Tightening Blocks in Complementary Analyses of Observational Studies: Optimization Algorithm and Examples

An observational block design has <i>I</i> blocks matched for covariates and <i>J</i> individuals per block, but treatments were not randomly assigned to individuals within blocks, as would have been done in an experiment. Tightening an observational block design means selecting J′&lt;J individuals from each block, and possibly I′≤I blocks, to construct a new observational block design that, in some way, addresses unmeasured biases from nonrandom treatment assignment. Tightening must preserve covariate balance while altering the design to achieve some additional objective. An optimization algorithm is introduced that achieves this while maintaining the block structure by finely balancing covariates across blocks and through optimal subset matching. An example is considered in detail, both to motivate and illustrate the tightening of an observational block design. Two tightened designs are built from a study of light daily alcohol consumption and its possible effects on HDL cholesterol. One tightened design adjusts for an outcome tentatively presuming it was unaffected by the treatment. The second tightened design uses a differential effect to remove bias from an unobserved general disposition that promotes several treatments. An R package tightenBlock implements the method, contains the data, and in that package the help-file for the function tighten reproduces the example.