Publication Bias in Meta-Analysis: Confidence Intervals for Rosenthal’s Fail-Safe Number
Table 2
Probability coverage of the different methods for confidence intervals (CI) according to the number of studies . The figure is organised as follows: the are drawn from four different distributions (standard normal distribution, half normal distribution, skew normal with negative skewness, and skew normal with positive skewness).
Draw from
Values of and from the standard normal distribution
Values of and from the half normal distribution HN(0, 1)
Values of and from the skew normal distribution with negative skewness SN()
Values of and from the skew normal distribution with positive skewness SN()
Standard normal distribution
Fixed
Distribution Based CI
0.948
0.950
0.948
0.952
0.994
0.110
0.002
0.000
0.985
0.999
1.000
1.000
0.982
0.998
1.000
1.000
Moments Based CI
0.933
0.996
1.000
1.000
0.529
0.088
0.005
0.000
0.842
0.686
0.337
0.120
0.842
0.686
0.337
0.120
Bootstrap CI
0.929
0.996
1.000
1.000
0.514
0.089
0.005
0.000
0.830
0.680
0.337
0.120
0.830
0.680
0.337
0.120
Random
Distribution Based CI
0.966
0.956
0.951
0.955
0.999
1.000
0.084
0.001
0.998
1.000
1.000
1.000
0.990
0.999
1.000
1.000
Moments Based CI
1.000
1.000
1.000
1.000
0.535
0.094
0.006
0.000
1.000
0.702
0.338
0.122
1.000
0.702
0.338
0.122
Bootstrap CI
0.929
0.996
1.000
1.000
0.429
0.074
0.004
0.000
0.804
0.649
0.322
0.115
0.804
0.649
0.322
0.115
Half normal distribution HN(0, 1)
Fixed
Distribution Based CI
0.635
0.021
0.000
0.000
0.945
0.952
0.951
0.948
0.864
0.624
0.279
0.053
0.841
0.483
0.142
0.014
Moments Based CI
0.861
0.187
0.000
0.000
0.771
0.880
0.911
0.927
0.885
0.657
0.126
0.003
0.885
0.657
0.126
0.003
Bootstrap CI
0.858
0.217
0.000
0.000
0.775
0.884
0.913
0.929
0.887
0.672
0.138
0.003
0.887
0.672
0.138
0.003
Random
Distribution Based CI
0.720
0.027
0.000
0.000
0.989
0.995
0.996
0.997
0.966
0.915
0.762
0.459
0.901
0.578
0.198
0.027
Moments Based CI
1.000
1.000
0.130
0.000
0.806
0.937
0.971
0.984
1.000
1.000
0.995
0.358
1.000
1.000
0.995
0.358
Bootstrap CI
0.858
0.217
0.000
0.000
0.715
0.859
0.899
0.920
0.885
0.698
0.152
0.004
0.885
0.698
0.152
0.004
Skew normal distribution with negative skewness SN()
Fixed
Distribution Based CI
0.872
0.666
0.399
0.174
0.980
0.472
0.184
0.048
0.953
0.970
0.977
0.981
0.944
0.948
0.949
0.957
Moments Based CI
0.917
0.979
0.944
0.858
0.597
0.375
0.200
0.074
0.845
0.860
0.882
0.895
0.845
0.860
0.882
0.895
Bootstrap CI
0.912
0.978
0.945
0.857
0.586
0.377
0.199
0.074
0.840
0.857
0.881
0.894
0.840
0.857
0.881
0.894
Random
Distribution Based CI
0.903
0.688
0.409
0.178
0.996
1.000
0.687
0.306
0.987
0.997
0.998
0.999
0.965
0.964
0.965
0.968
Moments Based CI
1.000
1.000
0.999
0.968
0.609
0.399
0.237
0.103
1.000
0.872
0.886
0.902
1.000
0.872
0.886
0.902
Bootstrap CI
0.912
0.978
0.945
0.857
0.514
0.342
0.181
0.066
0.818
0.845
0.874
0.889
0.818
0.845
0.874
0.889
Skew normal distribution with positive skewness SN()
