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I am mapping and modelling a disease of sheep. I have approx 4200 point locations in my dataset, each of which represents the centroid of a given sheep farm.

I have created a K-function difference plot (below) to assess whether my disease-positive farm density layer shows evidence of spatial dependence above and beyond that shown by my disease-negative farm density layer. From this plot I identified spatial dependence in my dataset out to a distance of 500m from a given disease-positive farm.

I have built a Poisson point process model and been through a model selection process. My model residuals appear to be relatively well behaved. See lurking variable plots below, raw and pearson residuals.

To assess the need (or not) for a spatial dependence/interaction term in my Poisson point process model, I created an inhomogeneous K-function plot from a density surface estimated from my final model. See inhomogeneous K-function plot below.

My questions:

1) Based on these plots, should I be including a spatial dependence/interaction term in my model? If so why?

2) Should the repulsion between points shown by the inhomogeneous K-function be accounted for in my Poisson point process model if it is not due to the disease itself? The inhomogeneous K-function plot shows no evidence that disease-positive farms cluster, but does show evidence consistent with repulsion. I believe this repulsion is an artifact of my data and not associated with the disease itself -I am using points to represent the area of a farm, so points can never be closer to each other than their farm borders.

Thanks in advance for any answers, I am very very appreciative!

K-function difference plot

Raw residuals lurking variable plot

Pearson residuals lurking variable plot

Inhomogeneous K-function plot

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  • $\begingroup$ The difference in K-functions seems to be very small judging by the y-axis values on your plot. I'd be wary of reading too much into that. It's an interesting choice to use point process models here - are you trying to model the location of sheep farms? In general for point process models we are interested in why the point is where it is. You are more interested in the characteristics of the points. Perhaps a geostats approach would be more appropriate i.e. treat your farms as fixed in location. $\endgroup$ – ASeaton Mar 4 at 15:14
  • $\begingroup$ Yes that is correct. I am trying to explain why disease occurs where it does - i.e trying to explain the spatial distribution of the disease. So far I have only accounted for first order effects in my model (things like temperature, rainfall, soil type). I am now trying to decide if it is necessary, based on these diagnostic plots, to include a term in my model to account for/explain second order effects (spatial auto-correlation/dependence). I am new to these models and this analysis. $\endgroup$ – Patrick Taggart Mar 4 at 20:50
  • $\begingroup$ Initially I just looked at model residuals (lurking variable plots) and from these decided that there was little clustering in model residuals and hence no need for a spatial dependence term. But, then someone suggested I needed to create the inhomogeneous K-function plot ........ and now I am not confident either way (to include or not to include spatial dependence term). $\endgroup$ – Patrick Taggart Mar 4 at 20:53
  • $\begingroup$ Remember that an inhomogeneous Poisson point process model still assumes independence. Hence, if you deem a point process model appropriate, then you should look into Cox/cluster models or Gibbs models (where the latter works for both clustering and repulsion and would allow you to model interaction explicitly...among other covariates that may result in spatially varying intensity). $\endgroup$ – coreydevinanderson Mar 14 at 20:21

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