Quantifying Contributions of Fixed and Spatial Effects in Poisson SPDE Model

[Nafsika Antoniadou]

In my Poisson model with both fixed effects and a spatial random effect modeled via the SPDE approach, is there a way to quantify the relative contribution of the fixed effects versus the spatial random effect to the total variance in the model?

[Finn Lindgren]

Hi,

there isn't one universal answer to this, but we provided a partial approach in Section 5.4 of

  https://projecteuclid.org/journals/annals-of-applied-statistics/volume-11/issue-4/Point-process-models-for-spatio-temporal-distance-sampling-data-from/10.1214/17-AOAS1078.full
The models there are Poisson point process models instead of Poisson count models, but the approach we discuss there works the same in both cases, as it considers the variability of the predictor expressions with and without the random field.

The idea is that if the full predictor is

  eta = fixed + field

then the pointwise posterior variance of eta at each location "s" is

  Var(eta(s)) = Var(fixed(s)) + Var(field(s)) + 2 Cov(fixed(s), field(s))
where all properties are in the posterior distribution,
and a non-zero covariance indicates some level of confounding (which one should always expect!).

One could then e.g. compute the spatial average ratios

  R_fixed = \int Var(fixed(s)) ds / \int eta(s) ds

and

  R_field = \int Var(field(s)) ds / \int eta(s) ds

These ratios will normally sum to more than 1, due to the confounding, so the question "what is the relative contribution" isn't a well-defined question.
One could of course consider

  R_fixed / (R_fixed + R_field)

but that masks some of the information; perhaps combine it with the correlation:
  C = \int Cov(fixed(s), field(s)) ds / \sqrt{ \int Var(fixed(s)) ds \int Var(field(s)) ds }

One can also combine this with separate model estimates;

  eta1 = fixed

  eta2 = fixed + field

and consider the differences between eta1 and eta2.

Finn

On Fri, 7 Nov 2025 at 12:33, Nafsika Antoniadou <antoniadou...@gmail.com> wrote:

In my Poisson model with both fixed effects and a spatial random effect modeled via the SPDE approach, is there a way to quantify the relative contribution of the fixed effects versus the spatial random effect to the total variance in the model?

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Finn Lindgren
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[Finn Lindgren]

See

  https://inlabru-org.github.io/inlabru/articles/2d_lgcp.html#estimating-abundance

for an example for how to compute posterior integrals, although you need to first compute the variances, and then integrate, and the example instead computes the expectation and std.dev. of the integra.

Finn