Another Look At the Clustering Perspective On Aggregation Problems

The clustering perspective, especially in conjunctien with a linear model one, is a fruitful approach to the problems of estimating correlation and regression coefficients from aggregated data. (1) When the correlation ratios for two variables are equal, then both the within-group and aggregate correlations between the variables will equal the total correlation for microunits if either does. When the correlation ratios are not equal, then one correlation must be greater than the total correlation even if the other equals it. In either case, the typically lower within-group correlation must result in a higher aggregate correlation. (2) For regression, the effects of aggregating on the within-group covariances between the residual and independent variables can be used to determine the effect of aggregation on the regression estimates. This approach is applied to the analysis of the effects of different types of aggregation, including some not previously discussed, on estimates for models that are correctly specified with homogeneous parameters and for models that are misspecified or have different parametric values for different subsets of the population. Contrary to previous conclusions, the effect of aggregation bias can be counter to that of specification bias in some quite probable cases.