The work of Gary King, in particular his book "A Solution to the Ecological Inference Problem" (the first two chapters are available here), would be of interest (as well as the accompanying software he uses for ecological inference). King shows in his book how the estimates of regression models using aggregate data can be improved by examining the potential bounds lower level groupings have based on available aggregate data. The fact that your data are mostly categorical groupings makes them amenable to this technique. (Although don't be fooled, it's not as much an omnibus solution as you might hope given the title!) More current work exists, but King's book is IMO the best place to start.
Another possibility would be just to represent the potential bounds of the data themselves (in maps or graphs). So for example you may have the sex distribution reported at the aggregate level (say 5,000 men and 5,000 women), and you know this aggregate level encompasses 2 different small area units of populations 9,000 and 1,000 individuals. You could then represent this as a contingency table of the form;
Unit1 ? ? 9000
Unit2 ? ? 1000
Although you don't have the information in the cells for the lower level aggregations, from the marginal totals we can construct minimum or maximum potential values for each cell. So, in this example the
Men X Unit1 cell can only take values inbetween 4,000 and 5,000 (Anytime the marginal distributions are more uneven the smaller the interval of possible values the cells will take). Apparently getting the bounds of the table is more difficult than I expected it to be (Dobra & Fienberg, 2000), but it appears a function is available in the
eiPack library in R (Lau et al., 2007, p. 43).
Multivariate analysis with aggregate level data is difficult, as aggregation bias inevitably occurs with this type of data. (In a nutshell, I would just describe aggregation bias as that many different individual level data generating processes could result in the aggregate level associations) A series of articles in the American Sociological Review in the 1970's are some of my favorite references for the topics (Firebaugh, 1978; Hammond, 1973; Hannan & Burstein, 1974) although canonical sources on the topic may be (Fotheringham & Wong, 1991; Oppenshaw, 1984; Robinson, 1950). I do think that representing the potential bounds that data could take could potentially be inciteful, although you are really hamstrung by the limitations of aggregate data for conducting multivariate analysis. That doesn't stop anyone from doing it though in the social sciences though (for better or for worse!)
Note, (as Charlie said in the comments) that King's "solution" has recieved a fair amount of critisicm (Anselin & Cho, 2002; Freedman et al., 1998). Although these critisicms aren't per say about the mathematics of King's method, more so in regards to what situations in which King's method still fails to account for aggregation bias (and I agree with both Freedman and Anselin in that the situations in which data for the social sciences are still suspect are far more common than those that meet King's assumptions). This is partly the reason why I suggest just examining the bounds (theres nothing wrong with that), but making inferences about individual level correlations from such data takes much more leaps of faith that are ultimately unjustified in most situations.
- Anselin, L. & W.K.T. Cho (2002). Spatial effects and ecological inference. Political Analysis 10(3): 276-297.
- Dobra A. & S.E. Fienberg (2000). Bounds for cell entries in contingency tables given marginal totals and decomposable graphs. Proceedings of the National Academy of Sciences 97(22): 11885-11892
- Firebaugh, G. (1978). A rule for inferring individual relationships from aggregate data. American Sociological Review 43(4): 557-572
- Fotheringham, A.S. & D.W. Wong (1991). The modifiable areal unit problem in multivariate statistical analysis. Environment and Planning A 23(7): 1025-1044
- Freedman, D.A., S.P. Klein, M. Ostland, & M.R. Roberts (1998). Reviewed Works: A Solution to the Ecological Inference Problem by G. King. Journal of the American Statistical Association 93(444): 1518-1522. (PDF here)
- Hammond, J.L. (1973) Two sources of error in ecological correlations. American Sociological Review 38(6): 764-777
- Hannan, M.T. & L. Burstein (1974). Estimation from grouped observations. American Sociological Review 39(3): 374-392
- King G. (1997). A Solution to the Ecological Inference Problem: Reconstructing Individual Behavior from Aggregate Data. Princeton: Princeton University Press.
- Lau O., R.T. Moore & M. Kellerman (2007). eiPack: R X C Ecological Inference and Higher-Dimension Data Management. R News 7(2): 43-47
- Oppenshaw, S. (1984). The Modifiable Areal Unit Problem. Norwich: Geo Books. (PDF here)
- Robinson, W.S. (1950). Ecological correlations and the behavior of individuals. American Sociological Review 15(3): 351-357. (PDF here)