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. Author manuscript; available in PMC: 2021 Aug 9.
Published in final edited form as: Stat Med. 2020 Jul 28;39(24):3347–3372. doi: 10.1002/sim.8670

TABLE 3.

Minimum and Median of mean RE from bias-corrected estimators

Mean REd MD-corrected estimators Mean REd FG-corrected estimators with d = 0.1 Mean REd FG-corrected estimators with d = 0.75
m n K PN (n) CVa PN (K) CVb CVc (r, ρ)e Minimum Mediane (r, ρ)e Minimum Mediane (r, ρ)e Minimum Mediane
20 20 20 1 0.22 (0.11, 0.34) 1 0.22 (0.12, 0.35) 0.32 (0.17, 0.46) (0.02, 0) 0.968 0.997 (0.06, 0.01) 0.986 0.998 (0.03, 0) 0.973 0.997
2 0.25 (0.10, 0.38) 0.34 (0.17, 0.53) (0.02, 0) 0.963 0.997 (0.05, 0.01) 0.984 0.998 (0.02, 0) 0.968 0.997
3 0.56 (0.31, 0.72) 0.61 (0.34, 0.87) (0.02, 0) 0.889 0.996 (0.05, 0) 0.940 0.997 (0.03, 0) 0.902 0.996
4 0.47 (0.37, 0.59) 0.53 (0.37, 0.69) (0.05, 0) 0.891 0.994 (0, 0.01) 0.934 0.995 (0.06, 0) 0.901 0.994
5 0.58 (0.46, 0.70) 0.63 (0.48, 0.79) (0.05, 0) 0.853 0.994 (0.07, 0) 0.906 0.995 (0.05, 0) 0.865 0.994
6 0.40 (0.29, 0.54) 0.47 (0.26, 0.68) (0.03, 0) 0.924 0.996 (0, 0.01) 0.959 0.997 (0.04, 0) 0.932 0.996
2 0.25 (0.12, 0.38) 1 0.22 (0.13, 0.35) 0.34 (0.18, 0.55) (0.03, 0) 0.964 0.996 (0.06, 0.01) 0.983 0.997 (0.03, 0) 0.970 0.996
2 0.25 (0.13, 0.38) 0.40 (0.20, 0.61) (0.03, 0) 0.962 0.996 (0.03, 0.01) 0.981 0.997 (0.04, 0) 0.970 0.996
3 0.55 (0.42, 0.73) 1 0.22 (0.12, 0.34) 0.60 (0.36, 0.89) (0.02, 0) 0.896 0.985 (0.13, 0.01) 0.936 0.988 (0.26, 0.01) 0.910 0.985
4 0.46 (0.36, 0.56) 1 0.22 (0.12, 0.33) 0.52 (0.38, 0.69) (0.16, 0.01) 0.904 0.974 (0.10, 0.01) 0.931 0.979 (0.16, 0.01) 0.911 0.974
5 0.58 (0.45, 0.70) 1 0.22 (0.11, 0.35) 0.63 (0.44, 0.82) (0.20, 0.01) 0.865 0.966 (0.10, 0.01) 0.904 0.973 (0.18, 0.01) 0.875 0.966
6 0.40 (0.25, 0.56) 1 0.22 (0.12, 0.33) 0.46 (0.28, 0.66) (0.22, 0.01) 0.935 0.985 (0.10, 0.01) 0.955 0.988 (0.15, 0.01) 0.939 0.985
20 20 5 1 0.22 (0.11, 0.34) 1 0.43 (0.25, 0.67) 0.49 (0.23, 0.84) (0.09, 0) 0.920 0.993 (0.06, 0.01) 0.948 0.995 (0, 0.01) 0.927 0.993
0.22 (0.11, 0.33) 2 0.45 (0.27, 0.70) 0.51 (0.25, 0.91) (0.08, 0) 0.915 0.993 (0.04, 0.01) 0.945 0.994 (0.01, 0.01) 0.923 0.993
0.22 (0.11, 0.32) 3 0.64 (0.38, 1.00) 0.68 (0.39, 1.02) (0.08, 0) 0.855 0.989 (0.04, 0.01) 0.904 0.991 (0.12, 0) 0.871 0.989
0.23 (0.14, 0.32) 4 0.56 (0.41, 0.70) 0.62 (0.41, 0.81) (0.11, 0) 0.871 0.989 (0.09, 0.01) 0.912 0.991 (0.01, 0.01) 0.883 0.989
0.23 (0.15, 0.32) 5 0.64 (0.49, 0.85) 0.69 (0.49, 0.92) (0.11, 0) 0.840 0.987 (0.06, 0.01) 0.891 0.989 (0.13, 0) 0.856 0.987
0.22 (0.13, 0.32) 6 0.52 (0.33, 0.84) 0.58 (0.32, 0.94) (0.08, 0) 0.889 0.991 (0.04, 0.01) 0.925 0.993 (0, 0.01) 0.900 0.991
2 0.25 (0.13, 0.38) 1 0.43 (0.25, 0.66) 0.51 (0.26, 0.89) (0.07, 0) 0.915 0.992 (0.02, 0.01) 0.945 0.994 (0.01, 0.01) 0.923 0.992
2 0.45 (0.25, 0.68) 0.55 (0.31, 0.95) (0, 0.01) 0.912 0.991 (0.02, 0.01) 0.942 0.993 (0, 0.01) 0.918 0.991
3 0.55 (0.42, 0.71) 1 0.43 (0.25, 0.66) 0.72 (0.43, 1.17) (0.06, 0) 0.844 0.978 (0.03, 0.01) 0.897 0.982 (0.08, 0.01) 0.865 0.978
4 0.46 (0.37, 0.56) 1 0.43 (0.25, 0.66) 0.66 (0.43, 0.97) (0.02, 0.01) 0.854 0.966 (0.02, 0.01) 0.897 0.972 (0.02, 0.01) 0.863 0.967
5 0.58 (0.45, 0.70) 1 0.43 (0.24, 0.67) 0.75 (0.48, 1.20) (0.03, 0.01) 0.818 0.958 (0, 0.01) 0.870 0.965 (0.03, 0.01) 0.830 0.958
6 0.40 (0.25, 0.56) 1 0.43 (0.24, 0.67) 0.61 (0.39, 1.06) (0, 0.01) 0.882 0.979 (0.04, 0.01) 0.918 0.983 (0, 0.01) 0.890 0.979
20 10 20 1 0.31 (0.18, 0.49) 1 0.22 (0.11, 0.34) 0.39 (0.21, 0.64) (0.03, 0) 0.954 0.989 (0.03, 0.01) 0.972 0.993 (0.06, 0.01) 0.960 0.989
2 0.25 (0.13, 0.38) 0.41 (0.23, 0.63) (0.01, 0) 0.949 0.989 (0.03, 0.01) 0.969 0.992 (0.05, 0.01) 0.957 0.989
3 0.55 (0.40, 0.73) 0.64 (0.39, 0.95) (0.02, 0) 0.874 0.987 (0, 0.01) 0.926 0.990 (0.02, 0) 0.891 0.987
4 0.47 (0.38, 0.59) 0.58 (0.40, 0.89) (0.05, 0) 0.876 0.983 (0.02, 0.01) 0.913 0.987 (0.05, 0) 0.889 0.983
5 0.58 (0.46, 0.71) 0.68 (0.46, 0.91) (0.03, 0) 0.837 0.983 (0.01, 0.01) 0.890 0.986 (0.05, 0) 0.854 0.983
6 0.40 (0.29, 0.54) 0.52 (0.32, 0.76) (0.04, 0) 0.909 0.986 (0.01, 0.01) 0.940 0.990 (0.04, 0) 0.920 0.986
2 0.34 (0.19, 0.51) 1 0.22 (0.11, 0.34) 0.41 (0.21, 0.64) (0.02, 0) 0.950 0.988 (0.03, 0.01) 0.968 0.991 (0.05, 0.01) 0.955 0.988
2 0.25 (0.13, 0.38) 0.46 (0.25, 0.70) (0.03, 0) 0.949 0.987 (0.02, 0.01) 0.965 0.991 (0.04, 0.01) 0.951 0.987
3 0.59 (0.40, 0.79) 1 0.22 (0.11, 0.34) 0.63 (0.36, 0.92) (0.01, 0) 0.884 0.966 (0.05, 0.01) 0.922 0.973 (0.12, 0.01) 0.895 0.967
4 0.48 (0.39, 0.62) 1 0.22 (0.11, 0.32) 0.54 (0.37, 0.68) (0.08, 0.01) 0.901 0.964 (0.03, 0.01) 0.930 0.973 (0.08, 0.01) 0.908 0.965
5 0.58 (0.47, 0.71) 1 0.22 (0.11, 0.32) 0.63 (0.49, 0.79) (0.10, 0.01) 0.868 0.952 (0.03, 0.01) 0.908 0.963 (0.10, 0.01) 0.877 0.953
6 0.44 (0.30, 0.63) 1 0.22 (0.11, 0.32) 0.50 (0.33, 0.76) (0.12, 0.01) 0.922 0.975 (0.04, 0.01) 0.946 0.981 (0.08, 0.01) 0.927 0.975
20 10 5 1 0.31 (0.18, 0.47) 1 0.43 (0.24, 0.74) 0.55 (0.28, 0.95) (0.03, 0.01) 0.905 0.981 (0.03, 0.02) 0.937 0.985 (0.03, 0.01) 0.911 0.981
2 0.45 (0.26, 0.82) 0.56 (0.32, 1.03) (0.01, 0.01) 0.898 0.980 (0.05, 0.02) 0.934 0.985 (0.04, 0.01) 0.907 0.981
0.31 (0.18, 0.49) 3 0.64 (0.38, 0.94) 0.73 (0.42, 1.30) (0.06, 0) 0.839 0.974 (0.04, 0.02) 0.892 0.979 (0.02, 0.01) 0.855 0.974
0.31 (0.21, 0.45) 4 0.56 (0.40, 0.71) 0.66 (0.42, 0.92) (0.10, 0) 0.857 0.974 (0.08, 0.02) 0.903 0.979 (0.04, 0.01) 0.869 0.974
0.32 (0.19, 0.46) 5 0.64 (0.49, 0.94) 0.74 (0.48, 1.13) (0.12, 0) 0.824 0.969 (0.10, 0.02) 0.880 0.975 (0.07, 0.01) 0.842 0.970
0.31 (0.18, 0.47) 6 0.53 (0.32, 0.80) 0.64 (0.39, 1.11) (0.02, 0.01) 0.872 0.977 (0.05, 0.02) 0.914 0.982 (0.05, 0.01) 0.883 0.977
2 0.33 (0.19, 0.50) 1 0.43 (0.24, 0.74) 0.56 (0.30, 1.05) (0.02, 0.01) 0.898 0.979 (0.04, 0.02) 0.934 0.984 (0.02, 0.01) 0.907 0.979
2 0.45 (0.26, 0.82) 0.60 (0.33, 1.13) (0.01, 0.01) 0.894 0.978 (0.04, 0.02) 0.931 0.983 (0.06, 0.01) 0.904 0.978
3 0.59 (0.40, 0.79) 1 0.43 (0.24, 0.74) 0.75 (0.43, 1.29) (0.06, 0) 0.832 0.955 (0.05, 0.02) 0.888 0.963 (0.06, 0.01) 0.847 0.956
4 0.48 (0.39, 0.62) 1 0.43 (0.26, 0.64) 0.67 (0.46, 0.96) (0.01, 0.01) 0.851 0.953 (0.03, 0.02) 0.898 0.962 (0.05, 0.01) 0.865 0.954
5 0.58 (0.47, 0.71) 1 0.43 (0.26, 0.62) 0.75 (0.48, 1.11) (0.01, 0.01) 0.818 0.940 (0.03, 0.02) 0.874 0.952 (0.01, 0.01) 0.834 0.941
6 0.44 (0.30, 0.63) 1 0.43 (0.25, 0.65) 0.64 (0.36, 1.14) (0.03, 0.01) 0.868 0.965 (0.02, 0.02) 0.912 0.972 (0.03, 0.01) 0.879 0.965
6 20 20 1 0.22 (0.03, 0.46) 1 0.21 (0.03, 0.49) 0.31 (0.04, 0.67) (0, 0) 0.870 0.997 (0.04, 0) 0.979 0.998 (0, 0) 0.911 0.997
2 0.36 (0.12, 0.63) 0.42 (0.08, 0.85) (0, 0) 0.773 0.997 (0.04, 0) 0.956 0.997 (0, 0) 0.842 0.997
3 0.54 (0.28, 0.84) 0.58 (0.12, 1.07) (0, 0) 0.633 0.996 (0.03, 0) 0.917 0.996 (0, 0) 0.733 0.996
4 0.54 (0.32, 0.82) 0.59 (0.36, 0.90) (0, 0) 0.543 0.991 (0.07, 0) 0.875 0.992 (0, 0) 0.686 0.991
5 0.74 (0.55, 0.92) 0.78 (0.54, 1.11) (0, 0) 0.308 0.989 (0.07, 0) 0.797 0.990 (0, 0) 0.510 0.989
6 0.81 (0.50, 1.16) 0.83 (0.30, 1.37) (0, 0) 0.421 0.994 (0.04, 0) 0.838 0.995 (0, 0) 0.551 0.995
2 0.36 (0.08, 0.64) 1 0.21 (0.03, 0.49) 0.42 (0.12, 0.88) (0, 0) 0.770 0.992 (0.23, 0.01) 0.959 0.993 (0, 0) 0.840 0.992
3 0.54 (0.18, 0.86) 1 0.21 (0.03, 0.49) 0.58 (0.17, 1.10) (0, 0) 0.633 0.985 (0.29, 0.01) 0.921 0.986 (0.01, 0) 0.732 0.985
4 0.53 (0.37, 0.69) 1 0.21 (0.03, 0.49) 0.57 (0.34, 0.85) (0.01, 0) 0.573 0.946 (0.10, 0.01) 0.891 0.960 (0.01, 0) 0.700 0.951
5 0.73 (0.51, 0.95) 1 0.21 (0.03, 0.48) 0.76 (0.48, 1.12) (0.01, 0) 0.327 0.918 (0.12, 0.01) 0.814 0.936 (0.01, 0) 0.522 0.925
6 0.82 (0.44, 1.15) 1 0.21 (0.03, 0.49) 0.83 (0.43, 1.41) (0.02, 0) 0.422 0.963 (0.32, 0.01) 0.842 0.967 (0.01, 0) 0.547 0.964
6 20 5 1 0.22 (0.04, 0.46) 1 0.41 (0.13, 0.85) 0.47 (0.11, 1.14) (0, 0) 0.722 0.994 (0, 0.01) 0.937 0.994 (0, 0) 0.804 0.994
0.22 (0.03, 0.46) 2 0.49 (0, 1.02) 0.54 (0.14, 1.08) (0, 0) 0.650 0.993 (0.01, 0.01) 0.918 0.993 (0, 0) 0.752 0.993
3 0.61 (0.13, 1.29) 0.65 (0.13, 1.27) (0, 0) 0.546 0.990 (0.09, 0) 0.887 0.991 (0, 0) 0.667 0.990
0.22 (0.03, 0.42) 4 0.58 (0.18, 1.03) 0.62 (0.19, 1.14) (0, 0) 0.541 0.988 (0.17, 0) 0.882 0.989 (0, 0) 0.677 0.988
0.21 (0.04, 0.44) 5 0.74 (0.40, 1.14) 0.77 (0.47, 1.29) (0, 0) 0.359 0.983 (0.15, 0) 0.824 0.985 (0, 0) 0.534 0.984
0.22 (0.03, 0.46) 6 0.83 (0.36, 1.39) 0.85 (0.35, 1.58) (0, 0) 0.406 0.986 (0.09, 0) 0.824 0.987 (0, 0) 0.538 0.986
2 0.36 (0.08, 0.64) 1 0.41 (0.13, 0.85) 0.55 (0.10, 1.18) (0, 0) 0.642 0.987 (0.04, 0.01) 0.915 0.988 (0, 0) 0.743 0.988
3 0.54 (0.18, 0.86) 1 0.41 (0.13, 0.85) 0.67 (0.15, 1.38) (0, 0) 0.534 0.979 (0.04, 0.01) 0.881 0.981 (0, 0) 0.653 0.980
4 0.53 (0.37, 0.69) 1 0.41 (0.13, 0.81) 0.68 (0.35, 1.16) (0.01, 0) 0.462 0.936 (0.02, 0.01) 0.854 0.954 (0.01, 0) 0.611 0.943
5 0.73 (0.51, 0.95) 1 0.41 (0.13, 0.81) 0.85 (0.43, 1.37) (0.01, 0) 0.276 0.905 (0, 0.01) 0.782 0.928 (0.01, 0) 0.476 0.914
6 0.82 (0.44, 1.15) 1 0.41 (0.13, 0.85) 0.88 (0.24, 1.65) (0, 0) 0.369 0.954 (0.13, 0.01) 0.806 0.961 (0.03, 0) 0.505 0.957
6 10 20 1 0.31 (0.06, 0.65) 1 0.21 (0.03, 0.49) 0.37 (0.07, 0.82) (0, 0) 0.809 0.989 (0.06, 0.01) 0.964 0.991 (0, 0) 0.871 0.989
2 0.36 (0.12, 0.63) 0.47 (0.08, 0.99) (0, 0) 0.720 0.988 (0.04, 0) 0.947 0.990 (0, 0) 0.805 0.988
3 0.54 (0.28, 0.84) 0.61 (0.16, 1.19) (0, 0) 0.597 0.986 (0.04, 0) 0.909 0.988 (0, 0) 0.705 0.987
4 0.54 (0.32, 0.82) 0.63 (0.38, 1.03) (0, 0) 0.502 0.975 (0.07, 0) 0.864 0.980 (0, 0) 0.651 0.977
0.30 (0.06, 0.65) 5 0.74 (0.55, 0.92) 0.81 (0.53, 1.29) (0, 0) 0.289 0.970 (0.07, 0) 0.789 0.975 (0, 0) 0.495 0.972
0.31 (0.06, 0.65) 6 0.81 (0.50, 1.16) 0.85 (0.25, 1.52) (0, 0) 0.404 0.982 (0.04, 0) 0.830 0.985 (0, 0) 0.539 0.983
2 0.42 (0.11, 0.80) 1 0.21 (0.03, 0.49) 0.47 (0.09, 1.02) (0, 0) 0.721 0.978 (0.08, 0.01) 0.942 0.982 (0, 0) 0.803 0.979
3 0.57 (0.19, 1.02) 1 0.21 (0.03, 0.49) 0.60 (0.19, 1.22) (0, 0) 0.603 0.963 (0.10, 0.01) 0.907 0.970 (0.01, 0) 0.711 0.966
4 0.54 (0.35, 0.85) 1 0.21 (0.03, 0.49) 0.57 (0.32, 0.96) (0.01, 0) 0.586 0.943 (0.07, 0.01) 0.898 0.958 (0.01, 0) 0.704 0.949
5 0.69 (0.42, 0.96) 1 0.22 (0.06, 0.48) 0.73 (0.40, 1.18) (0.03, 0) 0.393 0.911 (0.04, 0.01) 0.842 0.933 (0, 0) 0.563 0.919
6 0.81 (0.32, 1.32) 1 0.21 (0.03, 0.49) 0.83 (0.35, 1.48) (0.03, 0) 0.424 0.932 (0.13, 0.01) 0.837 0.945 (0.01, 0) 0.548 0.937
6 10 5 1 0.31 (0.06, 0.65) 1 0.41 (0.13, 0.85) 0.52 (0.12, 1.27) (0, 0) 0.676 0.980 (0, 0.02) 0.923 0.984 (0, 0) 0.768 0.982
2 0.49 (0, 1.02) 0.58 (0.14, 1.23) (0, 0) 0.609 0.978 (0.04, 0.01) 0.904 0.982 (0, 0) 0.718 0.980
3 0.61 (0.13, 1.29) 0.69 (0.16, 1.42) (0, 0) 0.515 0.972 (0.05, 0.01) 0.872 0.977 (0, 0) 0.640 0.974
0.31 (0.06, 0.59) 4 0.58 (0.18, 1.03) 0.66 (0.27, 1.27) (0, 0) 0.508 0.968 (0.11, 0.01) 0.870 0.975 (0.01, 0) 0.649 0.971
0.30 (0.09, 0.63) 5 0.74 (0.40, 1.14) 0.79 (0.45, 1.40) (0, 0) 0.342 0.959 (0.09, 0.01) 0.808 0.967 (0, 0) 0.518 0.962
0.31 (0.06, 0.65) 6 0.83 (0.36, 1.39) 0.87 (0.23, 1.72) (0, 0) 0.386 0.963 (0.02, 0.01) 0.811 0.970 (0, 0) 0.520 0.966
2 0.42 (0.11, 0.80) 1 0.41 (0.13, 0.85) 0.59 (0.12, 1.31) (0, 0) 0.602 0.968 (0, 0.01) 0.902 0.975 (0, 0) 0.712 0.970
3 0.57 (0.19, 1.02) 1 0.41 (0.13, 0.85) 0.69 (0.20, 1.48) (0, 0) 0.509 0.951 (0, 0.01) 0.867 0.961 (0.01, 0) 0.630 0.955
4 0.54 (0.35, 0.85) 1 0.40 (0.13, 0.74) 0.67 (0.32, 1.19) (0, 0) 0.503 0.931 (0.08, 0.02) 0.863 0.950 (0.02, 0) 0.629 0.939
5 0.69 (0.42, 0.96) 1 0.43 (0.13, 0.77) 0.82 (0.39, 1.40) (0, 0) 0.339 0.894 (0.13, 0.02) 0.810 0.923 (0.03, 0) 0.500 0.905
6 0.81 (0.32, 1.32) 1 0.41 (0.13, 0.85) 0.88 (0.20, 1.67) (0, 0) 0.369 0.917 (0.04, 0.01) 0.802 0.935 (0.05, 0) 0.511 0.924

K: provider size; m: number of practices; n: practice size; PN: pattern.

a

CV of practice size, mean (min, max).

b

CV of provider size, mean (min, max).

c

CV of cluster size, mean (min, max).

d

The mean RE among 1000 simulations is calculated for each (r, ρ).

e

The minimum and median of mean RE including the corresponding (r, ρ)s are identified across all values of (r, ρ).

Proposed Algorithm

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