  A fundamental problem in statistics is to compare the outcomes attained by
members of subpopulations. This problem arises in the analysis of randomized
controlled trials, in the analysis of A/B tests, and in the assessment of
fairness and bias in the treatment of sensitive subpopulations, especially when
measuring the effects of algorithms and machine learning. Often the comparison
makes the most sense when performed separately for individuals who are similar
according to certain characteristics given by the values of covariates of
interest; the separate comparisons can also be aggregated in various ways to
compare across all values of the covariates. Separating, segmenting, or
stratifying into those with similar values of the covariates is also known as
"conditioning on" or "controlling for" those covariates; controlling for age or
annual income is common.
  Two standard methods of controlling for covariates are (1) binning and (2)
regression modeling. Binning requires making fairly arbitrary, yet frequently
highly influential choices, and is unsatisfactorily temperamental in multiple
dimensions, with multiple covariates. Regression analysis works wonderfully
when there is good reason to believe in a particular parameterized regression
model or classifier (such as logistic regression). Thus, there appears to be no
extant canonical fully non-parametric regression for the comparison of
subpopulations, not while conditioning on multiple specified covariates.
Existing methods rely on analysts to make choices, and those choices can be
debatable; analysts can deceive others or even themselves. The present paper
aims to fill the gap, combining two ingredients: (1) recently developed
methodologies for such comparisons that already exist when conditioning on a
single scalar covariate and (2) the Hilbert space-filling curve that maps
continuously from one dimension to multiple dimensions.
