Table 1 Comparisons of the three Mendelian randomization (MR) methods

Theory

The weighted linear regression of \(\widehat{\beta }\)_YC and \(\widehat{\beta }\)_XG, as well as the forced intercept term is equal to zero

The weighted linear regression of \(\widehat{\beta }\)_YC and \(\widehat{\beta }\)_XG

Generalised least squares method of \(\widehat{\beta }\)_XY

R package function

MR-IVW function

MR-Egger function

GSMR function

Application conditions

IVs are satisfied with a correlation hypothesis, independence hypothesis and exclusivity hypothesis

IVs are satisfied with a correlation hypothesis and independence hypothesis

IVs are satisfied with a correlation hypothesis, independence hypothesis and exclusivity hypothesis

Advantages

i) When IVs lose pleiotropy, the estimation accuracy is high and the power of a test is high

i) Unbiased estimations are obtained when IVs display pleiotropy

ii) The average pleiotropy can be equalised by the intercept term

iii) Sensitivity analysis can be performed

i) When IVs lose pleiotropy, the estimation accuracy is high and the power of a test is high

ii) Can be tested by heterogeneity in dependent instruments (HEIDI) whether IVs have pleiotropy

Disadvantages

i) When IVs display pleiotropy, the estimation is biased

i) The estimation is biased, and the false positive error is inflated

ii) When there is no pleiotropy, the power of a test is lower than IVW and GSMR

i) When IVs display pleiotropy, the estimation is biased

limitations

Screen out the pleiotropic IVs

Require all IVs’ direction is the same

HEIDI delete pleiotropic IVs

Summary data

Homogenous population or the population after correcting the group structure