Table 4 Results of simulations for data generated from a mixed model as in Fig. 3 with uncorrelated random effects and an unobserved confounder between \(M_1\) and \(Y_1\)

100

0.700

0.100

0.704

0.119

0.006

0.185

0.317

0.086

0.898

0.886

0.956

0.257

0.907

0.250

-0.051

-0.029

0.410

0.173

0.888

0.862

1.064

0.384

0.966

0.322

-0.092

-0.163

0.446

0.239

0.878

0.832

1.123

0.527

0.984

0.355

-0.124

-0.326

0.464

0.311

0.862

0.732

1.162

0.660

0.990

0.370

-0.149

-0.440

0.478

0.398

0.844

0.604

1000

-

-

0.697

0.122

-0.004

0.221

0.106

0.035

0.856

0.754

-

-

0.899

0.255

-0.059

-0.006

0.148

0.053

0.816

0.860

-

-

0.957

0.327

-0.101

-0.149

0.181

0.090

0.750

0.716

-

-

0.974

0.359

-0.132

-0.318

0.211

0.185

0.674

0.282

-

-

0.979

0.373

-0.158

-0.435

0.237

0.299

0.618

0.056