Table 4.

Performance of multivariable IVW, univariable MR‐Egger, and multivariable MR‐Egger with ${\mathit{\beta }}_{{\mathit{X}}_{\mathit{k}}}$being correlated for all k

	Multivariable IVW		Univariable MR‐Egger			Multivariable MR‐Egger
	Mean ${\widehat{\theta }}_1$	Power,	Mean ${\widehat{\theta }}_1$	Power, %		Mean ${\widehat{\theta }}_1$	Power, %
	(mean SE)	%	(mean SE)	Intercept	Causal	(mean SE)	Intercept	Causal
Null causal effect: θ1=0
1. No pleiotropy, InSIDE satisfied
	0.000 (0.047)	4.0	0.099 (0.157)	4.3	10.1	0.000 (0.086)	4.4	4.6
2. Balanced pleiotropy, InSIDE satisfied
${\alpha }_j^{\prime }\sim \mathcal{N}$(0,0.004)	−0.001 (0.104)	4.7	0.093 (0.187)	4.5	7.4	−0.003 (0.169)	4.6	4.4
3. Directional pleiotropy, InSIDE satisfied
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.01,0.004)	0.043 (0.104)	7.0	0.099 (0.187)	5.8	8.0	0.001 (0.169)	5.9	4.8
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.05,0.004)	0.213 (0.105)	52.7	0.095 (0.187)	33.3	7.6	0.000 (0.169)	37.2	4.5
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.1,0.004)	0.426 (0.107)	96.3	0.096 (0.187)	84.5	7.6	−0.001 (0.169)	89.2	4.6
4. Directional pleiotropy, InSIDE violated
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.01,0.004)	0.062 (0.104)	9.5	0.184 (0.187)	4.6	17.9	0.078 (0.169)	4.7	7.6
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.05,0.004)	0.235 (0.104)	62.1	0.187 (0.187)	20.5	18.3	0.082 (0.169)	22.3	7.5
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.1,0.004)	0.448 (0.106)	97.9	0.181 (0.187)	73.3	17.8	0.077 (0.169)	80.3	7.2
Positive causal effect: θ1=0.3
1. No pleiotropy, InSIDE satisfied
	0.300 (0.047)	98.7	0.395 (0.158)	4.4	70.8	0.299 (0.087)	3.9	86.2
2. Balanced pleiotropy, InSIDE satisfied
${\alpha }_j^{\prime }\sim \mathcal{N}$(0,0.004)	0.300 (0.104)	81.5	0.399 (0.187)	4.4	58.0	0.301 (0.169)	4.6	44.4
3. Directional pleiotropy, InSIDE satisfied
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.01,0.004)	0.342 (0.104)	89.4	0.395 (0.187)	6.4	57.4	0.301 (0.169)	5.9	44.4
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.05,0.004)	0.513 (0.105)	99.4	0.394 (0.187)	33.0	57.4	0.296 (0.169)	38.0	43.4
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.1,0.004)	0.729 (0.107)	100.0	0.400 (0.187)	83.5	58.2	0.304 (0.169)	88.6	45.5
4. Directional pleiotropy, InSIDE violated
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.01,0.004)	0.365 (0.104)	92.1	0.489 (0.187)	4.2	74.0	0.382 (0.169)	4.6	63.2
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.05,0.004)	0.535 (0.104)	99.7	0.486 (0.187)	20.3	72.9	0.382 (0.169)	21.1	63.2
${\alpha }_j^{\prime }\sim \mathcal{N}$(0.1,0.004)	0.749 (0.106)	100.0	0.488 (0.187)	72.5	73.4	0.381 (0.169)	79.6	62.8

Abbreviations: InSIDE, Instrument Strength Independent of Direct Effect; IVW, inverse‐variance weighted; MR, Mendelian randomization; SE, standard error.