  Databases in domains such as healthcare are routinely released to the public
in aggregated form. Unfortunately, naive modeling with aggregated data may
significantly diminish the accuracy of inferences at the individual level. This
paper addresses the scenario where features are provided at the individual
level, but the target variables are only available as histogram aggregates or
order statistics. We consider a limiting case of generalized linear modeling
when the target variables are only known up to permutation, and explore how
this relates to permutation testing; a standard technique for assessing
statistical dependency. Based on this relationship, we propose a simple
algorithm to estimate the model parameters and individual level inferences via
alternating imputation and standard generalized linear model fitting. Our
results suggest the effectiveness of the proposed approach when, in the
original data, permutation testing accurately ascertains the veracity of the
linear relationship. The framework is extended to general histogram data with
larger bins - with order statistics such as the median as a limiting case. Our
experimental results on simulated data and aggregated healthcare data suggest a
diminishing returns property with respect to the granularity of the histogram -
when a linear relationship holds in the original data, the targets can be
predicted accurately given relatively coarse histograms.
