Background
Methods
Fractional polynomials and the hazard function
Network meta-analysis model for survival data using fractional polynomials
Illustrative example
Study 1 | Study 2 | Study 3 | Study 4 | Study 5 | Study 6 | Study 7 | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Lee et al., 2010 | Chang et al. 2006 | Kim et al., 2008 | Maruyama et al., 2008 | Hanna et al., 2004 | Cufer et al., 2006 | Shepherd et al., 2000 | |||||||||||||||||||||
Gefitinib | Docetaxel | BSC | Gefitinib | Gefitinib | Docetaxel | Gefitinib | Docetaxel | Pemtrexed | Docetaxel | Gefitinib | Docetaxel | BSC | Docetaxel | ||||||||||||||
r | n | r | n | r | N | r | n | r | n | r | n | r | n | r | n | r | n | r | n | r | n | r | n | r | n | r | n |
7 | 81 | 9 | 76 | 22 | 107 | 34 | 235 | 98 | 723 | 73 | 710 | 18 | 245 | 9 | 244 | 27 | 283 | 33 | 288 | 11 | 68 | 14 | 73 | 17 | 222 | 49 | 441 |
5 | 74 | 4 | 67 | 24 | 85 | 35 | 201 | 104 | 625 | 109 | 637 | 30 | 226 | 21 | 233 | 47 | 256 | 48 | 255 | 8 | 57 | 11 | 59 | 71 | 205 | 67 | 392 |
7 | 69 | 8 | 64 | 11 | 61 | 29 | 166 | 94 | 518 | 102 | 503 | 25 | 197 | 23 | 214 | 31 | 209 | 41 | 207 | 5 | 49 | 6 | 48 | 53 | 134 | 60 | 325 |
6 | 62 | 5 | 56 | 8 | 50 | 15 | 137 | 79 | 424 | 66 | 401 | 22 | 169 | 15 | 189 | 33 | 178 | 21 | 166 | 14 | 44 | 6 | 42 | 70 | 160 | 63 | 340 |
6 | 55 | 8 | 51 | 12 | 42 | 11 | 122 | 64 | 336 | 68 | 339 | 12 | 148 | 23 | 173 | 40 | 144 | 34 | 144 | 9 | 31 | 7 | 36 | 12 | 90 | 49 | 277 |
3 | 49 | 4 | 38 | 6 | 30 | 14 | 110 | 46 | 272 | 41 | 271 | 18 | 127 | 17 | 140 | 17 | 78 | 17 | 78 | 4 | 21 | 4 | 29 | 22 | 78 | 14 | 228 |
4 | 46 | 6 | 34 | 1 | 24 | 1 | 97 | 35 | 225 | 32 | 228 | 10 | 98 | 9 | 105 | 21 | 61 | 13 | 61 | 8 | 18 | 55 | 221 | ||||
3 | 42 | 2 | 28 | 30 | 190 | 29 | 196 | 8 | 77 | 10 | 87 | 8 | 40 | 18 | 48 | 41 | 166 | ||||||||||
3 | 38 | 4 | 26 | 14 | 131 | 18 | 139 | 6 | 63 | 15 | 69 | 11 | 33 | 9 | 30 | 30 | 125 | ||||||||||
3 | 35 | 3 | 22 | 14 | 117 | 16 | 121 | 4 | 47 | 4 | 44 | 10 | 21 | ||||||||||||||
5 | 29 | 3 | 18 | 14 | 83 | 13 | 89 | 3 | 35 | 5 | 35 | ||||||||||||||||
4 | 24 | 2 | 15 | 8 | 69 | 10 | 76 | 1 | 29 | 3 | 25 | ||||||||||||||||
3 | 21 | 1 | 13 | 5 | 50 | 6 | 46 | 1 | 25 | 2 | 18 | ||||||||||||||||
1 | 18 | 1 | 12 | 3 | 45 | 5 | 40 | 2 | 18 | 1 | 14 | ||||||||||||||||
3 | 17 | 2 | 10 | 6 | 31 | 4 | 24 | 1 | 10 | ||||||||||||||||||
3 | 14 | 2 | 8 | 3 | 20 | ||||||||||||||||||||||
1 | 12 |
Results
Illustrative example
Power p1
| Power p2
| Dbar | Dhat | pD | DIC |
---|---|---|---|---|---|
Fixed effects models | |||||
-2 | - | 904.5 | 884.6 | 19.9 | 924.4 |
-1 | - | 916.2 | 896.3 | 19.8 | 936.0 |
-0.5 | - | 925.5 | 905.6 | 19.9 | 945.3 |
0a
| - | 934.6 | 915.2 | 19.3 | 953.9 |
0.5 | - | 942.8 | 923.5 | 19.4 | 962.2 |
1b
| - | 948.7 | 929.2 | 19.4 | 968.1 |
2 | - | 957.3 | 937.5 | 19.9 | 977.2 |
3 | - | 964.5 | 944.3 | 20.2 | 984.7 |
-2 | -2 | 865.5 | 835.8 | 29.8 | 895.3 |
-2 | -1 | 857.2 | 827.0 | 30.2 | 887.4 |
-2 | -0.5 | 850.9 | 821.2 | 29.7 | 880.6 |
-2 | 0 | 844.4 | 815.2 | 29.1 | 873.5 |
-2 | 0.5 | 840.2 | 810.6 | 29.7 | 869.9 |
-2
|
1
|
837.1
|
807.1
|
30.1
|
867.2
|
-2 | 2 | 837.3 | 807.2 | 30.1 | 867.4 |
-2 | 3 | 837.8 | 808.2 | 29.6 | 867.4 |
-1 | -1 | 848.4 | 819.8 | 28.6 | 877.0 |
-1 | -0.5 | 849.0 | 820.1 | 28.9 | 877.9 |
-1 | 0 | 840.9 | 812.6 | 28.3 | 869.2 |
-1 | 0.5 | 840.2 | 811.4 | 28.8 | 869.0 |
-1 | 1 | 839.8 | 809.8 | 30.0 | 869.8 |
-1 | 2 | 844.3 | 814.0 | 30.2 | 874.5 |
-1 | 3 | 850.4 | 820.4 | 30.0 | 880.5 |
-0.5 | -0.5 | 840.5 | 811.6 | 28.9 | 869.4 |
-0.5 | 0 | 842.2 | 813.6 | 28.6 | 870.8 |
-0.5 | 0.5 | 842.4 | 812.3 | 30.1 | 872.5 |
-0.5 | 1 | 843.6 | 813.9 | 29.7 | 873.3 |
-0.5 | 2 | 851.9 | 822.9 | 29.1 | 881.0 |
-0.5 | 3 | 861.4 | 831.4 | 30.0 | 891.5 |
0 | 0 | 845.8 | 817.2 | 28.6 | 874.4 |
0 | 0.5 | 849.8 | 821.6 | 28.2 | 878.0 |
0 | 1 | 854.6 | 823.4 | 31.2 | 885.8 |
0 | 2 | 862.5 | 833.3 | 29.1 | 891.6 |
0 | 3 | 874.2 | 844.6 | 29.6 | 903.9 |
0.5 | 0.5 | 854.3 | 830.6 | 23.8 | 878.1 |
0.5 | 1 | 860.0 | 831.8 | 28.2 | 888.1 |
0.5 | 2 | 876.4 | 846.7 | 29.7 | 906.1 |
0.5 | 3 | 888.3 | 858.8 | 29.4 | 917.7 |
1 | 1 | 871.4 | 842.0 | 29.4 | 900.8 |
1 | 2 | 887.2 | 858.3 | 28.9 | 916.2 |
1 | 3 | 902.3 | 871.7 | 30.6 | 932.9 |
2 | 2 | 907.8 | 880.0 | 27.9 | 935.7 |
2 | 3 | 921.2 | 892.2 | 29.0 | 950.2 |
3 | 3 | 934.6 | 906.9 | 27.7 | 962.4 |
Random effects models
| |||||
-2 | - | 904.4 | 883.2 | 21.3 | 925.7 |
0a
| - | 934.4 | 911.4 | 23.0 | 957.4 |
-2c
| 1 | 835.9 | 803.9 | 32.0 | 867.9 |
-2
d
|
1
|
836.1
|
805.1
|
31.0
|
867.1
|
Fixed effects model, first order fractional polynomial | Fixed effects model, second order fractional polynomial | Random effects model (d0), second order fractional polynomial | |||||
---|---|---|---|---|---|---|---|
Median of posterior distribution | 95% Credible Interval | Median of posterior distribution | 95% Credible Interval | Median of posterior distribution | 95% Credible Interval | ||
p1
| power 1 | -2 | -2 | -2 | |||
p2
| power 2 | 1 | 1 | ||||
Pooled estimate for difference in β
0
| |||||||
d
0AB
| gefitinib vs. docetaxel | -0.003 | (-0.111; 0.112) | 0.158 | (-0.065; 0.409) | 0.144 | (-0.156; 0.411) |
d
0AC
| BSC vs. docetaxel | 0.770 | (0.598; 0.939) | 1.674 | (1.108; 2.193) | 1.653 | (1.039; 2.167) |
d
0AD
| pemetrexed vs. docetaxel | 0.020 | (-0.189; 0.243) | 0.141 | (-0.483; 0.744) | 0.142 | (-0.540; 0.761) |
Pooled estimate for difference in β
1
| |||||||
d
1AB
| gefitinib vs. docetaxel | 0.634 | (-0.429; 1.726) | -0.003 | (-1.470; 1.362) | 0.070 | (-1.330; 1.542) |
d
1AC
| BSC vs. docetaxel | -2.507 | (-4.281; -0.783) | -5.858 | (-8.503; -3.228) | -5.839 | (-8.265; -3.131) |
d
1AD
| pemetrexed vs. docetaxel | -0.995 | (-3.344; 1.381) | -1.359 | (-4.604; 1.885) | -1.368 | (-4.511; 1.814) |
Pooled estimate for difference in β
2
| |||||||
d
2AB
| gefitinib vs. docetaxel | -0.007 | (-0.016; 0.002) | -0.006 | (-0.015; 0.003) | ||
d
2AC
| BSC vs. docetaxel | -0.053 | (-0.082; -0.023) | -0.052 | (-0.079; -0.023) | ||
d
2AD
| pemetrexed vs. docetaxel | -0.005 | (-0.032; 0.023) | -0.005 | (-0.031; 0.022) | ||
σ
d0
| Standard deviation of d
0
; heterogeneity in difference of β
0
across comparisons | 0.060 | (0.002; 0.406) |
Fixed effects model, first order fractional polynomial | Fixed effects model, second order fractional polynomial | Random effects model (d0), second order fractional polynomial | ||||
---|---|---|---|---|---|---|
Median of posterior distribution | 95% Credible Interval | Median of posterior distribution | 95% Credible Interval | Median of posterior distribution | 95% Credible Interval | |
Docetaxel * | ||||||
β0A
| -2.422 | (-2.502; -2.348) | -2.689 | (-2.882; -2.476) | -2.694 | (-2.895; -2.513) |
β1A
| -2.174 | (-2.988; -1.401) | -1.224 | (-2.324; -0.198) | -1.193 | (-2.267; -0.206) |
β2A
| 0.013 | (0.004; 0.022) | 0.014 | (0.006; 0.022) | ||
Gefitinib | ||||||
β0B (β0A +d
0AB
) | -2.425 | (-2.515; -2.325) | -2.531 | (-2.776; -2.322) | -2.550 | (-2.777; -2.273) |
β1B (β1A +d
1AB
) | -1.540 | (-2.527; -0.631) | -1.227 | (-2.407; 0.082) | -1.124 | (-2.433; 0.010) |
β2B (β2A +d
2AB
) | 0.007 | (-0.002; 0.018) | 0.008 | (-0.002; 0.016) | ||
BSC | ||||||
β 0C (β0A +d
0AC
) | -1.652 | (-1.818; -1.461) | -1.015 | (-1.600; -0.480) | -1.041 | (-1.554; -0.462) |
β1C (β1A +d
1AC
) | -4.681 | (-6.564; -3.010) | -7.082 | (-9.677; -4.320) | -7.032 | (-9.700; -4.622) |
β2C (β2A +d
2AC
) | -0.040 | (-0.071; -0.008) | -0.039 | (-0.069; -0.010) | ||
Pemetrexed | ||||||
β0D (β0 A +d
0AD
) | -2.402 | (-2.616; -2.188) | -2.548 | (-3.101; -1.954) | -2.552 | (-3.093; -1.978) |
β1D (β1A +d
1AD
) | -3.169 | (-5.445; -0.673) | -2.583 | (-5.634; 0.308) | -2.561 | (-5.329; 0.167) |
β2D (β2A +d
2AD
) | 0.009 | (-0.018; 0.033) | 0.009 | (-0.014; 0.028) | ||
Expected survival (in months) | ||||||
docetaxel | 12.5 | (11.9; 13.3) | 13.0 | (12.2; 13.8) | 13.0 | (12.2; 13.9) |
gefitinib | 12.1 | (11.3; 13.0) | 12.2 | (11.3; 13.2) | 12.2 | (10.6; 14.0) |
BSC | 7.2 | (6.5; 8.1) | 6.2 | (5.1; 7.6) | 6.2 | (4.8; 7.9) |
pemetrexed | 12.7 | (10.9; 14.7) | 12.7 | (10.5; 15.0) | 12.9 | (9.5; 17.4) |
Difference in expected survival (in months) | ||||||
gefitinib vs docetaxel | -0.4 | (-1.4; 0.6) | -0.8 | (-1.9; 0.3) | -0.8 | (-2.6; 1.1) |
BSC vs docetaxel | -5.3 | (-6.2; -4.3) | -6.8 | (-8.0; -5.4) | -6.8 | (-8.4; -5.0) |
pemetrexed vs docetaxel | 0.1 | (-1.8; 2.2) | -0.3 | (-2.5; 2.0) | -0.2 | (-3.6; 4.4) |
BSC vs gefitinib | -4.9 | (-6.0; -3.8) | -6.0 | (-7.4; -4.5) | -6.0 | (-8.1; -3.9) |
pemetrexed vs gefitinib | 0.5 | (-1.6; 2.8) | 0.4 | (-2.0; 3.0) | 0.6 | (-3.2; 5.3) |
pemetrexed vs. BSC | 5.4 | (3.4; 7.6) | 6.4 | (3.9; 9.1) | 6.7 | (3.0; 11.4) |