Controlling for multiple covariates
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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.
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