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Generalized projection tests for function-valued parameters with applications to testing structural causal assumptions

Rui Wang, Albert Osom, Bo Zhang

Abstract

Structural assumptions are central to the causal inference literature. In practice, it is often crucial to assess their validity or to test implications that follow from them. In many settings, such tests can be framed as evaluating whether a function-valued parameter equals zero. In this paper, we propose a class of generalized projection tests based on series estimators for function-valued parameters. We establish conditions under which the proposed tests are valid and illustrate their applicability through examples from the data fusion and instrumental variables literature. Our approach accommodates flexible machine learning methods for estimating nuisance parameters. In contrast to many existing approaches, the limiting distribution of the proposed test statistics is straightforward to compute under the null hypothesis. We apply our method to test the equality of conditional COVID-19 risk across vaccine arms in the COVID-19 Variant Immunologic Landscape (COVAIL) trial.

Generalized projection tests for function-valued parameters with applications to testing structural causal assumptions

Abstract

Structural assumptions are central to the causal inference literature. In practice, it is often crucial to assess their validity or to test implications that follow from them. In many settings, such tests can be framed as evaluating whether a function-valued parameter equals zero. In this paper, we propose a class of generalized projection tests based on series estimators for function-valued parameters. We establish conditions under which the proposed tests are valid and illustrate their applicability through examples from the data fusion and instrumental variables literature. Our approach accommodates flexible machine learning methods for estimating nuisance parameters. In contrast to many existing approaches, the limiting distribution of the proposed test statistics is straightforward to compute under the null hypothesis. We apply our method to test the equality of conditional COVID-19 risk across vaccine arms in the COVID-19 Variant Immunologic Landscape (COVAIL) trial.
Paper Structure (27 sections, 23 theorems, 153 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 27 sections, 23 theorems, 153 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Testing structural causal assumptions; related work
  3. Our contribution
  4. Notation
  5. A generalized projection test for function-valued parameters
  6. A fixed-dimension projection test
  7. The generalized projection test
  8. Debiasing nuisance function estimation
  9. Hypothesis testing algorithms
  10. Theoretical properties of the proposed tests
  11. Limiting behavior under the null hypothesis
  12. Limiting behavior under fixed and local alternatives
  13. Testing structural assumptions in causal inference: two examples
  14. Testing the mean exchangeability assumption in data fusion problems
  15. Testing a compatibility condition in instrumental variable problems
  16. ...and 12 more sections

Key Result

Theorem 1

Under Assumptions Assumption: basis to Assumption: nuisance, and an additional condition that we have where $\tau_1,...,\tau_{J_n}$ are eigenvalues of

Figures (1)

  • Figure 1: Kaplan-Meier estimates of the cumulative incidence among Stage 1 Moderna participants who received an Omicron-containing vaccine (Omicron + Beta, Omicron + Delta, Omicron, Omicron + Prototype; $n = 407$ in total), Stage 1 Moderna participants who received a Prototype vaccine ($n = 97$), Stage 2 Pfizer--BioNTech participants who received an Omicron-containing vaccine (Omicron + Beta, Omicron, Omicron + Prototype; $n = 156$ in total), and Stage 2 Pfizer--BioNTech participants who received a Prototype vaccine ($n = 47$).

Theorems & Definitions (48)

  • Definition 1
  • Theorem 1: Weighted chi-square approximation of the unstandardized test statistic
  • proof
  • Remark 1: Comparison to breunig2015goodness
  • Theorem 2: Uniform Type I error control of the standardized test
  • Theorem 3
  • Theorem 4
  • Example 1: Parametric model specification
  • Example 2: Conditional covariance
  • Proposition S1
  • ...and 38 more