Correlations between each pair of mammogram risk scores
Table 2 shows that for all pairs of mammogram risk scores, the within-twin within-trait correlations ranged from 0.22 to 0.59, the within-twin cross-trait correlation ranged from 0.28 to 0.81, and the cross-twin cross-trait correlations ranged from 0.28 to 0.61 (all P < 0.05).
Table 2
The within-twin within-trait correlations, the within-twin cross-trait correlation, and the cross-twin cross-trait correlations between the mammogram risk scores (95% confidence intervals in parentheses)
| Light areas | Bright areas | Brightest areas | Cirrus |
Light areas | 0.59 (0.54,0.64) | 0.81 (0.79,0.84) | 0.49 (0.44,0.55) | 0.38 (0.32,0.44) |
Bright areas | 0.52 (0.47,0.57) | 0.53 (0.47,0.58) | 0.69 (0.65,0.72) | 0.45 (0.39,0.51) |
Brightest areas | 0.35 (0.29,0.41) | 0.40 (0.34,0.46) | 0.34 (0.27,0.40) | 0.43 (0.37,0.49) |
Cirrus | 0.28 (0.21,0.35) | 0.33 (0.26,0.39) | 0.28 (0.22,0.35) | 0.48 (0.42,0.53) |
| Bright areas a | Brightest areas a | | |
Bright areas a | 0.41 (0.35,0.47) | 0.28 (0.21,0.34) | | |
Brightest areas a | 0.61 (0.57,0.66) | 0.22 (0.15,0.29) | | |
Note: Bold indicates the within-twin within-trait correlations, the upper triangle above the diagonal (in bold) indicates the within-twin cross-trait correlations, the lower triangle below the diagonal (in bold) indicates the cross-twin cross-trait correlations. |
a adjusted for age at mammography and Cirrus |
Causal Inference For Pairs Of Mammogram Risk Scores
Table 3 shows the ICE FALCON results of model fitting and inference about proportions of familial confounding and causation assigned to the associations between the pairs of mammogram risk scores; similar analyses of using the Cumulus and Altocumulus measures can be found in Supplementary material 1; see Table S2.
Table 3
The relationships between Cirrus and mammographic density measures analysed by using the ICE FALCON method
Assignment of X-Y | | Model 1 | | Model 2 | | Model 3 | | Change* | | Conclusion from ICE FALCON |
| Coef.(se) | P | Coef.(se) | P | Coef.(se) | P | Coef.(se) | P | Familial confounding | Causal effect |
Light areas-bright areas | Self | 0.802(0.030) | 2×10− 161 | | | 0.770(0.033) | 5×10− 117 | 0.032(0.013) | 0.02 | Yes (11%) | Light areas cause bright areas (89%) |
Co-twin | | | 0.513(0.041) | 4×10− 35 | 0.070(0.027) | 0.01 | 0.443(0.037) | 8×10− 34 |
Bright areas-light areas | Self | 0.770(0.025) | 4×10− 210 | | | 0.727(0.026) | 10×10− 174 | 0.043(0.011) | 5×10− 5 | Yes (42%) | Bright areas cause light areas (58%) |
Co-twin | | | 0.404(0.045) | 10− 19 | 0.144(0.025) | 8×10− 9 | 0.259(0.035) | 10− 13 |
Light areas-Cirrus | Self | 0.352(0.041) | 6×10− 18 | | | 0.328(0.041) | 3×10− 15 | 0.024(0.011) | 0.02 | Yes (63%) | Light areas cause Cirrus (37%) |
Co-twin | | | 0.176(0.038) | 4×10− 06 | 0.087(0.037) | 0.02 | 0.089(0.020) | 10− 5 |
Cirrus-light areas | Self | 0.285(0.035) | 2×10− 16 | | | 0.303(0.034) | 6×10− 19 | -0.018(0.008) | 0.02 |
Co-twin | | | 0.091(0.033) | 6×10− 03 | 0.135(0.030) | 9×10− 6 | -0.045(0.015) | 3×10− 3 |
Bright areas- Cirrus | Self | 0.398(0.038) | 6×10− 26 | | | 0.367(0.039) | 2×10− 21 | 0.031(0.010) | 2×10− 3 | Yes (72%) | Bright areas cause Cirrus (28%) |
Co-twin | | | 0.215(0.038) | 10− 08 | 0.139(0.036) | 10− 4 | 0.076(0.021) | 2×10− 4 |
Cirrus- bright areas | Self | 0.361(0.034) | 5×10− 26 | | | 0.355(0.034) | 10− 25 | 0.006(0.008) | 0.46 |
Co-twin | | | 0.165(0.036) | 6×10− 06 | 0.163(0.033) | 7×10− 7 | 0.002(0.017) | 0.92 |
Brightest areas-Cirrus | Self | 0.351(0.040) | 2×10− 18 | | | 0.356(0.039) | 2×10− 20 | -0.005(0.008) | 0.5 | Yes (66%) | Cirrus causes brightest areas (34%) |
Co-twin | | | 0.138(0.035) | 9×10− 05 | 0.166(0.032) | 3×10− 7 | -0.028(0.016) | 0.08 |
Cirrus-brightest areas | Self | 0.389(0.039) | 1×10− 23 | | | 0.360(0.040) | 9×10− 20 | 0.028(0.010) | 4×10− 3 |
Co-twin | | | 0.191(0.038) | 5×10− 07 | 0.114(0.036) | 2×10− 03 | 0.077(0.018) | 3×10− 5 |
Light areas-brightest areas | Self | 0.470(0.036) | 5×10− 38 | | | 0.424(0.041) | 3×10− 25 | 0.046(0.017) | 8×10− 3 | Yes (45%) | Light areas cause brightest areas (55%) |
Co-twin | | | 0.284(0.040) | 2×10− 12 | 0.103(0.037) | 5×10− 3 | 0.181(0.025) | 9×10− 13 |
Brightest areas-light areas | Self | 0.347(0.035) | 7×10− 23 | | | 0.395(0.033) | 4×10− 33 | -0.048(0.012) | 6×10− 5 |
Co-twin | | | 0.113(0.034) | 8×10− 4 | 0.221(0.030) | 10− 13 | -0.108(0.018) | 3×10− 9 |
Bright areas- brightest areas | Self | 0.680(0.025) | 10− 159 | | | 0.653(0.029) | 3×10− 109 | 0.027(0.014) | 6×10− 2 | Yes (15%) | Bright areas cause brightest areas (85%) |
Co-twin | | | 0.378(0.039) | 2×10− 22 | 0.058(0.031) | 0.07 | 0.320(0.031) | 5×10− 25 |
Brightest areas-bright areas | Self | 0.619(0.032) | 2×10− 84 | | | 0.606(0.029) | 4×10− 94 | 0.012(0.010) | 0.21 |
Co-twin | | | 0.136(0.038) | 4×10− 4 | 0.202(0.027) | 8×10− 14 | -0.065(0.027) | 0.02 |
Light areas a -brightest areas a | Self | 0.386(0.038) | 9×10− 25 | | | 0.365(0.041) | 6×10− 9 | 0.020(0.016) | 0.217 | Yes (33%) | Light areas cause brightest areas not through Cirrus (67%) |
Co-twin | | | 0.198(0.043) | 5×10− 6 | 0.051(0.040) | 0.201 | 0.147(0.024) | 5×10− 10 |
Brightest areas a -light areas a | Self | 0.287(0.034) | 10− 17 | | | 0.339(0.034) | 2×10− 23 | -0.052(0.012) | 2×10− 5 |
Co-twin | | | 0.046(0.036) | 0.202 | 0.165(0.033) | 8×10− 7 | -0.119(0.018) | 7×10− 11 |
Bright areas a -brightest areas a | Self | 0.608(0.030) | 4×10− 90 | | | 0.597(0.033) | 3×10− 75 | 0.011(0.011) | 0.321 | Yes (8%) | Bright areas cause brightest areas not through Cirrus (92%) |
Co-twin | | | 0.268(0.044) | 2×10− 9 | 0.033(0.033) | 0.312 | 0.234(0.032) | 2×10− 13 |
Brightest areas a - bright areas a | Self | 0.562(0.032) | 6×10− 69 | | | 0.569(0.031) | 3×10− 77 | -0.008(0.008) | 0.326 |
Co-twin | | | 0.058(0.044) | 0.186 | 0.153(0.031) | 9×10− 7 | -0.096(0.028) | 0.001 |
se standard error |
a adjusted for Cirrus and age at mammography, |
Model 1: Yself = βselfXself + ε1, Model 2: Yself = βcotwinXcotwin + ε2, Model 3: Yself = βselfXself + βcotwinXcotwin + ε3. |
*change refers to the change in coefficients from Model 1 to Model 3 for woman herself, the change in coefficients from Model 2 to Model 3 for woman’s cotwin. |
The Bright Areas And Light Areas
With the light areas as the predictor and the bright areas as the outcome, there was a decrease of 4% (P = 0.02) from βself = 0.802 (P = 10− 161) in Model 1 to b’self = 0.770 (P = 10− 117) in Model 3, and a decrease of 86% (P = 10− 34) from βco−twin = 0.513 (P = 10− 34) in Model 2 to b’co−twin = 0.070 (P = 0.01) in Model 3. These results are consistent with the light areas having a causal effect on the bright areas that accounts for 89% of their association, with marginal evidence for familial confounding.
With the bright areas as the predictor and the light areas as the outcome, there was a decrease of 6% (P = 10− 5) from βself = 0.770 (P = 10− 210) in Model 1 to b’self = 0.727 (P = 10− 174) in Model 3, and a decrease of 64% (P = 10− 33) from βco−twin = 0.404 (P = 10− 19) in Model 2 to b’co−twin = 0.144 (P = 10− 9) in Model 3. These results are consistent with the bright areas having a causal effect on the light areas that accounts for 58% of their association, and familial confounding which accounts for 42%.
Therefore, the ICE FALCON results suggest the existence of familial confounding and bidirectional causality between the light areas and bright areas. To avoid confusion arising from the potential bidirectional causation, we conducted analyses separately for the light areas and bright areas.
Cirrus And The Light Areas
With Cirrus as the outcome and the light areas as the predictor, there was a decrease of 7% (P = 0.03) from βself = 0.352 (P = 10− 17) in Model 1 to b’self = 0.328 (P = 10− 16) after adjusting for co-twin’s light areas in Model 3. There was a decrease of 51% (P = 10− 7) from βco−twin = 0.176 (P = 10− 5) in Model 2 to b’co−twin = 0.087 (P = 0.02) after adjusting for the twin’s light areas in Model 3. These results are consistent with there being a combination of familial confounding that accounts for 37% of the association between the two risk scores as well as the light areas having a causal effect on Cirrus that accounts for 63% of their association.
We then reversed the predictor and outcome roles by assigning the light areas to be the outcome and Cirrus as the predictor. There was an increase from βself = 0.285 (P = 10− 15) in Model 1 to β’self = 0.303 (P = 10− 20) in Model 3 (P = 0.02). There was also an increase in the co-twin’s coefficient from βco−twin = 0.091 (P = 0.006) in Model 2 to β’co−twin = 0.135 (P = 10− 5) in Model 3 (P = 0.003). These results are consistent with the findings above that the association is due to a combination of familial confounding and the light areas having a causal effect on Cirrus.
Cirrus And The Bright Areas
With Cirrus as the outcome and the bright areas as the predictor, there was a decrease of 8% (P = 0.002) from βself = 0.398 (P = 10− 27) in Model 1 to β’self = 0.367 (P = 10− 21) in Model 3. There was a decrease of 35% (P = 10− 4) from βco−twin = 0.215 (P = 10− 8) in Model 2 to β’co−twin = 0.139 (P = 10− 4) in Model 3. These results are consistent with there being a combination of familial confounding that accounts for 72% of the association between the two risk scores, and the bright areas having a causal effect on Cirrus that accounts for 28% of their association.
When we reversed the predictor and outcome roles by assigning the bright areas to be the outcome and Cirrus as the predictor, βself was not significantly different from β’self after adjusting for co-twin’s Cirrus (P = 0.5), and a similar statement applies to the lack of a difference between βco−twin and β’co−twin (P = 0.9). These results were not consistent with Cirrus having a causal effect on the bright areas.
Cirrus And The Brightest Areas
With brightest areas as the outcome and Cirrus as the predictor, there was a decrease of 7% (P = 10− 3) from βself = 0.389 (P = 10− 23) in Model 1 to β’self = 0.360 (P = 10− 19) in Model 3 (P = 0.004), while there was a decrease of 40% (P = 10− 5) from βco−twin = 0.191 (P = 10− 6) in Model 2 to β’co−twin = 0.114 (P = 0.002) in Model 3 (P = 10− 5). These results are consistent with Cirrus having a causal effect on the brightest areas, which accounts for 34% of their association, as well as there being familial confounding which accounts for 64%.
When we reversed the predictor and outcome roles by assigning brightest areas to be the predictor and Cirrus as the outcome, there was no difference between βself and β’self (P = 0.5), while there was an increase from βco−twin = 0.138 (P = 10− 5) in Model 2 to β’cotwin = 0.166 (P = 10− 7) in Model 3 that was not nominally significant (P = 0.08). These results were not consistent with the brightest areas having a causal effect on Cirrus.
The Light Areas And Brightest Areas
With the light areas as the predictor and the brightest areas as the outcome, there was a decrease of 10% (P = 0.008) from βself = 0.470 (P = 10− 38) in Model 1 to β’self = 0.424 (P = 10− 25) in Model 3, and a decrease of 64% (P = 10− 13) from βco−twin = 0.284 (P = 10− 12) in Model 2 to β’co−twin = 0.103 (P = 0.005) in Model 3. These results are consistent with the light areas having a causal effect on the brightest areas that accounts for 55% of their association, and familial confounding which accounts for 45%.
With the brightest areas as the predictor and the light areas as the outcome, there was an increase from βself = 0.347 (P = 10− 23) in Model 1 to 0.395 (P = 10− 33) in Model 3 (P = 10− 5), and an increase from βco−twin = 0.113 (P = 10− 4) in Model 2 to β,co−twin = 0.221 (P = 10− 13) in Model 3 (P = 10− 9). These results are not consistent with the brightest areas having a causal effect on the light areas.
The Bright Areas And Brightest Areas
With the bright areas as the predictor and the brightest areas as the outcome, there was a marginally significant decrease of 4% (P = 0.06) from βself = 0.680 (P = 10− 159) in Model 1 to β’self = 0.653 (P = 10− 109) in Model 3 (P = 0.06), while there was a decrease of 85% (P = 10− 25) from βco−twin = 0.378 (P = 10− 22) in Model 2 to b’co−twin = 0.058 (P = 0.07) in Model 3 of 85% (P = 10− 25). These results are consistent with the bright areas having a causal effect on the brightest areas that accounts for 85% of their association, and familial confounding which accounts for 15%.
With the brightest areas as the predictor and the bright areas as the outcome, there was no difference between βself and β’self (P = 0.21), while there was an increase in βco−twin = 0.136 (P = 10− 4) in Model 2 to β’co−twin = 0.202 (P = 10− 14) in Model 3 (P = 0.02). These results are not consistent with the brightest areas having a causal effect on the bright areas.
The above results are consistent with the existence of causal pathways from both the light areas and the bright areas to the brightest areas, and to Cirrus, and from Cirrus to the brightest areas.
Causal Inference For The Light Areas And Brightest Areas Not Through Cirrus
With the light areas adjusted for Cirrus as the predictor and the brightest areas adjusted for Cirrus as the outcome, there was no difference between βself in Model 1 and β,self in Model 3 (P = 0.2), while there was a decrease from βco−twin = 0.198 (P = 10− 6) in Model 2 to β’co−twin = 0.051 (P = 0.2) in Model 3 by 74% (P = 10− 10). These results are consistent with a causal effect from the light areas to the brightest areas, and not through Cirrus, that accounts for 67% of their association and with familial confounding that accounts for 33%.
With brightest areas adjusted for Cirrus as the predictor and light areas adjusted for Cirrus as the outcome, βself increased from 0.287 (P = 10− 17) in Model 1 to 0.339 (P = 10− 23) after adjusting for co-twin’s brightest areas in Model 3 (P = 10− 5), βco−twin increased from 0.046 (P = 0.2) in Model 2 to β’co−twin = 0.165 (P = 10− 7) after adjusting for the twin’s brightest areas in Model 3 (P = 10− 11). These results are consistent with a combination of a direct causal effect from the light areas to the brightest areas not through Cirrus, and familial confounding.
Causal Inference For The Bright Areas And Brightest Areas Not Through Cirrus
With the bright areas adjusted for Cirrus as the predictor and the brightest areas adjusted for Cirrus as the outcome, βself decreased, but not significantly (P = 0.3), from 0.608 (P = 10− 90) in Model 1 to β’self = 0.597 (P = 10− 75) after adjusting for co-twin’s bright areas in Model 3, βco−twin decreased from 0.268 (P = 10− 9) in Model 2 to the null (P = 0.3) after adjusting for co-twin’s bright areas in Model 3 by 87% (P = 10− 13). These results are consistent with the bright areas having a causal effect on the brightest areas not through Cirrus, which accounts for 92% of their association, and the familial confounding accounting for 8% of the association.
With the brightest areas adjusted for Cirrus as the predictor and the bright areas adjusted for Cirrus as the outcome, βself did not change significantly (P = 0.3) after adjusting for co-twin’s brightest areas, βco−twin increased from 0.08 (P = 0.186) in Model 2 to β’co−twin 0.153 (P = 10− 6) after adjusting for the twin’s brightest areas in Model 3 (P = 0.001). These results are consistent with a combination of a causal effect, from the bright areas to the brightest areas not through Cirrus, and familial confounding.
Figure 2 shows the possible causal pathways between the three mammographic density measures and Cirrus, based on the results presented.