The many weak instruments problem and Mendelian randomization

Instrumental variable estimates of causal effects can be biased when using many instruments that are only weakly associated with the exposure. We describe several techniques to reduce this bias and estimate corrected standard errors. We present our findings using a simulation study and an empirical application. For the latter, we estimate the effect of height on lung function, using genetic variants as instruments for height. Our simulation study demonstrates that, using many weak individual variants, two-stage least squares (2SLS) is biased, whereas the limited information maximum likelihood (LIML) and the continuously updating estimator (CUE) are unbiased and have accurate rejection frequencies when standard errors are corrected for the presence of many weak instruments. Our illustrative empirical example uses data on 3631 children from England. We used 180 genetic variants as instruments and compared conventional ordinary least squares estimates with results for the 2SLS, LIML, and CUE instrumental variable estimators using the individual height variants. We further compare these with instrumental variable estimates using an unweighted or weighted allele score as single instruments. In conclusion, the allele scores and CUE gave consistent estimates of the causal effect. In our empirical example, estimates using the allele score were more efficient. CUE with corrected standard errors, however, provides a useful additional statistical tool in applications with many weak instruments. The CUE may be preferred over an allele score if the population weights for the allele score are unknown or when the causal effects of multiple risk factors are estimated jointly.