Evidence factors from multiple, possibly invalid, instrumental variables

Valid instrumental variables enable treatment effect inference even when selection into treatment is biased by unobserved confounders. When multiple candidate instruments are available, but some of them are possibly invalid, the previously proposed reinforced design enables one or more nearly independent valid analyses that depend on very different assumptions. That is, we can perform evidence factor analysis. However, the validity of the reinforced design depends crucially on the order in which multiple instrumental variable analyses are conducted. Motivated by the orthogonality of balanced factorial designs, we propose a balanced block design to offset the possible violation of the exclusion restriction by balancing the instruments against each other in the design, and demonstrate its utility for constructing approximate evidence factors under multiple analysis strategies free of the order imposition. We also propose a novel stratification method using multiple, nested candidate instruments, in which case the balanced block design is not applicable. We apply our proposed methods to evaluate (a) the effect of education on future earnings using instrumental variables arising from the disruption of education during World War II via the balanced block design, and (b) the causal effect of malaria on stunting among children in Western Kenya using three nested instruments.