Agglomerative Hierarchical Clustering for Selecting Valid Instrumental Variables

Abstract

We propose an instrumental variable (IV) selection procedure which combines the agglomerative hierarchical clustering method and the Hansen-Sargan overidentification test for selecting valid instruments for IV estimation from a large set of candidate instruments. Some of the instruments may be invalid in the sense that they may fail the exclusion restriction.We show that under the plurality rule, our method can achieve oracle selection and estimation results. Compared to the previous IV selection methods, our method has the advantages that it can deal with the weak instruments problem effectively, and can be easily extended to settings where there are multiple endogenous regressors and heterogenous treatment effects. We conduct Monte Carlo simulations to examine the performance of our method, and compare it with two existing methods, the Hard Thresholding method (HT) and the Confidence Interval method (CIM). The simulation results show that our method achieves oracle selection and estimation results in both single and multiple endogenous regressors settings in large samples when all the instruments are strong. Also, our method works well when some of the candidate instruments are weak, outperforming HT and CIM. We apply our method to the estimation of the effect of immigration on wages in the US.

Using clustering of genetic variants in Mendelian randomization to interrogate the causal pathways underlying multimorbidity

“…For a single SNP 𝐺 𝑗 , let 𝛽 𝑗 be its Wald ratio estimate for outcome 𝑌 1 , and 𝜃 𝑗 for 𝑌 2 . The extension of Algorithm 1 and 2 can be summarized as extending two quantities to allow for both 𝛽 𝑗 , 𝜃 𝑗 : the weighted squared Euclidean distance defined in (12), and the Q statistic defined in (10). Let the covariance between the two SNP-specific estimates be 𝐶𝑜𝑣( 𝛽 𝑗 , 𝜃 𝑗 ) = 𝜌 and assume that 𝜌 is the same across all 𝑗 = 1, ...., 𝑝.…”

“…This motivates the aforementioned extension of MR-AHC, which classifies the SNPs into distinct clusters based on their SNP-specific causal estimates for both outcomes. and the Q statistic defined in (10). Let the covariance between the two SNP-specific estimates be 𝐶𝑜𝑣( 𝛽 𝑗 , 𝜃 𝑗 ) = 𝜌 and assume that 𝜌 is the same across all 𝑗 = 1, ...., 𝑝.…”

“…In the presence of mechanistic heterogeneity, an important observation is that the causal estimates obtained with all SNPs will be heterogeneous, and only SNPs belonging to the same causal pathway have similar, or clustered, causal estimates. To discover such variant clusters that drive distinct causal mechanisms, we extend the Agglomerative Hierarchical Clustering (AHC) method in Apfel and Liang [10], as proposed in the field of econometrics to detect is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) (iii) weight. We also propose a two-step extension of the method to account for outliers in the Wald estimates, which further improves the clustering accuracy.…”

“…In the presence of mechanistic heterogeneity, an important observation is that the causal estimates obtained with all SNPs will be heterogeneous, and only SNPs belonging to the same causal pathway have similar, or clustered, causal estimates. To discover such variant clusters that drive distinct causal mechanisms, we extend the Agglomerative Hierarchical Clustering (AHC) method in Apfel and Liang [10], as proposed in the field of econometrics to detect instrument clusters, from the individual-level data setting to the two-sample summary data MR setting, which we call ‘MR-AHC’. Specifically, we propose an adjusted version of Ward’s minimum variance criterion for choosing the pair of clusters to merge at each step to account for the uncertainty of the clustering objects (i.e.…”

“…We propose to achieve this aim by adapting the AHC method developed in Apfel and Liang [10] to the two-sample summary data MR, called 'MR-AHC', which is introduced in the next section.…”