A statistic used to estimate the variance explained in dependent variable by the instrumental variable (IV).

In MR analyses, this is typically used to estimate the variance explained in the exposure of interest by the genetic IV. Coupled with various other statistics describing the sample and parameters being used to estimate the causal effect of the exposure on the outcome (e.g., sample size, type-1 error rate, the hypothesised regression coefficient of the causal relationship of the exposure and outcome, and the variance of the continuous exposure and continuous outcome of interest or the proportion of cases vs. controls for binary outcomes), the R-squared can be used to estimate statistical power in MR analyses. The R-squared can be calculated with individual-level data using linear or logistic regression of the exposure and the genetic IV. The R-squared can also be estimated using summary-level data for two-sample MR, which typically uses one or more single nucleotide polymorphisms (SNPs) associated with the exposure as the genetic IV, if the required statistics are available for each SNP (e.g., the beta coefficient, standard error and sample size of the association between the SNP and exposure and the minor allele frequency of the SNP).

## References

- Yarmolinsky J, Bonilla C, Haycock PC, Langdon RJQ, Lotta LA, Langenberg C, et al. Circulating Selenium and Prostate Cancer Risk: A Mendelian Randomization Analysis. J Natl Cancer Inst. 2018; 110: 1035-1038.
- Palmer TM, Lawlor DA, Harbord RM, et al. Using multiple genetic variants as instrumental variables for modifiable risk factors. Stat Methods Med Res 2012; 21: 223-42.
- Brion MJ, Shakhbazov K, Visscher PM. Calculating statistical power in Mendelian randomization studies. Int J Epidemiol 2013; 42: 1497-501.

## Other terms in 'Sources of bias and limitations in MR':

- Assortative mating
- Canalization
- Collider
- Collider bias
- Conditional F-statistic for multiple exposures
- Confounding
- Exclusion restriction assumption
- F-statistic
- Harmonization (in two-sample MR)
- Homogeneity Assumption
- Horizontal Pleiotropy
- Independence assumption
- INstrument Strength Independent of Direct Effect (InSIDE) assumption
- Intergenerational (or dynastic) effects
- Monotonicity assumption
- MR for testing critical or sensitive periods
- MR for testing developmental origins
- No effect modification assumption
- NO Measurement Error (NOME) assumption
- Non-linear MR
- Non-overlapping samples (in two-sample MR)
- Overfitting
- Pleiotropy
- Population stratification
- Regression dilution bias (attenuation by errors)
- Relevance assumption
- Reverse causality
- Same underlying population (in two-sample MR)
- Statistical power and efficiency
- Vertical pleiotropy
- Weak instrument bias
- Winner's curse