Table 5 Estimated values of fixed effects and measures of heterogeneity for the two-level logistic regression, under random effects variance values 1.0 and 1.5

\(\mathbf {\sigma ^{2}_{u}=1.0}\)

True values:

0.500

-1.500

0.300

-0.300

0.2331

2.596

0.4160

0.1209

4.538

10

5

0.562

-1.508

0.291

-0.290

0.259

2.800

0.424

0.109

5.408

20

0.620

-1.501

0.312

-0.502

0.231

2.581

0.361

0.100

3.683

50

0.542

-1.459

0.300

-0.276

0.230

2.575

0.421

0.128

4.624

100

0.594

-1.517

0.320

-0.467

0.243

2.669

0.376

0.099

4.059

250

0.502

-1.509

0.308

-0.396

0.231

2.580

0.389

0.112

4.114

50

5

0.477

-1.509

0.299

-0.315

0.213

2.461

0.407

0.133

4.060

20

0.537

-1.491

0.297

-0.368

0.221

2.515

0.394

0.120

4.007

50

0.475

-1.506

0.301

-0.306

0.229

2.565

0.413

0.123

4.421

100

0.503

-1.494

0.300

-0.287

0.230

2.572

0.419

0.125

4.539

250

0.495

-1.494

0.295

-0.324

0.228

2.558

0.408

0.122

4.310

100

5

0.522

-1.504

0.324

-0.268

0.222

2.519

0.423

0.133

4.430

20

0.521

-1.494

0.300

-0.295

0.229

2.568

0.417

0.124

4.469

50

0.513

-1.494

0.299

-0.332

0.232

2.588

0.407

0.118

4.370

100

0.501

-1.505

0.307

-0.331

0.236

2.616

0.408

0.116

4.474

250

0.496

-1.491

0.297

-0.318

0.232

2.586

0.411

0.120

4.434

300

5

0.514

-1.476

0.312

-0.338

0.211

2.445

0.399

0.131

3.911

20

0.509

-1.505

0.294

-0.297

0.231

2.581

0.416

0.123

4.503

50

0.496

-1.498

0.292

-0.290

0.234

2.602

0.419

0.122

4.607

100

0.479

-1.504

0.303

-0.274

0.235

2.608

0.424

0.123

4.706

250

0.491

-1.503

0.301

-0.287

0.234

2.618

0.417

0.123

4.607

500

5

0.494

-1.467

0.304

-0.283

0.209

2.433

0.415

0.140

4.080

20

0.493

-1.506

0.304

-0.275

0.228

2.563

0.422

0.127

4.542

50

0.497

-1.501

0.302

-0.293

0.233

2.595

0.418

0.122

4.565

100

0.506

-1.494

0.306

-0.287

0.230

2.574

0.419

0.125

4.524

250

0.489

-1.504

0.301

-0.285

0.235

2.607

0.420

0.122

4.643

\(\mathbf {\sigma ^{2}_{u}=1.5}\)

True values:

0.500

-1.500

0.300

-0.300

0.3131

3.216

0.4312

0.0805

6.819

10

5

0.437

-1.521

0.335

-0.224

0.307

3.166

0.448

0.092

7.213

20

0.505

-1.484

0.272

-0.440

0.326

3.331

0.403

0.066

6.360

50

0.498

-1.530

0.294

-0.129

0.314

3.229

0.471

0.097

6.248

100

0.502

-1.486

0.302

-0.314

0.304

3.143

0.427

0.083

6.459

250

0.435

-1.516

0.304

-0.166

0.315

3.245

0.464

0.096

6.809

50

5

0.507

-1.487

0.307

-0.325

0.273

2.885

0.418

0.097

5.438

20

0.538

-1.515

0.300

-0.350

0.324

3.314

0.422

0.073

6.864

50

0.526

-1.503

0.307

-0.235

0.305

3.147

0.445

0.090

6.996

100

0.556

-1.503

0.307

-0.369

0.321

3.285

0.418

0.073

6.675

250

0.416

-1.495

0.298

-0.260

0.304

3.139

0.439

0.088

6.807

100

5

0.488

-1.455

0.284

-0.268

0.286

2.989

0.434

0.096

6.150

20

0.460

-1.491

0.305

-0.253

0.311

3.198

0.442

0.086

7.066

50

0.527

-1.511

0.308

-0.352

0.318

3.258

0.420

0.075

6.641

100

0.432

-1.499

0.306

-0.231

0.311

3.200

0.447

0.087

7.258

250

0.510

-1.498

0.296

-0.235

0.311

3.197

0.446

0.087

7.201

300

5

0.479

-1.503

0.312

-0.266

0.298

3.085

0.437

0.091

6.528

20

0.479

-1.501

0.306

-0.275

0.314

3.223

0.437

0.082

7.033

50

0.518

-1.495

0.297

-0.312

0.312

3.206

0.428

0.080

6.698

100

0.487

-1.503

0.299

-0.314

0.312

3.208

0.428

0.080

6.696

250

0.491

-1.503

0.301

-0.287

0.300

3.101

0.432

0.088

6.459

500

5

0.507

-1.495

0.301

-0.309

0.292

3.042

0.426

0.089

6.079

20

0.499

-1.505

0.300

-0.328

0.309

3.178

0.424

0.080

6.484

50

0.516

-1.500

0.298

-0.299

0.312

3.202

0.431

0.081

6.779

100

0.529

-1.501

0.301

-0.328

0.313

3.216

0.425

0.078

6.631

250

0.510

-1.497

0.228

-0.228

0.310

3.192

0.447

0.088

7.225