# ANOVA with dependent variable(s) as 2-dimensional distribution? Is it possible?

Feel free to direct me to where an answer can be found, but I've banged my head against this for a while and turn to you good people:

Scenario:

I have a bunch of non-exclusive two-dimensional data (location data) for several thousand unique observations of independent subjects:

Subject1, CategoryA, TypeB, two-dimensional_distribution
Subject2, CategoryB, TypeB, two-dimensional_distribution
Subject3, CategoryA, TypeA, two-dimensional_distribution
etc...


The two dimensional distribution is quite simple containing only three possible x and y coordinates with presence(1)/absence(0) for each. For example:

C----0-----1-----1
B----1-----1-----0
A----1-----0-----0
-----X-----Y-----Z


If I combine all the obs, the fallout is something like

C---15%---62%---25%
B---10%---35%---12%
A----9%---11%----7%
-----X-----Y-----Z


How can I go about performing an ANOVA/MANOVA to look at the influence of "category" and "type" on the distributions and determine whether there is a significant difference in distributions among groups.

Tricky bit:

• The frequency distributions don't equal to 100% (as you each observation can have presence in more than one location.)

Ultimately I would like to run this through R - so if you have any ideas in that direction...

• Perhaps this problem would look simpler if you recoded the data in the form subject, categoryA, typeB, X, Y with one record for each presence. (The example would produce 5 records of the form (-,-,X,A), (-,-,X,B), (-,-,Y,B), (-,-,Y,C), (-,-,Z,C).) This amounts to a four-way table with testing for significance of the first two components (category and type). – whuber Nov 22 '11 at 17:36
• Interesting - so essentially obs1 would become 5 separate obs of subject1, obs2 would become 4 (or six or whatever) obs of subject2... On the outset, I like it... 2 questions: 1) is this really legal? and 2) it may work here, but it doesn't seen scale-able. Say if the x and y axis were larger scales (100 or 1000). Just for my theoretical understanding... – Trees4theForest Nov 23 '11 at 10:48
• "Legality" really gets to the heart of the issue: it concerns whether this way of viewing the data is appropriate for your analysis. I cannot comment on that because you haven't disclosed enough details about the nature of these observations. Scalability might be poor if you had to create physical records for each observation. However, it should suffice to record counts in each 4D cell: for each combination of subject, category, X value, and Y value, indicate the total number of times it has been observed. – whuber Nov 23 '11 at 16:01