Spatial Survival: Leukemia Hazards Across Districts
Survival analysis models the time until an event, and a complication is that some observations are only right-censored: the patient was still alive when the study ended, so we know the survival time exceeds some value but not by how much. This tutorial works the canonical example, the Leukemia dataset of Henderson et al. (2002): 1043 patients in North-West England, each with a survival time, a censoring indicator, four covariates, and the district they lived in. On top of a survival regression we add a Besag (1974) spatial frailty over the 24 districts, which lets the model pick up residual geographic variation in hazard after adjusting for the covariates.
We fit the same model two ways and check that they agree with each other and with an external R-INLA reference:
the Poisson piecewise-exponential trick, the textbook reformulation of a survival model as Poisson regression, which runs on Latte's built-in Poisson likelihood with no custom code; and
a hand-written Weibull survival likelihood, a few lines of
logpdfthat handle censoring directly, dropped into an@lattemodel.
The first route reuses a built-in likelihood; the second shows what it takes to supply a parametric likelihood that isn't built in.
using Latte
using GaussianMarkovRandomFields: BesagModel
using Distributions
using LinearAlgebra
using SparseArrays
using Statistics
using CSV
using DataFrames
using CairoMakieThe data
status = 1 marks an observed death and status = 0 a right-censored patient. The continuous covariates (age, white-blood-cell count wbc, Townsend deprivation index tpi) are standardized; sex is 0/1. The data and the district adjacency graph ship with the tutorial, originally from R-INLA's Leuk example.
data_dir = joinpath(@__DIR__, "data")
leuk = CSV.read(joinpath(data_dir, "leuk.csv"), DataFrame)
edges = CSV.read(joinpath(data_dir, "leuk_graph.csv"), DataFrame)
n_district = maximum(leuk.district)
# Symmetric 0/1 adjacency for the Besag prior.
Is, Js = Int[], Int[]
for r in eachrow(edges)
push!(Is, r.i, r.j)
push!(Js, r.j, r.i)
end
W = sparse(Is, Js, ones(length(Is)), n_district, n_district)
@info "Leukemia survival data" patients = nrow(leuk) districts = n_district events = sum(leuk.status) censored = sum(leuk.status .== 0)┌ Info: Leukemia survival data
│ patients = 1043
│ districts = 24
│ events = 879
└ censored = 164The covariate matrix used by both models (the spatial frailty is added separately, indexed by district).
const COVNAMES = ["age", "sex", "wbc", "tpi"]
covmat = hcat(leuk.age_z, Float64.(leuk.sex), leuk.wbc_z, leuk.tpi_z)
covmat[1:4, :] # first four patients (standardized age, sex, wbc, tpi)4×4 Matrix{Float64}:
0.0149561 0.0 -0.347337 -0.631948
0.833096 0.0 5.64945 -1.0249
0.724011 0.0 1.58476 -1.45357
0.996724 1.0 6.33606 -0.478067Route 1: the Poisson piecewise-exponential trick
A proportional-hazards survival model with hazard
Then
# Interval cut points from the quantiles of the observed event times.
event_times = leuk.time[leuk.status .== 1]
K = 12
cuts = quantile(event_times, range(0, 1; length = K + 1))
cuts[1] = 0.0
cuts[end] = maximum(leuk.time) + 1.0
# Expand to person-period records.
pp_district = Int[]
pp_interval = Int[]
pp_exposure = Float64[]
pp_event = Int[]
pp_cov = Vector{Float64}[]
for r in eachrow(leuk)
for j in 1:K
lo, hi = cuts[j], cuts[j + 1]
lo >= r.time && break # patient already left follow-up
exposure = min(r.time, hi) - lo
exposure <= 0 && continue
push!(pp_district, r.district)
push!(pp_interval, j)
push!(pp_exposure, exposure)
push!(pp_event, (r.status == 1 && lo < r.time <= hi) ? 1 : 0)
push!(pp_cov, [r.age_z, Float64(r.sex), r.wbc_z, r.tpi_z])
end
end
n_pp = length(pp_event)
# Design matrix: one indicator per interval (the piecewise baseline) followed
# by the four covariates. The interval indicators carry the intercept, so no
# separate intercept column is needed.
Xpp = zeros(n_pp, K + 4)
for r in 1:n_pp
Xpp[r, pp_interval[r]] = 1.0
Xpp[r, (K + 1):(K + 4)] = pp_cov[r]
end
log_exposure = log.(pp_exposure)
@info "Person-period dataset" rows = n_pp intervals = K┌ Info: Person-period dataset
│ rows = 7498
└ intervals = 12The model is a plain Poisson regression with the log-exposure offset and a Besag frailty indexed by district.
@latte function leuk_poisson(y, X, offset, district, W)
τ ~ PCPrior.Precision(1.0, α = 0.01)
β ~ MvNormal(zeros(size(X, 2)), 100.0 * I(size(X, 2))) # K baselines + 4 covariates
u ~ BesagModel(W)(τ = τ)
for r in eachindex(y)
y[r] ~ Poisson(exp(offset[r] + X[r, :] ⋅ β + u[district[r]]))
end
end
lgm_pois = leuk_poisson(pp_event, Xpp, log_exposure, pp_district, W)
result_pois = inla(lgm_pois, pp_event; progress = false)
# `latent_marginals(result, :name)` returns the marginals for a named latent
# block. Here `:β` holds the K piecewise baselines followed by the four
# covariate log-hazard-ratios, so the covariate effects are its last four entries.
pois_β = latent_marginals(result_pois, :β)
pois_coef = [(mean(pois_β[K + k]), std(pois_β[K + k])) for k in 1:4]
for (nm, (m, s)) in zip(COVNAMES, pois_coef)
println(" $nm: ", round(m, digits = 3), " ± ", round(s, digits = 3))
end age: 0.579 ± 0.04
sex: 0.07 ± 0.068
wbc: 0.224 ± 0.033
tpi: 0.101 ± 0.035Route 2: a hand-written Weibull survival likelihood
The Poisson trick leaves the baseline hazard nonparametric. If instead we want a parametric Weibull baseline, there is no built-in survival observation model, so we write the likelihood ourselves. With a Weibull proportional-hazards model the cumulative hazard is
We encode that as a small Distribution whose logpdf branches on the event indicator. The Weibull shape α ~ LogNormal(...) prior in the model below.
# Independent type parameters for `η` and `α` let the latent-derived `η` (an
# AD dual number) and the `α` hyperparameter (a Float64) carry different
# types, which is all it takes to be AD-ready; no promoting constructor is
# needed. `@latte` only needs `logpdf`, so there is nothing else to define.
struct WeibullSurv{A, B} <: ContinuousUnivariateDistribution
η::A
α::B
event::Int
end
function Distributions.logpdf(d::WeibullSurv, t::Real)
logt = log(t)
log_cumhaz = d.α * logt + d.η
log_haz = log(d.α) + (d.α - 1) * logt + d.η
return d.event == 1 ? log_haz - exp(log_cumhaz) : -exp(log_cumhaz)
endThe @latte model carries an intercept plus the four covariates in β, the same Besag frailty, and the Weibull shape as a hyperparameter. The prior goes directly on the natural parameter with α ~ LogNormal(0, 1), so @latte reads α as a declared hyperparameter: its positive support follows from the prior, and its marginal comes back in natural (shape) space.
@latte function leuk_weibull(t, X, district, status, W)
α ~ LogNormal(0.0, 1.0)
τ ~ PCPrior.Precision(1.0, α = 0.01)
β ~ MvNormal(zeros(size(X, 2)), 100.0 * I(size(X, 2))) # intercept + 4 covariates
u ~ BesagModel(W)(τ = τ)
for i in eachindex(t)
η = X[i, :] ⋅ β + u[district[i]]
t[i] ~ WeibullSurv(η, α, status[i])
end
end
Xw = hcat(ones(nrow(leuk)), covmat) # intercept + covariates
lgm_weib = leuk_weibull(leuk.time, Xw, leuk.district, leuk.status, W)LatentGaussianModel
├─ hyperparameters (2)
│ α ~ LogNormal(μ=0.0, σ=1.0) [log]
│ τ ~ PCPrior.Precision(λ=4.605)
├─ latent field · 29 dims
│ β 5 FixedEffects
│ u 24 Besag
└─ likelihood
custom logpdf · 1043 observationsDeclaring the natural-parameter prior leaves nothing for @latte to hoist, so this model runs under the default ADStrategy without extra knobs. We pass FiniteDiffStrategy() here purely for speed: with only a handful of hyperparameters and a custom likelihood, the finite-difference inner Laplace step runs about twice as fast as the autodiff default and gives the same results.
result_weib = inla(lgm_weib, leuk.time; diff_strategy = FiniteDiffStrategy(), progress = false)
# `:β` is intercept + 4 covariates, so the covariate effects are entries 2:5.
weib_β = latent_marginals(result_weib, :β)
weib_coef = [(mean(weib_β[1 + k]), std(weib_β[1 + k])) for k in 1:4] # skip intercept
# `α` is a declared hyperparameter (the prior is on the natural shape), so its
# marginal already lives in natural space. Read it back through the accessor
# by name; for a scalar hyperparameter the block holds a single marginal.
α_mean = mean(hyperparameter_marginals(result_weib, :α)[1])
for (nm, (m, s)) in zip(COVNAMES, weib_coef)
println(" $nm: ", round(m, digits = 3), " ± ", round(s, digits = 3))
end
println(" Weibull shape α ≈ ", round(α_mean, digits = 3))[ Info: AutoDiffLikelihood: `pointwise_loglik_func` provided without `diagonal_hessian_safe=true`. `loghessian` will compute the full Hessian via `DI.hessian`. Set `diagonal_hessian_safe=true` only if every pointwise term `i` depends solely on `x[i]` (e.g. `y[i] ~ Family(x[i])`).
┌ Warning: Hyperparameter mode optimization did not converge for any start (n_starts = 1)
└ @ Latte ~/work/Latte.jl/Latte.jl/src/laplace/mode_finding.jl:438
┌ Warning: Accumulator: using Monte Carlo for AutoDiffLikelihood (no `linear_predictor_marginals` method). Results carry MC error; tune `n_samples` or set `fallback = :error` to require analytic support.
└ @ Latte ~/work/Latte.jl/Latte.jl/src/posterior/accumulators/accumulator_core.jl:284
age: 0.589 ± 0.04
sex: 0.078 ± 0.069
wbc: 0.221 ± 0.033
tpi: 0.093 ± 0.035
Weibull shape α ≈ 0.589The two routes agree, and match R-INLA
Both formulations recover the same covariate log-hazard-ratios. As an external check, the table also reports R-INLA's weibullsurv fit (variant 0, the proportional-hazards parameterization) on the same data and priors. The numbers line up to two decimals, including the posterior standard deviations.
# R-INLA weibullsurv (variant 0) reference: (mean, sd).
rinla_coef = [(0.592, 0.04), (0.079, 0.069), (0.22, 0.033), (0.094, 0.035)]
println("\ncovariate │ Poisson (route 1) │ Weibull (route 2) │ R-INLA")
for (k, nm) in enumerate(COVNAMES)
a, b, c = pois_coef[k], weib_coef[k], rinla_coef[k]
println(
rpad(nm, 9), "│ ", rpad(string(round(a[1], digits = 3), " ± ", round(a[2], digits = 3)), 18),
"│ ", rpad(string(round(b[1], digits = 3), " ± ", round(b[2], digits = 3)), 18),
"│ ", round(c[1], digits = 3), " ± ", round(c[2], digits = 3)
)
end
covariate │ Poisson (route 1) │ Weibull (route 2) │ R-INLA
age │ 0.579 ± 0.04 │ 0.589 ± 0.04 │ 0.592 ± 0.04
sex │ 0.07 ± 0.068 │ 0.078 ± 0.069 │ 0.079 ± 0.069
wbc │ 0.224 ± 0.033 │ 0.221 ± 0.033 │ 0.22 ± 0.033
tpi │ 0.101 ± 0.035 │ 0.093 ± 0.035 │ 0.094 ± 0.035A coefficient plot makes the agreement easy to see: each covariate's posterior mean ± one standard deviation, for all three fits.
let
fig = Figure(size = (680, 360))
ax = Axis(
fig[1, 1], xlabel = "log hazard ratio", yticks = (1:4, COVNAMES),
title = "Covariate effects: two Latte routes vs. R-INLA"
)
series = [
("Poisson trick", pois_coef, :dodgerblue, -0.18),
("Weibull likelihood", weib_coef, :firebrick, 0.0),
("R-INLA", rinla_coef, :black, 0.18),
]
for (lab, coef, col, dy) in series
ys = (1:4) .+ dy
ms = [c[1] for c in coef]
ss = [c[2] for c in coef]
errorbars!(ax, ms, ys, ss; direction = :x, color = col, whiskerwidth = 8)
scatter!(ax, ms, ys; color = col, markersize = 11, label = lab)
end
vlines!(ax, 0.0; color = (:gray, 0.5), linestyle = :dash)
axislegend(ax; position = :rt, framevisible = false)
fig
end
All three agree: higher age, higher white-blood-cell count, and greater deprivation each raise the hazard, while the sex effect is small and its interval covers zero. The Weibull route also estimates a shape
The spatial frailty
After the covariates, what does geography add? The Besag frailty
# The spatial frailty is its own named block `:u`, one entry per district.
weib_u = latent_marginals(result_weib, :u)
frailty = exp.([mean(weib_u[d]) for d in 1:n_district])
mapdf = CSV.read(joinpath(data_dir, "leuk_map.csv"), DataFrame)
polys = Vector{Point2f}[]
poly_region = Int[]
for g in groupby(mapdf, :poly) # one ring per `poly`; some districts have several
push!(polys, Point2f.(g.x, g.y))
push!(poly_region, first(g.region))
end
let
crange = (minimum(frailty), maximum(frailty))
fig = Figure(size = (560, 620))
ax = Axis(
fig[1, 1], aspect = DataAspect(),
title = "Posterior spatial frailty exp(u) (hazard multiplier)"
)
hidedecorations!(ax)
hidespines!(ax)
poly!(
ax, polys; color = frailty[poly_region], colorrange = crange,
colormap = :balance, strokewidth = 0.5, strokecolor = (:black, 0.4)
)
Colorbar(
fig[1, 2]; colorrange = crange, colormap = :balance,
label = "relative hazard exp(u)"
)
fig
end
The frailty is mild but real: a few districts carry a residual excess hazard, and others a deficit, that the covariates do not explain. This is the kind of structure a spatial random effect absorbs; left out, it would leak into biased covariate estimates or overconfident intervals.
Takeaway
The Poisson piecewise-exponential trick turns a proportional-hazards model into a Poisson GLM that runs on the built-in likelihood, with no custom code.
When a parametric likelihood isn't built in, you can supply it. A five-line
WeibullSurvlogpdfthat branches between the density and the survival function to handle censoring drops into@latte, with posterior uncertainty over the shape, the coefficients, and the field.Both routes agree with each other and with the R-INLA reference to two decimals, and the spatial frailty uses the same GMRF machinery as the rest of these tutorials.
References
The canonical spatial survival example: leukemia survival for 1043 patients across districts of North-West England, modeled with covariates and a spatially structured frailty.
The foundational paper on conditional autoregressive (CAR) models, the intrinsic Gaussian Markov random field that the spatial frailty here is built from.
The original INLA paper: deterministic approximate Bayesian inference for latent Gaussian models via nested Laplace approximations and numerical integration over the hyperparameters.
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