Measuring homogeneity across different spatial aggregations of data I'm working with dataset of individual households that I aggregate into 'areas' using several different spatial configurations, from smaller to bigger.
These areas are then characterized by four variables (two categorical, two continuous).
I'd like to see what effects these different aggregations have on the dataset. Particularly, I'd like to estimate what the differences in homogeneity are when I move from one spatial resolution to another.
What would be the best way to approach this problem? 
Is there any measure I could use for this purpose?
 A: There are many ways you can characterize homogeneity, so there could be many answers to your question. One of the most intuitive ways I have seen it displayed is in a book chapter, "Spatial Analysis of Regional Income Inequality" by Sergio Rey in the book Spatially Integrated Social Science (PDF). The approach Rey takes in that chapter is to visualize the change in a metric called Theil's Index. Particularly this is intutitive as the Theil index can be broken down into the "between" unit variation and the "within" unit variation. Subsequently Rey examines the change in the components of Theil's index between different census aggregations across time. (As a note, I find Rey's notation of the Theil index far easier to follow than the Wikipedia page)
This metric is only applicable to continuous variables, so a different approach would be necessary for the categorical variables. A prolific listing of commonly used indices to measure racial segregation are provided in this paper (Massey and Denton, 1988). All of those metrics can be used with categorical variables. Ones I have come across in Criminology/Sociology are the index of qualitative variation and diversity indices. 
A: Homogeneity Definition: To start, let's define homogeneity as the degree to which households grouped in same area are like one another for some attribute.
MAUP Approach
Paraphrasing the stated problem: We are uncertain how homogeneity changes as we decrease the spatial resolution of the design of how we group households into areas. 
For this problem @AndyW answer is solid. In the field of geography, your problem can be classed within the modifiable areal unit problem (MAUP).You can search the index for 'MAUP' at this site. 
Alternative Clustering Approach
An alternative problem: Given that we want to maximise areal homogeneity when aggregating households, we are uncertain of the optimal spatial configuration of how we should group the households. 
With the p-regions clustering algorithm, you can visually explore different structures of homogeneity within your data by creating different maps of household areas by playing with these 2 parameters: 


*

*changing the different attributes
for maximising area homogeneity

*changing the number of households
required to build an areal group

