Significant Test for Joint Probability Distributions

Question

I am familiar with significant tests and distance measure between standard probability distributions. However, I am looking for a significance test between joint probability distributions (JPD) for a set of ecological data. (The data is discrete and categorical).

As an example, we have the following two joint probability distributions, ${P_1}$ and ${P_2}$. (Categories in rows; attributes in columns).

$${P_1}= \begin{pmatrix} 0.203 & 0.203 & 0.020 \\ 0.033 & 0.229 & 0.033 \\ 0.059 & 0.072 & 0.150 \\ \end{pmatrix} $$

$${P_2}= \begin{pmatrix} 0.159 & 0.051 & 0.025 \\ 0.080 & 0.239 & 0.051 \\ 0.040 & 0.188 & 0.167 \\ \end{pmatrix} $$

with the respective sample sizes being ${N_{P_1}=153}$; ${N_{P_2}=276}$

The respective, if not obvious, hypotheses are: ${{H_0}: {P_1}={P_2}}$ and ${{H_A}: {P_1}\neq{P_2}}$.

Is there a significance test for this situation? My assumption there is such a test (after all, there seems to be a test for everything) and that I have simply not come across yet. Any pointers would be most welcome.

(Note: I have already done a range of other tests (e.g. Chi-squared test, entropy) for each JPD. I have also calculated the Hellinger distance between each JPD. This question relates explicitly to a significance test between two JPD).

Answer 1

Treat each joint distribution as a simple categorical one with 9 levels, this way you can compare them with distances like Hellinger, Jensen-Shannon etc. and perform Chi square or G test.

This is clearly the best option for your case, if you want to consider the joint distribution in particular, without giving any weight to affine combinations of categories, like marginal distributions do.