# Regression With Mean Rates?

Today, I was looking at the National Census data in my country. In my country, we have something very similar to "States" (Divisions) and "ZIP Codes" (Subdivisions).

During the Census, data is collected from each Subdivision and - for example, based on a 20% random sample in each Subdivision, information is collected such income, employment rate, age, etc. However, when this data is published, we can only see the average and median value for each of these variables within each Subdivision. We also know the number of people within each Subdivision. The data would look something like this:

  Division Subdivision Average_Income Median_Income Average_Age Median_Age Subdivision_Population
1        A           1       51265.41      48585.45          52         42                  10698
2        A           2       51067.56      49606.41          46         62                   9781
3        A           3       48662.40      49297.64          34         37                  10217
4        B           1       50097.79      49819.19          53         40                  10257
5        B           2       50855.28      49005.76          70         53                   9912
6        C           1       51002.73      49697.60          67         50                  10225
7        C           2       48742.88      50573.90          49         52                   9871
8        C           3       50831.21      50410.04          38         40                   9945
9        C           4       50649.13      46939.21          71         63                  11214



I was interested in learning about if Regression Models can still be applied in such a context (i.e. when individual subject-level data is not provided and only the aggregate level data is provided). As an example, study the impact of income on age.

Initially, I had thought that perhaps I could just "get away with" using a standard Linear Regression Model as if I had access to the individual subject-level data. However, when reading more about this topic, I learned of a new topic called "Ecological Statistics" (the name was a bit misleading - I thought "Ecological Statistics" was about Statistics applied in the Environment). I found some interesting references which show why a standard Linear Regression Model can NOT be used in such a context:

In general, I was interested in learning more about what types of Statistical and Regression Models can be used for problems such as mine. For example, in the problem I described, I do NOT have access to the Standard Errors - is it still possible to create a Regression Model for this problem and generate meaningful estimates for the beta-coefficients? Or could I still fit a regression model to the entire data and find the mean effect of income on age?

References:

Note : I just realized - with this approach, it is likely impossible to create a random-effects nodel because we only have one observation per group!

• You might be interested to read about ecological fallacy as well. Commented Jan 12, 2023 at 9:42
• It might also be worth asking yourself: If I did have data about age and income at the household level, would I be able to say something meaningful about "what is the effect of Age on Income"? Commented Jan 12, 2023 at 10:01
• You might also be interested in the atomistic fallacy. Commented Jan 14, 2023 at 7:20
• Technically, we can create (a regression) model for any tabular data or data frame within mean rates better on the data sequences. Apart from Note, I found some workaround over Census Data Analysis, which could make sense for your question and reach meaningful outputs. Commented Jan 14, 2023 at 10:52
• "if Regression Models can still be applied ... as an example, study the impact of income on age." You can always apply a regression model, but the interpretation of the model may differ. Regression models are not the same as causal models and your data sample is not the same as a direct relationship. What do you mean with 'the impact'? Commented Jan 20, 2023 at 11:32