5.5 Spatio-temporal small area disease risk estimation

$$Y_{ij}\sim{Po(E_{ij}\phi_{ij})}$$

A Spatio-temporal model with parametric time trends

$$log(\phi_{ij}=\alpha+\mathit{u_i}+\mathit{v_i}+(\beta+\delta_i) \times \mathit{t_j}$$

$$Y\sim f(bym)+f(iid)+ \text{idtime}$$

formula <- Y ~ 
  f(idarea,    # area index
    model = "bym", 
    graph = g) + 
  f(idarea1,   # area index
    idtime,   
    model = "iid") + 
  idtime.      # time index

• f(idarea,model = "bym", graph = g) is the area random effect $\mathit{u_i}+\mathit{v_i}$.

• f(idarea1, idtime, model = "iid"). is the differential time trend $\delta_i \times \mathit{t}_j$

• idtime is the global trend $\beta \times \mathit{t}_j$

An alternative is from Knorr-Held (2000) specify models with interaction between space and time:

formula <- Y ~ f(idarea, model = "bym", graph = g) +
  f(idtime, model = "rw2") +
  f(idtime1, model = "iid") +
  f(idareatime, model = "iid")

# f(idareatime, model = "iid")

f(idtime,
  model = "iid",
  group = idarea, 
  ontrol.group = list(model = "besag", graph = g)
)

f(idarea,
  model = "iid",
  group = idtime, control.group = list(model = "rw2")
)

f(idarea,
  model = "besag", graph = g,
  group = idtime, control.group = list(model = "rw2")
)