I would like to perform an individual-level multivariate analysis at small levels of geographic aggregation (Australian census collection districts). Clearly, the census isn't available at these small levels of aggregation for privacy reasons so I am investigating other alternatives. Almost all the variables of interest are categorical. I have two datasets at my disposal:

  • The 1% census sample is available at a much greater level of spatial aggregation (an area with a population of ~190,000 and vast spatial segregation of demographics).

  • Frequency tables for the variables I am interested in at the small area level (500 small areas, mean pop = 385, sd = 319, median = 355).

How can I use these two datasets to simulate a population distribution at the small area level that is as close as possible to the actual population of the small area?

I appreciate that there may well be routine methods for doing this; if so a pointer to a textbook or relevant journal articles would be vastly appreciated.

  • possibly related (I’m having a similar problem): stats.stackexchange.com/questions/14399/… Gibbs sampling might be what is needed here. – mzuba Sep 20 '11 at 8:32
  • You might want to ask your question on SRMSNET mailing list of the American Statistical Association. If you are in Australia, I would approach Ray Chambers -- I guess nobody knows SAE better than him in the Southern hemisphere :). – StasK Sep 20 '11 at 15:39
  • This problem is closely related to "dasymetric mapping." – whuber Sep 20 '11 at 22:36
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    I agree with @whuber, and dasymetric mapping may be of interest to fmark given the subject material as well. Unfortunately, it is largely separate from the ecological inference literature I cited in my answer (I don't want to pile on more literature more!) What do you think fmark? – Andy W Sep 21 '11 at 0:29
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    Some dasymetric mapping techniques have begun to use ancillary data to attempt to interpolate data to smaller areas. The goals of ecological inference and dasymetric mapping are somewhat different (somewhat analogous to the difference between prediction/forecasting and inference). I will write another post on those sources I have collected that I think would be of interest as well. Unfortunately I can't give much more useful advice than cite a bunch of literature. It is a popular contemporary topic, and hopefully you can contribute to it! – Andy W Sep 21 '11 at 12:40
up vote 5 down vote accepted

Dasymetric mapping is mainly focused on interpolating population estimates to smaller areas than available in currently disseminated data (see this question for a host of useful references on the topic). Frequently this was done by simply identifying areas (based on land characteristics) in which obviously no population exists, and then re-estimating population densities (ommitting those areas). An example might be if there is a body of water in a city, another might be if you identify industrial land parcels which can not have any residential population. More recent approaches to dasymetric mapping incorporate other ancillary data in a probabilistic framework to allocate population estimates (Kyriakidis, 2004; Liu et al., 2008; Lin et al., 2011; Zhang & Qiu, 2011).

Now it is easy to see the relation to your question at hand. You want the population estimates of the small areas. But, it should also be clear how it may fall short of your goals. You not only want the population data, but characteristics of those populations as well. One of the terms used to describe this situation is the change of support problem (Cressie, 1996; Gotway & Young, 2002). Borrowing from the geostatistical literature in which one tries to make predictions of a certain characteristic over a wide area from point samples, recent work has attempted to interpolate areal data to different target zones. Much of the work of Pierre Goovaerts focuses on such area-to-point kriging methods, a recent article in the journal Geographical Analysis has several examples of the method applied different subject materials (Haining et al., 2010), and one of my favorite applications of it is in this article (Young et al., 2009).

What I cite should hardly be viewed as a panacea to the problem though. Ultimately many of the same issues with ecological inference and aggregation bias apply to the goals of areal interpolation as well. It is likley many of the relationships between the micro level data are simply lost in the aggregation process, and such interpolation techiques will not be able to recover them. Also the process through which the data is empirically interpolated (through estimating variograms from the aggregate level data) is often quite full of ad-hoc steps which should make the process questionable (Goovaerts, 2008).

Unfortunately, I post this in a separate answer as the ecological inference literature and the literature on dasymetric mapping and area-to-point kriging are non-overlapping. Although the literature on ecological inference has many implications for these techniques. Not only are the interpolation techniques subject to aggregation bias, but the intelligent dasymetric techniques (which use the aggregate data to fit models to predict the smaller areas) are likely suspect to aggregation bias. Knowledge of the situations in which aggregation bias occurs should be enlightening as to the situations in which areal interpolation and dasymetric mapping will largely fail (especially in regards to identifying correlations between different variables at the disaggregated level).


Citations

  • Thanks for the useful starting point into the contemporary literature - I wasn't aware of dasymetric methods that did more than reweight population densities, so I'll look into this eagerly. – fmark Sep 21 '11 at 23:11

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;

       Men     Women
Unit1   ?        ?    9000
Unit2   ?        ?    1000
       5000   5000 

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.


Citations

  • Note that others have been critical of King's approach to the ecological fallacy issue; David Freedman is a notable example. Here is the response that Freedman and his coauthors give to King's book cited above: citeseerx.ist.psu.edu/viewdoc/… Of course, King has a response and Freedman et al. have a response to the response to the response... I can't quite figure out what you're trying to do and what data you have, but I am generally very skeptical of ecological inference-type analyses. – Charlie Sep 20 '11 at 18:48
  • Yes @Charlie I agree (and I particularly enjoy Freedman's take on the issue in general). That is partly the reason I point to the general literature on aggregation bias at the end of my post. I'm not quite sure what you mean by the statement "I can't quite figure out what you're trying to do and what data you have, but I am generally very skeptical of ecological inference-type analyses", is this in regards to the King and Freedman both complaining about not sharing data? – Andy W Sep 20 '11 at 18:51
  • @Andy, Is this technique the same to what economists know as partially identified distributions (springer.com/statistics/statistical+theory+and+methods/book/…)? – StasK Sep 20 '11 at 19:25
  • @Andy, No, sorry for the ambiguity. I was actually speaking to the OP. If he has frequency tables at the small area and wants to get statistics at the small area level, what is missing? I guess that he must have just the margins, but not the cell contents, as you suggest in your post. – Charlie Sep 20 '11 at 19:26
  • @StasK, I don't know. I will check to see if King makes any reference to Manski later this week when I have access to the book. There is likely some overlap given ecological inference is mentioned in the abstract. Another potential (free) source to explore the association between the two may be the reader King edited on "New Methods in Ecological Inference" (posted in entirety on his website) – Andy W Sep 20 '11 at 19:37

I am not sure a well-defined answer exists in the literature for this, given that Google search gives basically three usable references on multivariate small area estimation. Pfeffermann (2002) discusses discrete response variables in section 4 of the paper, but these will be univariate models. Of course, with hierarchical Bayesian methods (Rao 2003, Ch. 10), you can do any sort of wonders, but if in the end you find yourself just replicating your priors (because you have so little data), this would be a terrible outcome of your simulation exercise. Besides, Rao only treats continuous variables.

I guess the biggest challenge will be the decomposition of the covariance matrix into the between- and within-small-area components. With 1% sample, you will only have 3 observations from your SAE, so it might be hard to get a stable estimate of the within-component.

If I were in your shoes, I would try a multivariate extension of Pfeffermann's model with a multivariate random effect of the small area. You may indeed end up with a hierarchical Bayesian model for this, if nothing design-based works.

UPDATE (to address Andy's comment to this answer): the bootstrap methods for small area estimation (Lahiri 2003) specifically recreate a plausible population from the study. While the focus of the bootstrap exercise is to estimate the variances of the small area estimates, the procedures should be of interest and relevance to the posted problem.

  • I wouldn't evaluate the status of a literature based on a google search. I'm not quite sure small area estimation is what the author would be looking for in this instance. As far as I understand, that literature is focused on either making predictions of characteristics in small areas (Kriegler & Berk, 2010), or estimating parameters in multi-level models based on a sparse number of samples per aggregate unit. – Andy W Sep 20 '11 at 17:29
  • I'm not sure small area estimation is what I want to do. As I understand it, small area estimation seeks to go from a few samples in a small area to aggregate summary statistics. I'm seeking the opposite (to go from a large area population distribution and small area aggregate summary statistics to a simulated small area population). Lahiri 2003 looks like a good starting point here. – fmark Sep 21 '11 at 0:51

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