Testing inhomogeneous patterns in spatstat

I applied the permutation test on my data to test if they are inhomogeneous, present on page. 689 of the spatstatbook. As an example of the bronze filter data. To do so, I unmarked my points and ran the two tests (I also divided my area into 6 quadrants: 2 of 20x150m and 4 of 15x150m -> total area 100x150m), which showed that my general data are more or less homogeneous (I did the test via image of the behaviors of kscaled and kinho, where both had practically the same behavior). My local tests gave locTest(T=1.3437, p-value=0.225), corrTest (T = 2.3059, p-value = 0.052), which concludes that my overall data is more or less homogeneous.

Although I have unmarked my data to do the analysis, as in the example, I have many marks (sp, and many functional traits). My question is, should I apply the permutation test for each mark type? In the case of categorical, for each level? Or does the general test alone suffice for the assumption of homogeneity?

This is not a correct interpretation of the permutation test described on pages 689-694 of the spatstat book.

If you're unsure whether a point pattern is homogeneous, the first step should be to test whether the intensity is homogeneous, using something like quadrat.test or anova.ppm.

Usually the function studpermu.test described on pages 689-694 is applied when we already know that the point pattern is not homogeneous and we're trying to decide what kind of inhomogeneity it exhibits.

It's true that, if you used studpermu.test with summaryfunction=Kest (the default) then this would perform a test of second order homogeneity (more precisely - a test of whether the different subsets of the pattern have the same K-function).

However, from your question, it appears that you followed the code used in the example on pages 689-694. That code used the bronzefilter dataset, which is very obviously inhomogeneous, and the question is what kind of inhomogeneity assumptions are appropriate.

The two tests locTest and corTest you mention were presumably carried out by calling studpermu.test with summaryfunction=Kscaled and summaryfunction=Kinhom respectively (following the code in the book). The p-value that you get for locTest is 0.225 which is not significant, meaning that the null hypothesis of a locally scaled point process is accepted. The p-value obtained for corTest is 0.052 which is very close to a formal rejection of the null hypothesis of a correlation-stationary process. Therefore, these tests suggest that your unmarked point pattern comes from an inhomogeneous, locally-scaled point process -- not a homogeneous point process.