfase[Prates, M. O., Azevedo, D. R. M., Godoy, L. C., & Bandyopadhyay, D. (2021)]. **Can Transformed Gaussian Random Fields Handle Spatial Confounding?**. _Journal of Indian Statistical Association, Special Edition - Spatial Statistics (To be submitted)_.

+ .enfase[Azevedo, D. R. M., & Prates, M. O. (2021)]. **RASCO: An R package to Alleviate the Spatial Confounding**. _Journal of Statistical Software (Under development)_.

---

## References

.enfase[Reich, B. J., Hodges, J. S., and Zadnik, V. (2006)]. **Effects of residual smoothing on the posterior of the fixed effects in disease‐mapping models**. _Biometrics, 62(4), 1197-1206_.

.enfase[Prates, M. O., Assunção, R. M., and Rodrigues, E. C. (2019)]. **Alleviating Spatial Confounding for Areal Data Problems by Displacing the Geographical Centroids**. _Bayesian Analysis 14.2 (2019): 623-647_.

.enfase[Hughes, J., and Murali H. (2013)]. **Dimension reduction and alleviation of confounding for spatial generalized linear mixed models**. _Journal of the Royal Statistical Society: Series B (Statistical Methodology) 75.1 (2013): 139-159_.

.enfase[Hanks, E. M., Schliep, E. M., Hooten, M. B., and Hoeting, J. A. (2015)]. **Restricted spatial regression in practice: geostatistical models, confounding, and robustness under model misspecification**. _Environmetrics 26.4 (2015): 243-254_.

.enfase[Knorr‐Held, L., and Best, N. G. (2001)]. **A shared component model for detecting joint and selective clustering of two diseases**. _Journal of the Royal Statistical Society: Series A (Statistics in Society), 164(1), 73-85_.

.enfase[Banerjee, S., Melanie M. W., and Bradley P. C. (2003)]. **Frailty modeling for spatially correlated survival data, with application to infant mortality in Minnesota**. _Biostatistics 4.1 (2003): 123-142_.

---
class: slide-page

## References

.enfase[Hefley, T. J. and Hooten, M. B., Hanks, E. M. and Russell, R. E., and Walsh, D. P. (2017)]. **The Bayesian group lasso for confounded spatial data**. _Journal of Agricultural, Biological and Environmental Statistics 22.1 (2017): 42-59_.
 
.enfase[Thaden, H., and Thomas K. (2018)]. **Structural equation models for dealing with spatial confounding**. _The American Statistician 72.3 (2018): 239-252_.
 
.enfase[Whittle, P. (1954)]. **On stationary processes in the plane.*"**. _Biometrika (1954): 434-449_.
 
.enfase[Ord, K. (1975)]. **Estimation methods for models of spatial interaction.**. _Journal of the American Statistical Association 70.349 (1975): 120-126_.
 
.enfase[Besag, J., Jeremy Y., and Annie M. (1991)]. **Bayesian image restoration, with two applications in spatial statistics**. _Annals of the institute of statistical mathematics 43.1 (1991): 1-20_
 
.enfase[Leroux, B. G., Xingye, L., and Norman B. (2000)]. **Estimation of disease rates in small areas: a new mixed model for spatial dependence**. _Statistical models in epidemiology, the environment, and clinical trials. Springer, New York, NY, 2000. 179-191_.
 
.enfase[Datta, A., Banerjee, S., Hodges, J. S., and Gao, L. (2019)]. **Spatial disease mapping using directed acyclic graph auto-regressive (DAGAR) models**. _Bayesian Analysis 14.4 (2019):1221-1244_.
 
.enfase[Rodrigues, E. C., and Renato, M. A. (2012)]. **Bayesian spatial models with a mixture neighborhood structure**. _Journal of Multivariate Analysis 109 (2012): 88-102_.

---

<h1 style="color:#378fbb; font-size:90px">
 THANK YOU!
</h1>

<h5>
 require-r.com
 douglas-mesquita
 DouglasMesquita
 douglas-mesquita
</h5>

---

<div class="my-section">
 <h2>
 Extras
 </h2>
</div>

---
class: slide-page

## RSCM - application

+ .enfase[New cases] of .enfase[lung and bronchus cancer] for .enfase[men] and .enfase[women] in California (58 counties)
+ Same priors as in the simulation study
+ Covariates:

---
class: slide-page

## RSCM - application

`$\mathcal{M}_1$` - .enfase[Univariate non-spatial model]:
.mathbox[
`$Y_{id} \sim P(E_{id}\theta_{id}),$`

`$\log(\theta_{id}) = \beta_{d0} + X_{id}\beta,$`

`$Y_{1} \perp Y_{2}.$`
]

`$\mathcal{M}_2$` - .enfase[Univariate spatial model]:
.mathbox[
`$Y_{id} \sim P(E_{id}\theta_{id}),$`

`$\log(\theta_{id}) = \beta_{d0} + X_{id}\beta + \psi_{id},$`

`$\psi_1 \sim \text{ICAR}(W, \tau_{\psi_1}Q); ~~ \psi_2 \sim \text{ICAR}(W, \tau_{\psi_2}Q),$`

`$Y_{1} \perp Y_{2}$`
]
---
class: slide-page

## RSCM - application

`$\mathcal{M}_3$` - .enfase[Shared Component model without specific spatial term]:

`$\log(\theta_{i1}) = \beta_{10} + X_{i1}\beta + \delta\psi_{i}; ~~ \log(\theta_{i2}) = \beta_{20} + X_{i2}\beta + \frac{1}{\delta}\psi_{i},$`

`$\psi \sim \text{ICAR}(W, \tau_{\psi}Q).$`
]

`$\mathcal{M}_4$` - .enfase[Shared Component model with specific spatial term]:
.mathbox[
`$Y_{id} \sim P(E_{id}\theta_{id}),$`

`$\log(\theta_{i1}) = \beta_{10} + X_{i1}\beta + \delta\psi_{i} + \phi_{1i}; ~~ \log(\theta_{i2}) = \beta_{20} + X_{i2}\beta + \frac{1}{\delta}\psi_{i} + \phi_{2i},$`

`$\psi \sim \text{ICAR}(W, \tau_{\psi}Q); ~~ \phi_1 \sim \text{ICAR}(W, \tau_{\phi_1}Q); ~~ \phi_2 \sim \text{ICAR}(W, \tau_{\phi_2}Q).$`
]

---
class: slide-page

## RSCM - application

+ Model results