Restricted Shared Component model

Fit a Restricted Shared Component model for two diseases

rscm(data, formula1, formula2, family = c("poisson", "poisson"),
            E1 = NULL, E2 = NULL, area = NULL, neigh = NULL,
            proj = "none", nsamp = 1000,
            priors = list(prior_gamma = c(0, 0.35),
                          prior_prec = list(tau_s = c(0.01, 0.01),
                                            tau_1 = c(0.01, 0.01),
                                            tau_2 = c(0.01, 0.01))),
            random_effects = list(shared = TRUE,
                                  specific_1 = TRUE,
                                  specific_2 = TRUE),
            ...)

Arguments

data	a data frame or list containing the variables in the model.
formula1	an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted for disease 1.
formula2	an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted for disease 2.
family	a vector of size two with two families. Some allowed families are: poisson, nbinomial, zeroinflatedpoisson0, zeroinflatednbinomial0. See INLA::inla.list.models() for more information.
E1	known component, for disease 1, in the mean for the Poisson likelihoods defined as E = exp($\eta$), where $\eta$ is the linear predictor. Default = 1.
E2	known component, for disease 2, in the mean for the Poisson likelihoods defined as E = exp($\eta$), where $\eta$ is the linear predictor. Default = 1.
area	areal variable name in `data`.
neigh	neighborhood structure. A `SpatialPolygonsDataFrame` or `sf` object.
proj	"none" or "spock".
nsamp	number of samples. Default = 1000.
priors	a list containing: prior_gamma: a vector of size two containing mean and precision for the normal distribution applied for $\gamma$ prior_prec: a list with: tau_s: a vector of size two containing shape and scale for the gamma distribution applied for $\tau_s$ tau_1: a vector of size two containing shape and scale for the gamma distribution applied for $\tau_1$ tau_2: a vector of size two containing shape and scale for the gamma distribution applied for $\tau_2$
random_effects	a list determining which effects should we include in the model. Default: list(shared = TRUE, specific_1 = TRUE, specific_2 = TRUE).
...	other parameters used in ?INLA::inla

Value

$sample

a sample of size nsamp for all parameters in the model

$summary_fixed

summary measures for the coefficients

$summary_hyperpar

summary measures for hyperparameters

$summary_random

summary measures for random quantities

$out

INLA output

$time

time elapsed for fitting the model

Details

The fitted model is given by $$Y_1 ~ Poisson(E_1\theta_1),$$ $$Y_2 ~ Poisson(E_2\theta_2),$$

$$log(\theta_1) = X\beta + \gamma\psi + \phi_1,$$ $$log(\theta_2) = X\beta + \psi + \phi_2,$$

$$\psi ~ ICAR(\tau_s); \phi_1 ~ ICAR(\tau_1); \phi_2 ~ ICAR(\tau_2).$$

$$\delta = \sqrt\gamma$$

Examples

library(spdep)
#> Loading required package: sp
#> Loading required package: spData
#> To access larger datasets in this package, install the spDataLarge
#> package with: `install.packages('spDataLarge',
#> repos='https://nowosad.github.io/drat/', type='source')`
#> Loading required package: sf
#> Linking to GEOS 3.6.2, GDAL 2.2.3, PROJ 4.9.3

set.seed(123456)

##-- Spatial structure
data("neigh_RJ")

##-- Parameters
alpha_1 <- 0.5
alpha_2 <- 0.1
beta_1 <- c(-0.5, -0.2)
beta_2 <- c(-0.8, -0.4)
tau_s <- 1
tau_1 <- tau_2 <- 10
delta <- 1.5

##-- Data
data <- rshared(alpha_1 = alpha_1, alpha_2 = alpha_2,
                beta_1 = beta_1, beta_2 = beta_2,
                delta = delta,
                tau_1 = tau_1, tau_2 = tau_2, tau_s = tau_s,
                confounding = "linear",
                neigh = neigh_RJ)

##-- Models
scm_inla <- rscm(data = data,
                 formula1 = Y1 ~ X11 + X12,
                 formula2 = Y2 ~ X21 + X12,
                 family = c("nbinomial", "poisson"),
                 E1 = E1, E2 = E2,
                 area = "reg", neigh = neigh_RJ,
                 priors = list(prior_prec = list(tau_s = c(0.5, 0.05)), prior_gamma = c(0, 0.5)),
                 proj = "none", nsamp = 1000,
                 random_effects = list(shared = TRUE, specific_1 = TRUE, specific_2 = TRUE))

rscm_inla <- rscm(data = data,
                  formula1 = Y1 ~ X11 + X12,
                  formula2 = Y2 ~ X21 + X12,
                  family = c("nbinomial", "poisson"),
                  E1 = E1, E2 = E2,
                  area = "reg", neigh = neigh_RJ,
                  priors = list(prior_prec = list(tau_s = c(0.5, 0.05)), prior_gamma = c(0, 0.5)),
                  proj = "spock", nsamp = 1000,
                  random_effects = list(shared = TRUE, specific_1 = TRUE, specific_2 = TRUE))

##-- Summary
scm_inla$summary_fixed
#>             mean    median      mode       sd     lower     upper
#> alpha1  0.454628  0.453513  0.453410 0.066026  0.324790  0.583147
#> alpha2  0.099061  0.097661  0.096875 0.052867 -0.002203  0.202202
#> X11_1  -0.529634 -0.527243 -0.524312 0.094617 -0.725151 -0.360070
#> X12_1  -0.096918 -0.099301 -0.090591 0.712265 -1.437652  1.300094
#> X21_2  -0.821991 -0.821873 -0.825052 0.053160 -0.924361 -0.721452
#> X12_2  -0.315113 -0.308277 -0.308639 0.355988 -0.940467  0.398512
rscm_inla$summary_fixed
#>             mean    median      mode       sd     lower     upper
#> alpha1  0.486263  0.488156  0.487102 0.068773  0.353212  0.625193
#> alpha2  0.101960  0.102878  0.102827 0.052721  0.010352  0.210354
#> X11_1  -0.450881 -0.448670 -0.446537 0.092258 -0.653287 -0.288857
#> X12_1  -0.200463 -0.204132 -0.211575 0.166593 -0.516788  0.136881
#> X21_2  -0.819529 -0.820090 -0.821525 0.048604 -0.908102 -0.723579
#> X12_2  -0.294968 -0.295732 -0.295278 0.087644 -0.473432 -0.129857

scm_inla$summary_hyperpar
#>                                                              mean    median
#> size for the nbinomial observations (1/overdispersion) 102.159337 73.338553
#> Precision for psi                                        1.624693  1.593234
#> Precision for phi1                                      20.131572 14.180947
#> Precision for phi2                                      16.421046 15.845955
#> Beta for psi_gamma                                       2.192927  2.182752
#> Delta                                                    1.479839  1.475757
#> Precision for psi SC                                     0.739540  0.729550
#>                                                             mode        sd
#> size for the nbinomial observations (1/overdispersion) 52.844186 96.965589
#> Precision for psi                                       1.567703  0.262432
#> Precision for phi1                                     10.922866 19.075469
#> Precision for phi2                                     15.171983  5.720999
#> Beta for psi_gamma                                      2.174132  0.109452
#> Delta                                                   1.469693  0.035434
#> Precision for psi SC                                    0.718342  0.104168
#>                                                           lower      upper
#> size for the nbinomial observations (1/overdispersion) 4.585551 285.715222
#> Precision for psi                                      1.155745   2.159096
#> Precision for phi1                                     2.143624  55.320532
#> Precision for phi2                                     6.204610  27.587251
#> Beta for psi_gamma                                     1.991282   2.414290
#> Delta                                                  1.418131   1.550095
#> Precision for psi SC                                   0.552672   0.952562
rscm_inla$summary_hyperpar
#>                                                              mean    median
#> size for the nbinomial observations (1/overdispersion) 233.451915 76.311841
#> Precision for psi                                        0.632703  0.622718
#> Precision for phi1                                      26.830775 18.309253
#> Precision for phi2                                      23.987323 19.677618
#> Beta for psi_gamma                                       2.184501  2.179755
#> Delta                                                    1.477570  1.476902
#> Precision for psi SC                                     0.289227  0.283742
#>                                                             mode         sd
#> size for the nbinomial observations (1/overdispersion) 33.075785 585.792918
#> Precision for psi                                       0.613547   0.122101
#> Precision for phi1                                     11.596291  27.735642
#> Precision for phi2                                     15.702163  17.051720
#> Beta for psi_gamma                                      2.176957   0.124036
#> Delta                                                   1.474474   0.042099
#> Precision for psi SC                                    0.282195   0.051091
#>                                                           lower      upper
#> size for the nbinomial observations (1/overdispersion) 5.276925 898.357883
#> Precision for psi                                      0.405141   0.875612
#> Precision for phi1                                     0.435416  79.900368
#> Precision for phi2                                     2.346459  57.307706
#> Beta for psi_gamma                                     1.945202   2.430674
#> Delta                                                  1.397345   1.556711
#> Precision for psi SC                                   0.195223   0.388073

Restricted Shared Component model

Arguments

Value

Details

Examples

Contents