Introduction

In this chapter, we consider the effect of weak instruments on instrumental variable (IV) analyses. Weak instruments are those that do not explain a large proportion of the variation in the exposure, and so the statistical association between the IV and the exposure is not strong. This is of particular relevance in Mendelian randomization studies since the associations of genetic variants with exposures of interest are often weak. This chapter focuses on the impact of weak instruments on the bias and coverage of IV estimates.

Key points from chapter

- Bias from weak instruments can result in seriously misleading estimates of causal effects. Studies with instruments having large expected F statistics are less biased on average. However, if a study by chance has a larger observed F statistic than expected, then the causal estimate will be more biased.
- Coverage levels with weak instruments can be poorly estimated by methods which rely on assumptions of asymptotic normality.
- Data-driven choice of instruments or analysis can exacerbate bias. In particular, any threshold guideline such as ensuring that an observed F statistic is greater than 10 is misleading. Methods, instruments, and data to be used should be specified prior to data analysis. Meta-analyses based on study-specific estimates of causal effect are susceptible to bias.
- Bias can be alleviated by use of measured covariates and parsimonious modelling of the genetic association (such as a per allele additive SNP model rather than one coefficient per genotype). This should be accompanied by sensitivity analyses to assess potential bias, for example from model misspecification.
- Bias can be reduced substantially by using LIML, Bayesian and allele score (see next chapter) methods rather than 2SLS, and bias in practice with a single IV should be minimal. Nominal coverage levels can be maintained by the use of Fiellerâ€™s theorem with a single IV, and confidence intervals from the Andersonâ€“Rubin test statistic or Bayesian MCMC methods with multiple IVs.

Relevant papers to chapter:

S. Burgess, S.G. Thompson. Bias in causal estimates from Mendelian randomization studies with weak instruments. Statist. Med. 2011; 30(11):1312-1323.

S. Burgess, S.G. Thompson, CRP CHD Genetics Collaboration. Avoiding bias from weak instruments in Mendelian randomization studies. Int. J. Epidemiol. 2011; 40(3):755-764.

N.M. Davies, S.v.H.K. Scholder, H. Farbmacher, S. Burgess, F. Windmeijer, G. Davey Smith. The many weak instruments problem and Mendelian randomization. Statist. Med. 2014.