PaperlayerView exact revision on arXiv

Source-linked AI summary

Instrumental Variables Estimation with Some Invalid Instruments and its Application to Mendelian Randomization

Hyunseung Kang, Anru Zhang, T. Tony Cai, Dylan S. Small

arXiv:1401.5755v3stat.ME

TL;DR

Conventional instrumental-variable estimation requires knowing which instruments are valid, a difficult requirement in Mendelian randomization because genetic instruments may have pleiotropic effects. The paper develops sisVIVE to estimate causal effects without that knowledge and shows that identification is possible when fewer than half the instruments are invalid. Simulations and a Mendelian randomization analysis support sisVIVE as a robust alternative to TSLS when instruments may be invalid.

  • Problem

    Instrumental-variable estimation requires validity knowledge that is often incomplete in Mendelian randomization because genetic markers may have pleiotropic effects.

  • Method

    The paper develops sisVIVE, a fast penalized ℓ1 estimation procedure for causal effects with possibly invalid instruments and without a predefined set of valid instruments.

  • Results

    Simulations and data analysis show that sisVIVE is a robust alternative to TSLS when instruments may be invalid.

  • Takeaways & Limitations

    Knowing only that a proportion of instruments is valid, without knowing which ones, can be sufficient for causal-effect identification and estimation when fewer than half are invalid.

  • Takeaways & Limitations

    The authors note that when constants differ across sets of valid instruments, their theorem does not apply and what sisVIVE estimates remains unclear.

Abstract

from arXiv · show

Instrumental variables have been widely used for estimating the causal effect between exposure and outcome. Conventional estimation methods require complete knowledge about all the instruments' validity; a valid instrument must not have a direct effect on the outcome and not be related to unmeasured confounders. Often, this is impractical as highlighted by Mendelian randomization studies where genetic markers are used as instruments and complete knowledge about instruments' validity is equivalent to complete knowledge about the involved genes' functions. In this paper, we propose a method for estimation of causal effects when this complete knowledge is absent. It is shown that causal effects are identified and can be estimated as long as less than $50$% of instruments are invalid, without knowing which of the instruments are invalid. We also introduce conditions for identification when the 50% threshold is violated. A fast penalized $\ell_1$ estimation method, called sisVIVE, is introduced for estimating the causal effect without knowing which instruments are valid, with theoretical guarantees on its performance. The proposed method is demonstrated on simulated data and a real Mendelian randomization study concerning the effect of body mass index on health-related quality of life index. An R package \emph{sisVIVE} is available online.

1 INTRODUCTION

Mendelian randomization often lacks complete knowledge of genetic instruments’ validity because pleiotropy and other biological phenomena can violate instrumental-variable assumptions. The paper shows that causal effects can still be identified and estimated when fewer than half of instruments are invalid, without knowing which instruments are invalid, and introduces sisVIVE for estimation.

  • Motivation: Valid instruments must relate to the exposure, have no direct pathway to the outcome, and be unrelated to unmeasured variables affecting exposure and outcome.
  • Motivation: Pleiotropy, linkage disequilibrium, and population stratification can violate instrumental-variable assumptions in Mendelian randomization.
  • Motivation: Incomplete knowledge of genetic markers’ biological functions makes it difficult to verify instrument validity, motivating methods that tolerate possible invalid instruments.
  • Prior limitations: Earlier approaches either required at least one known valid instrument or imposed restrictive structure on direct effects, limiting their applicability to Mendelian randomization.
  • Contributions: Less than 50% invalid instruments is sufficient for identification and estimation without knowing which instruments are invalid; with four instruments, at most one may be invalid.
  • Contributions: The paper develops sisVIVE, a fast penalized ℓ1 procedure with theoretical estimation guarantees, and evaluates it in simulations and a BMI–HRQL Mendelian randomization study.

VARIABLES

The paper models observed outcomes and exposures using instruments, potential outcomes, direct instrument effects, confounding, and a causal exposure effect. Instrument validity is represented through the support of α∗, while the framework also allows covariates, nonlinear transformations, and heterogeneous effects.

  • Observed variables: Observed data comprise n independent outcome, exposure, and instrument vectors, with Z an n by L instrument matrix.The framework defines standard vector and matrix notation, including projections and residual projections.
  • Model: The ALICE framework defines β∗ as the causal effect of changing exposure by one unit.φ∗ captures direct instrument effects, while ψ∗ represents confounding effects involving instruments and outcomes.
  • Instrument validity: α∗ combines direct instrument effects and confounding effects, so α∗=0 corresponds to instruments without either effect.An instrument is valid when its coefficient is outside supp(α∗), and invalid when its coefficient is in that support.
  • Extensions: The model can residualize outcomes, exposures, and instruments on measured covariates before applying the paper’s results.It can also incorporate multiple valid or invalid instruments, basis transformations for nonlinear models, and heterogeneous effects under additional conditions.
  • Instrument validity: For multiple instruments, validity is defined relative to the full set of instruments being considered.An instrument may be valid in one considered set even when instruments are associated with or causally affect another instrument.

3 ESTIMATION OF CAUSAL EFFECT

The paper studies identification and estimation of causal effects when the identities of invalid instruments are unknown. It establishes consistency-based identification conditions, a guaranteed threshold below 50% invalid instruments, and the sisVIVE estimator with a fast equivalent algorithm.

  • 3.1 Identifiability of Model: Identification requires a unique solution for α∗ and β∗ from observed instrument–outcome and instrument–exposure moments.These moments identify Γ∗ and γ∗, after which α∗ and β∗ are identified when the mapping to the observed quantities is unique.
  • 3.1 Identifiability of Model: At least one known valid instrument is sufficient for a unique β∗, but knowing only that valid instruments exist does not guarantee identification.When the valid set is unknown, multiple values of β∗ can satisfy the moment equation.
  • 3.1 Identifiability of Model: An upper bound U on the number of invalid instruments can identify α∗ and β∗ without identifying the invalid instruments individually.Theorem 1 requires all candidate valid subsets of size L−U+1 to produce the same causal-effect value q_m.
  • 3.1 Identifiability of Model: When the 50% condition is violated, identification can still hold if candidate valid subsets yield consistent causal-effect estimates.Different subset estimates, such as q1=1 and q2=2, violate the criterion and produce no unique solution.
  • 3.1 Identifiability of Model: 50%: if U ≤ L/2, the consistency criterion is automatically satisfied and equation (7) has a unique solution.This guarantees identification when fewer than half of the candidate instruments are invalid, regardless of γ∗ and Γ∗ values.
  • Estimation and computation: sisVIVE uses an equivalent penalized ℓ1 formulation and a two-step algorithm whose solution is identical to sisVIVE and practically as fast as ordinary least squares.The first step is a standard Lasso problem, and LARS solves the penalized problem across tuning parameters efficiently.

4 SIMULATION STUDY

The simulations evaluate sisVIVE against OLS and TSLS under varying endogeneity, instrument strength, correlation, and numbers of invalid instruments. sisVIVE generally outperforms naive TSLS and approaches oracle TSLS when fewer than half the instruments are invalid.

  • Varying endogeneity: When endogeneity is small, OLS has lower estimation error than the other methods, especially with weak instruments.With sufficient endogeneity, oracle TSLS performs best, while sisVIVE remains close to oracle TSLS; their gap grows as absolute instrument strength weakens.
  • Varying endogeneity: Naive TSLS has high error regardless of instrument strength because it ignores bias from invalid instruments.It treats all instruments as valid, unlike sisVIVE, which estimates effects while allowing some instruments to be invalid.
  • Varying invalid instruments: At s = 0, sisVIVE is nearly identical to naive and oracle TSLS, while oracle TSLS performs much better than the other methods for any s.SisVIVE deteriorates slightly with weak absolute strength as instrument correlation increases.
  • Varying invalid instruments: When s < L/2 = 5, sisVIVE performs comparably to oracle TSLS and better than naive TSLS.At the identification boundary, its performance becomes similar to naive TSLS regardless of absolute and relative instrument strength.
  • Overall comparison: Cross-validated sisVIVE generally outperforms naive TSLS and is comparable to oracle TSLS.It also performs better than sisVIVE using the tuning parameter selected by Corollary 2 in most cases.

5 DATA ANALYSIS

The data analysis applies sisVIVE to estimate the effect of censored BMI on health-related quality of life using obesity-associated SNPs from the Wisconsin Longitudinal Study. SisVIVE agrees with TSLS when no instrument is suspected and excludes rs6265 when pleiotropy makes it suspect.

  • Data and measures: The candidate instruments are rs1421085, rs1501299, and rs2241766, previously associated with obesity.The authors are uncertain about validity because they lack complete knowledge of the SNPs’ biological functions.
  • Three-instrument analysis: −0.013 (SE: 0.0010) is the OLS estimate for the decrease in HUI3 associated with a one-unit increase in censored BMI.This agrees with prior evidence that obese individuals have lower HUI3 scores.
  • Three-instrument analysis: −0.00019 is the causal-effect estimate from both TSLS and sisVIVE using the three candidate instruments.SisVIVE selects no SNP as invalid and requires at least two of the three SNPs to be valid under the stated assumption.
  • Four-instrument analysis: 0.00091 (SE: 0.022) is the four-instrument TSLS estimate, compared with −0.00011 from sisVIVE after adding rs6265.SisVIVE excludes rs6265, which may affect health-related quality of life through mechanisms other than obesity.
  • Overall comparison: SisVIVE matches TSLS in the first analysis and removes the suspected instrument in the second, remaining robust to possibly invalid instruments.TSLS assumes all instruments are valid in both analyses.

6 DISCUSSION

The paper argues that causal-effect estimation remains possible with possibly invalid instruments, without knowing which instruments are valid, and proposes sisVIVE as an alternative to TSLS. It also identifies limitations involving treatment-effect heterogeneity and strict conditions for accommodating more invalid instruments.

  • Knowing only that a proportion of instruments is valid can suffice for estimation without identifying which instruments are valid.
  • sisVIVE dominates naive TSLS in almost every aspect while performing similarly to oracle TSLS.
  • Simulation and data analysis show that sisVIVE is a robust alternative to TSLS with possibly invalid instruments.
  • The current model assumes constant treatment effects, leaving treatment-effect heterogeneity for future work.
  • Future work includes relaxing Corollary 2 conditions to allow more invalid instruments and deriving identification tests.

A.1 Numerical Example

The numerical examples illustrate that identification depends on whether candidate valid-instrument subsets produce consistent causal-effect solutions. Small perturbations can change identification status, while the structural-parameter result follows from uniqueness under the consistency criterion.

  • With four instruments and γ∗=(1, 2, 3, 4), two candidate valid sets yield different q values, so identification is not possible.
  • Perturbing Γ∗ to (1, 2, 6, 8 + ϵ) leaves one identifying subset for any ϵ>0, despite arbitrarily small distance from the unidentified case.
  • Γ∗=(1, 2, 7, 9) has one identifying subset and remains identifiable under small perturbations.
  • Γ∗=(1, 2, 6, 8) lies at an identification boundary where small perturbations can change whether the model is identified.
  • Under the modeling assumptions, structural parameters are identified if and only if the consistency criterion holds.

B.1 Values of ρ and µ

The supplementary simulations assess whether Corollary 2’s conditions hold and examine sisVIVE under alternative instrument-correlation structures. The conditions are often strict, while estimator performance depends strongly on endogeneity and the proportion of invalid instruments.

  • Corollary 2 requires conditions based on ρ and µ, including the upper bound min(1/(12µ), 1/(10ρ^2)) on s.
  • In most settings, Corollary 2 is satisfied only when s=0, so it cannot characterize sisVIVE when invalid instruments are present.
  • Theorem 2 generalizes Corollary 2’s result at the expense of stronger requirements.
  • When endogeneity is sufficient, oracle TSLS performs best, sisVIVE resembles it, and naive TSLS performs worst.
  • When s<L/2=5, sisVIVE is comparable to oracle TSLS and better than naive TSLS; at the identification boundary, it becomes similar to naive TSLS.
  • sisVIVE’s median absolute deviation is as small as oracle TSLS when s<5 but as large as naive TSLS when s≥5.
  • Correctly selecting invalid instruments closely corresponds to estimation accuracy, whereas incorrectly excluding valid instruments mainly reduces efficiency.

B.4 Varying Instrument Strength

The simulations vary instrument-strength patterns and show that sisVIVE generally matches or outperforms naive two-stage least squares, often approaching oracle performance. Performance degrades most when invalid instruments are stronger, although it remains no worse than naive two-stage least squares.

  • Under strong instruments, sisVIVE performs as well as the oracle, while under weak instruments its relative performance depends on the strength pattern.
  • sisVIVE performs much better than naive two-stage least squares when valid instruments are stronger than invalid instruments, in both strong and weak settings.
  • sisVIVE’s estimation error is more closely related to correctly selecting invalid instruments than valid instruments.
  • sisVIVE performs better than naive two-stage least squares in the vast majority of simulation cases and often similarly to the oracle.
  • When invalid instruments are stronger, sisVIVE deviates from the oracle despite identification being guaranteed at s = 4, but remains no worse than naive two-stage least squares.
  • The Corollary 2 condition on s is strict in these simulations and may not hold even when sisVIVE remains empirically competitive.

B.6 Choice of λ

This section compares cross-validation and theory-based choices of λ for sisVIVE. Cross-validation generally performs better, especially when overall instrument strength is weak, although theory offers the stated formal basis.

  • Cross-validation usually selects a smaller λ than Theorem 2, leaving more nonzero estimated coefficients and declaring more instruments invalid than valid.
  • In most strong-instrument settings, sisVIVE with cross-validated λ performs just as well as with theory-based λ.
  • For weak instruments, cross-validated λ performs better than the theory-based choice in nearly all examined strength patterns, sometimes drastically better.
  • Cross-validated λ generally yields better sisVIVE estimation performance than the theory-based λ, especially for weak instruments.
  • The empirical advantage of cross-validation is not strongly supported by theory because little theory justifies cross-validation for ℓ1 penalization settings.
  • An alternative λ-selection procedure is proposed for the always-identified region, but its investigation is left for future research.

C Additional Discussion about Theorem 2

The discussion compares sample-size requirements for RIP and MIP conditions relevant to Theorem 2. RIP can hold in regimes where the stronger MIP condition does not.

  • Theorem 2 uses a restricted-isometry-type condition, while Corollary 2 uses the mutual incoherence property condition.
  • The RIP condition implies the MIP condition, but the converse does not necessarily hold.
  • For iid standard Normal instrument matrices, n ≥ C s log(L/s) ensures the RIP condition under the stated result.
  • When n is between s log(L/s) and s^2 log L, the instrument matrix can satisfy RIP with high probability without satisfying MIP.

D Wisconsin Longitudinal Data

The Wisconsin Longitudinal Study analysis uses genetic instruments to estimate reduced-form relationships for BMI and HUI3. With three or four SNPs, overidentification tests do not reject the hypothesis that the instruments are valid.

  • The analysis uses Wisconsin Longitudinal Study data and genetic instruments, with BMI and HUI3 as dependent variables in reduced-form OLS estimates.
  • For three SNPs, the Sargan test gives χ^2 = 0.12 and p-value = 0.94, retaining the null hypothesis that all instruments are valid at the 0.05 level.
  • For four SNPs, the Sargan overidentification test gives χ^2 = 2.49 and p-value = 0.48.
  • The proof establishes uniqueness of β* when α* is unique, using the moment equation and the corresponding parameter representations.
  • When s < U, the proof uses the fact that there are more than L − U valid instruments to construct subsets supporting the subspace condition.
  • The subspace restriction condition holds automatically whenever U ≥ L/2.

E.3 Proof of Theorem 2

The proof reinterprets the estimator as a two-step procedure and derives error bounds using Lasso inequalities and restricted-isometry properties. It also notes a possible multi-endogenous-variable generalization that is not pursued for clarity.

  • Two-step estimator: The proof first analyzes the Lasso error h = α̂_λ − α* and then uses that result to bound the second estimation step.This decomposition separates sparse instrument-related estimation from the subsequent causal-effect estimate.
  • Error bounds: Theorem 2 bounds the largest-s coordinates of the Lasso error by λ√s divided by a restricted-isometry expression.The displayed bound involves δ^-_2s(Z), δ^+_2s(Z), and δ^+_2s(P_ḊZ).
  • Error bounds: Restricted orthogonal constants and a sparse-vector decomposition convert the bound on h_max(s) into a bound on ∥P_ḊZh∥_2.The proof represents the remainder as a weighted mean of s-sparse vectors before applying the restricted-isometry relations.
  • Scope: The proof could generalize from a vector D to multiple endogenous variables, but the paper does not explore that extension.The restriction is presented as a choice made for clarity of presentation.
  • Conditions: The resulting bounds require a positive denominator involving restricted-isometry constants and impose an additional sparsity constraint on s.The proof also states that the bound can potentially be improved by handling these constraints more carefully.
  • Two-step estimator: The original estimator is equivalent to a Lasso problem with outcome P_Ḋ⊥P_ZY and design matrix P_Ḋ⊥Z.The first objective term can be made zero by choosing β appropriately, leaving the Lasso minimizer unchanged.

F Figures

The figures evaluate instrument-selection proportions and median absolute estimation error across endogeneity, instrument strength, invalid-instrument count, and correlation structures. Their layouts vary the strength and correlation conditions while using repeated simulations to summarize performance.

  • Estimation performance: Figures 4–7 report median absolute estimation error after 500 simulations for ten instruments under within-group or between-group correlation.Figures 4 and 6 vary endogeneity with three invalid instruments, while Figures 5 and 7 vary the invalid-instrument count.
  • Experimental factors: Across the figure panels, columns vary absolute and relative instrument strength, including Strong versus Weak and Equal versus Variable settings.Relative strength is varied through γ* while holding absolute strength fixed.
  • Instrument selection: Figures 8–9 report average proportions of correctly selected valid and invalid instruments for ten instruments when correlation exists between all instruments.Figure 8 varies endogeneity with s = 3, whereas Figure 9 varies s.
  • Instrument selection: Figures 10–13 evaluate correct-selection proportions under correlations restricted within valid or invalid groups or occurring between those groups.These figures use ten instruments, fix s = 3 when varying endogeneity, and otherwise vary the number of invalid instruments.
Loading 1401.5755v3…