6. Covariates measured on a different support • SDALGCP2

Covariates are often observed on a different support from the outcome: air quality at a few monitoring stations, temperature on a coarse climate grid, deprivation on census tracts that do not match the health districts. SDALGCP2 aligns them by predicting the covariate to the candidate points of the outcome regions and entering it with an uncertainty-aware (Berkson) correction. This tutorial is self-contained.

The method

The intensity-scale offset (Tutorial 2) needs the covariate $z(x_{ik})$ at the candidate points $x_{ik}$ . When $z$ is observed elsewhere we (1) model it as a Gaussian process and krige it to those points, obtaining a predictive mean $\mu_z(x_{ik})$ and variance $v_z(x_{ik})$ ; (2) plug these into a Berkson-corrected offset $b_i(\beta)=\log\sum_k w_{ik}\, \exp\!\Big\{\mu_z(x_{ik})^\top\beta+\tfrac12\,\beta^\top V_z(x_{ik})\,\beta\Big\}.$ The $\tfrac12\beta^\top V_z\beta$ term propagates the prediction uncertainty (plugging in the kriged mean alone would attenuate $\beta$ ); as $v_z\to0$ it recovers the raster case. Derivation: math/confounding-and-misalignment.pdf.

Point support: monitoring stations

The covariate z is observed at 40 scattered monitors (an sf of points with a z column); the outcome is counts over an 8×8 lattice.

library(SDALGCP2)
library(sf)

set.seed(123)
zf  <- function(xy) 1.5 * sin(xy[, 1] / 4) + 1.2 * cos(xy[, 2] / 5) + 0.5 * xy[, 1] / 20
reg <- st_sf(geometry = st_make_grid(
  st_as_sfc(st_bbox(c(xmin = 0, ymin = 0, xmax = 20, ymax = 20))), n = c(8, 8)))
N <- nrow(reg)

# simulate counts from the point-level intensity model (true beta_z = 0.8)
pts <- sda_points(reg, delta = 1.0, method = 3); w <- lapply(pts, function(p) p$weight)
b_true <- sapply(seq_len(N), function(i) {
  Z <- cbind(1, zf(as.matrix(pts[[i]]$xy))); log(sum(w[[i]] * exp(as.numeric(Z %*% c(-6, 0.8))))) })
reg$pop   <- round(runif(N, 800, 5000))
reg$cases <- rpois(N, reg$pop * exp(b_true))

# the covariate, observed ONLY at 40 monitors
mon <- st_as_sf(data.frame(x = runif(40, 0, 20), y = runif(40, 0, 20)), coords = c("x", "y"))
mon$z <- zf(st_coordinates(mon))

Pass the monitors through covariates =; reg does not need a z column:

fit <- sdalgcp(cases ~ z + offset(log(pop)), data = reg, covariates = list(z = mon))
fit$beta_opt["z"]
#> 0.89   (true 0.8)

Internally the covariate is kriged from the monitors to every candidate point:

The covariate’s own spatial model (range, nugget, variance) is estimated by maximum likelihood; SDALGCP2 then fits the outcome model with the Berkson-corrected offset.

Areal support: a different partition

If the covariate is reported as averages over other polygons (e.g. a coarser grid or census tracts), supply those polygons. SDALGCP2 detects the geometry and uses aggregated areal kriging, reusing the same C++ kernels that build the outcome correlation:

# covariate observed as averages over a coarser 4x4 partition
covpoly <- st_sf(geometry = st_make_grid(
  st_as_sfc(st_bbox(c(xmin = 0, ymin = 0, xmax = 20, ymax = 20))), n = c(4, 4)))
cpts <- sda_points(covpoly, delta = 1.0, method = 3)
covpoly$z <- sapply(cpts, function(p) mean(zf(as.matrix(p$xy))))   # areal averages

fit_areal <- sdalgcp(cases ~ z + offset(log(pop)), data = reg, covariates = list(z = covpoly))
fit_areal$beta_opt["z"]

#> True z effect: 0.80
#> Point support  (40 monitors):           0.89
#> Areal support  (25 different polygons):  0.85

In both cases the covariate effect is recovered from data that were never observed on the outcome’s own units — point monitors, or polygons that do not match the outcome regions.

Notes

The candidate-point grid that discretises the outcome regions doubles as the common support on which outcome and covariate are aligned.
Set berkson = FALSE in SDALGCP2_misaligned() for the naive kriged-mean plug-in (no uncertainty propagation).
Currently the covariate model’s parameter uncertainty is plugged in (its predictive uncertainty is propagated); a fully joint two-stage model is a natural extension. ```