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Table 1 Simulation results with partly interval-censored data, including the estimated bias (Bias), the sample standard error (SSE) of the estimates, the average of the standard error estimates (SEE), and the 95% empirical coverage probability (CP)

From: A pairwise pseudo-likelihood approach for regression analysis of left-truncated failure time data with various types of censoring

   

Proposed method

 

CL method

 

Ignoring truncation

n

Par

True

Bias

SSE

SEE

CP

 

Bias

SSE

SEE

CP

 

Bias

SSE

SEE

CP

\(A^{*}\) follows the uniform distribution

                

100

\(\beta _1\)

1

0.041

0.246

0.254

96.6

 

0.061

0.290

0.278

92.7

 

0.192

0.273

0.311

94.1

 

\(\beta _2\)

1

0.045

0.403

0.408

95.6

 

0.055

0.492

0.467

93.7

 

0.180

0.474

0.445

90.8

 

\(\Lambda (0.4)\)

0.16

0.009

0.094

0.087

95.6

 

0.002

0.092

 

-0.084

0.043

 

\(\Lambda (0.8)\)

0.64

-0.037

0.165

0.161

93.9

 

-0.048

0.164

 

-0.245

0.103

 

\(\Lambda (1.2)\)

1.44

-0.038

0.240

0.248

93.5

 

-0.067

0.240

 

-0.445

0.209

300

\(\beta _1\)

1

0.008

0.134

0.129

93.5

 

0.013

0.156

0.151

93.1

 

0.120

0.148

0.306

95.9

 

\(\beta _2\)

1

0.012

0.212

0.212

94.6

 

0.025

0.248

0.253

95.2

 

0.146

0.244

0.305

92.4

 

\(\Lambda (0.4)\)

0.16

0.020

0.067

0.064

94.2

 

0.020

0.067

 

-0.081

0.026

 

\(\Lambda (0.8)\)

0.64

0.020

0.104

0.105

95.3

 

0.020

0.107

 

-0.240

0.061

 

\(\Lambda (1.2)\)

1.44

-0.028

0.179

0.189

96.5

 

-0.024

0.182

 

-0.436

0.116

500

\(\beta _1\)

1

0.014

0.101

0.099

95.4

 

0.018

0.115

0.117

96.0

 

0.154

0.101

0.197

79.6

 

\(\beta _2\)

1

0.014

0.161

0.163

94.6

 

0.020

0.191

0.193

95.0

 

0.146

0.193

0.219

85.7

 

\(\Lambda (0.4)\)

0.16

0.012

0.048

0.048

96.7

 

0.012

0.048

 

-0.081

0.018

 

\(\Lambda (0.8)\)

0.64

0.010

0.076

0.075

95.0

 

0.009

0.077

 

-0.244

0.044

 

\(\Lambda (1.2)\)

1.44

-0.012

0.133

0.131

94.6

 

-0.012

0.135

 

-0.443

0.094

\(A^{*}\) follows the exponential distribution

                

100

\(\beta _1\)

1

0.045

0.242

0.251

94.9

 

0.062

0.272

0.266

93.1

 

0.146

0.266

0.294

95.3

 

\(\beta _2\)

1

0.047

0.396

0.405

95.3

 

0.071

0.451

0.453

95.3

 

0.149

0.435

0.435

90.5

 

\(\Lambda (0.4)\)

0.16

0.009

0.083

0.080

95.5

 

0.009

0.085

 

-0.068

0.046

 

\(\Lambda (0.8)\)

0.64

-0.036

0.159

0.160

93.7

 

-0.038

0.159

 

-0.178

0.114

 

\(\Lambda (1.2)\)

1.44

-0.042

0.234

0.240

92.7

 

-0.042

0.235

 

-0.280

0.251

300

\(\beta _1\)

1

0.011

0.131

0.133

95.9

 

0.016

0.147

0.148

94.9

 

0.084

0.137

0.250

97.4

 

\(\beta _2\)

1

-0.001

0.210

0.217

95.9

 

0.007

0.228

0.246

96.7

 

0.082

0.229

0.294

95.6

 

\(\Lambda (0.4)\)

0.16

0.017

0.053

0.053

96.5

 

0.017

0.054

 

-0.065

0.025

 

\(\Lambda (0.8)\)

0.64

0.014

0.097

0.093

93.8

 

0.013

0.100

 

-0.173

0.066

 

\(\Lambda (1.2)\)

1.44

-0.016

0.184

0.182

94.5

 

-0.015

0.184

 

-0.300

0.140

500

\(\beta _1\)

1

0.012

0.100

0.101

95.2

 

0.016

0.115

0.113

94.4

 

0.069

0.116

0.263

92.3

 

\(\beta _2\)

1

0.010

0.165

0.167

94.7

 

0.008

0.187

0.188

94.7

 

0.097

0.173

0.247

92.3

 

\(\Lambda (0.4)\)

0.16

0.014

0.044

0.045

95.3

 

0.015

0.044

 

-0.067

0.020

 

\(\Lambda (0.8)\)

0.64

0.013

0.075

0.073

94.6

 

0.013

0.077

 

-0.174

0.061

 

\(\Lambda (1.2)\)

1.44

-0.003

0.136

0.138

97.5

 

-0.003

0.138

 

-0.292

0.118

  1. Note: “Proposed method” denotes the proposed pairwise pseudo-likelihood method, “CL method” denotes the conditional likelihood method, and “Ignoring truncation” denotes the NPMLE approach that ignores the existence of left truncation