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Testing significance of Strauss parameters in mppm model

I have a follow up question from my previous post.

Upon creating mppm models like these:

Str <- hyperframe(str=with(simba, Strauss(mean(nndist(Points)))))

fit0 <- mppm(Points ~ group, simba)
fit1 <- mppm(Points ~ group, simba, interaction=Str, 
           iformula = ~str + str:id)

Using anova.mppm to run a likelihood ratio test shows that the interaction is highly significant as a whole, but I would also like to test:

  1. whether each individual id shows significant regularity.

  2. whether some groups of ids show significantly stronger inhibition than other groups, for example, whether ids 1-7 are are significantly more regular than ids 8-10.

  3. perform pairwise comparisons of regularity between different ids.

I am aware I could build separate ppm models for each id to test for significant regularity in each id, but I am not sure this is the best approach. Also, I do not think the "summary output" with the p-values for each Strauss interaction parameter can be used for pairwise comparisons other than to the reference level.

Any advice is greatly appreciated.

Thank you!

Solution

  • First let me explain that, for Gibbs models, the likelihood is intractable, so anova.mppm performs the adjusted composite likelihood ratio test, not the likelihood ratio test. However, you can essentially treat this as if it were the likelihood ratio test based on deviance differences.

    1. whether each individual id shows significant regularity

    I am aware I could build separate ppm models for each id to test for significant regularity in each id, but I am not sure this is the best approach.

    This is appropriate. Use ppm to fit a Strauss model to an individual point pattern, and use anova.ppm to test whether the Strauss interaction is statistically significant.

    1. whether some groups of ids show significantly stronger inhibition than other groups, for example, whether ids 1-7 are are significantly more regular than ids 8-10.

    Introduce a new categorical variable (factor) f, say, that separates the two groups that you want to compare. In your model, add the term f:str to the interaction formula; this gives you the alternative hypothesis. The null and alternative models are identical except that the alternative includes the term f:str in the interaction formula. Now apply anova.mppm. Like all analyses of variance, this performs a two-sided test. For the one-sided test, inspect the sign of the coefficient of f:str in the fitted alternative model. If it has the sign that you wanted, report it as significant at the same p-value. Otherwise, report it as non-significant.

    1. perform pairwise comparisons of regularity between different ids.

    This is not yet supported (in theory or in software).

    [Congratulations, you have reached the boundary of existing methodology!]