Background
Methods
Approach I
Population | Input parameters | Estimated parameters | |||||||
---|---|---|---|---|---|---|---|---|---|
ρ | Normal (μ, σ) | Adjusted OR | Cases | Controls | SD of β
0
+ ∑β
i
X
i
| AUC | |||
ρ | (μ, σ) | ρ | (μ, σ) | ||||||
A | 0.2 |
μ: (0, 0);
\( \sigma \): (1, 1) | (1.5, 1.5) | 0.17 |
μ: (0.37, 0.35);
\( \sigma \): (0.97, 0.97) | 0.17 |
μ: (-0.09, -0.09);
\( \sigma \): (0.98, 0.98) | 0.61 | 0.663 |
B | -0.1 | ,, | ,, | -0.12 |
μ: (0.27, 0.28);
\( \sigma \): (0.98, 0.98) | -0.12 |
μ: (-0.07, -0.07);
\( \sigma \): (0.99, 0.99) | 0.54 | 0.645 |
C | - 0.2 | ,, | ,, | -0.22 |
μ: (0.26, 0.24);
\( \sigma \): (0.99, 0.99) | -0.22 |
μ: (-0.06, -0.06)
\( \sigma \): (0.99, 0.99) | 0.51 | 0.639 |
D | 0.1 | ,, | ,, | 0.07 |
μ: (0.33, 0.33)
\( \sigma \): (0.99, 0.99) | 0.07 |
μ: (-0.08, -0.08)
\( \sigma \): (0.99, 0.99) | 0.59 | 0.660 |
E | 0.4 | ,, | ,, | 0.37 |
μ: (0.42, 0.42)
\( \sigma \): (0.97, 0.97) | 0.37 |
μ: (-0.10, -0.10)
\( \sigma \): (0.98, 0.98) | 0.67 | 0.676 |
F | 0.2 | ,, | (1.5, 1.2) | 0.18 |
μ: (0.34, 0.20)
\( \sigma \): (0.98, 0.98) | 0.19 |
μ: (-0.08, -0.05)
\( \sigma \): (0.99, 0.99) | 0.47 | 0.629 |
G | ,, | ,, | (1.2, 1.2) | 0.19 |
μ: (0.16, 0.17)
\( \sigma \): (1, 1) | 0.19 |
μ: (-0.04, -0.04)
\( \sigma \): (1, 1) | 0.27 | 0.575 |
H | ,, | ,, | (1.5, 3) | 0.10 |
μ: (0.41, 0.76)
\( \sigma \): (0.97, 0.97) | 0.14 |
μ: (-0.10, -0.19)
\( \sigma \): (0.98, 0.98) | 1.25 | 0.789 |
I | ,, | ,, | (0.8, 0.8) | 0.20 |
μ: (-0.20, -0.20)
\( \sigma \): (1, 1) | 0.19 |
μ: (0.05, 0.05)
\( \sigma \): (0.99, 0.99) | 0.33 | 0.593 |
J | -0.1 | ,, | (1.5, 0.8) | -0.09 |
μ: (0.37, -0.20);
\( \sigma \): (0.99, 0.99) | -0.08 |
μ: (-0.08, 0.05);
\( \sigma \): (0.98, 0.98) | 0.49 | 0.632 |
K | 0.2 | ,, | ,, | 0.21 |
μ: (0.28, -0.11)
\( \sigma \): (0.99, 0.99) | 0.21 |
μ: (-0.07, 0.03)
\( \sigma \): (0.99, 0.99) | 0.42 | 0.616 |
L | 0.4 | ,, | ,, | 0.40 |
μ: (0.24, -0.05);
\( \sigma \): (0.99, 0.99) | 0.41 |
μ: (-0.06, 0.01);
\( \sigma \): (0.99, 0.99) | 0.37 | 0.603 |
M | ,, | Mean: (0, 0); SD: (1, 3) | (1.5, 1.5) | 0.10 |
μ: (0.41, 2.47)
\( \sigma \): (0.97, 0.97) | 0.14 |
μ: (-0.10, -0.62)
\( \sigma \): (0.98, 0.98) | 1.37 | 0.804 |
N | - 0.2 | ,, | ,, | -0.25 |
μ: (0.11, 2.25)
\( \sigma \): (1, 1) | -0.24 |
μ: (-0.03, -0.56)
\( \sigma \): (1, 1) | 1.21 | 0.781 |
O | 0.1 | ,, | ,, | 0.01 |
μ: (0.34, 2.38)
\( \sigma \): (0.98, 0.98) | 0.04 |
μ: (-0.08, -0.59)
\( \sigma \): (0.99, 0.99) | 1.31 | 0.795 |
P | 0.4 | ,, | ,, | 0.30 |
μ: (0.56, 2.56)
\( \sigma \): (0.94, 0.94) | 0.33 |
μ: (-0.14, -0.63)
\( \sigma \): (0.96, 0.96) | 1.42 | 0.810 |
Approach II
Population | Input parameters for cases and controls | Estimated parameters for the population | ||||||
---|---|---|---|---|---|---|---|---|
ρ | Normal (μ, σ) | ρ | (μ, σ) | Unadjusted OR * | Adjusted OR ** | SD of β
0
+ ∑β
i
X
i
| AUC | |
A | Cases = 0.2 Ctrls = 0.2 |
μCases: (1, 2); Ctrls: (0, 0)
\( \sigma \)Cases: (2, 2); Ctrls: (2, 2) | 0.25 |
μ: (0.2, 0.4); \( \sigma \): (2.04, 2.15) | 1.28, 1.65 | 1.17, 1.60 | 1.13 | 0.770 |
B | Cases = 0.2 Ctrls = 0.4 | ,, | 0.40 | ,, | ,, | 1.09, 1.60 | 1.09 | 0.765 |
C | Cases = 0.2 Ctrls = - 0.2 | ,, | -0.04 | ,, | ,, | 1.34, 1.68 | 1.25 | 0.785 |
D | Cases = 0.1 Ctrls = 0.1 | ,, | 0.16 | ,, | ,, | 1.22, 1.62 | 1.17 | 0.777 |
E | Cases = - 0.1 Ctrls = - 0.1 | ,, | -0.02 | ,, | ,, | 1.35, 1.70 | 1.28 | 0.795 |
F | Cases = 0.2 Ctrls = 0.2 |
μCases: (1, 3); Ctrls: (0, 0) SD Cases: (2, 2); Ctrls: (2, 2) | 0.27 |
μ: (0.2, 0.6); \( \sigma \): (2.04, 2.33) | 1.28, 2.12 | 1.11, 2.07 | 1.77 | 0.858 |
G | ,, |
μCases: (1, 3); Ctrls: (0, 2) SD Cases: (2, 2); Ctrls: (2, 2) | 0.23 |
μ: (0.2, 0.2); \( \sigma \): (2.04, 2.04) | 1.28, 1.28 | 1.23, 1.23 | 0.67 | 0.676 |
H | ,, |
μCases: (1, 2); Ctrls: (0, 0) SD Cases: (2, 3); Ctrls: (2,3) | 0.24 |
μ: (0.2, 0.4); \( \sigma \): (2.04, 3.10) | 1.28, 1.25 | 1.21, 1.22 | 0.80 | 0.705 |
I | ,, |
μCases: (1, 2); Ctrls: (0, 0) SD Cases: (2, 1); Ctrls: (2, 1) | 0.27 |
μ: (0.2, 0.4); \( \sigma \): (2.04, 1.28) | 1.28, 7.39 | 1.05, 7.23 | 2.56 | 0.922 |
Analyses
Results
Model development
Approach I
Approach II
Model validation
Validated in population | Developed in population | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A | B | C | D | E | F | G | H | I | J | K | L | M | N | 0 | P | |
A |
0.663
| * | * | * | * | 0.656 | 0.663 | 0.652 | 0.663 | 0.556 | ** | ** | * | * | * | * |
B | 0.645 |
0.645
| * | * | * | 0.632 | 0.645 | 0.631 | 0.644 | 0.534 | ** | ** | * | * | * | * |
C | 0.639 | * |
0.639
| * | * | 0.626 | 0.639 | 0.622 | 0.639 | 0.534 | ** | ** | * | * | * | * |
D | 0.660 | * | * |
0.660
| * | 0.649 | 0.660 | 0.649 | 0.659 | 0.545 | ** | ** | * | * | * | * |
E | 0.676 | * | * | * |
0.676
| 0.670 | 0.676 | 0.670 | 0.676 | 0.570 | ** | ** | * | * | * | * |
F | 0.624 | * | * | * | * |
0.629
| 0.624 | 0.602 | 0.625 | 0.578 | ** | ** | * | * | * | * |
G | 0.575 | * | * | * | * | 0.571 |
0.575
| 0.570 | 0.575 | 0.526 | ** | ** | * | * | * | * |
H | 0.770 | * | * | * | * | 0.728 | 0.770 |
0.789
| 0.767 | 0.502 | ** | ** | * | * | * | * |
I | 0.593 | * | * | * | * | 0.590 | 0.593 | 0.587 |
0.593
| 0.534 | ** | ** | * | * | * | * |
J | 0.531 | * | * | * | * | 0.529 | 0.530 | 0.531 | 0.530 |
0.632
| ** | ** | * | * | * | * |
K | 0.540 | * | * | * | * | 0.571 | 0.540 | 0.502 | 0.543 | 0.615 |
0.616
| ** | * | * | * | * |
L | 0.542 | * | * | * | * | 0.541 | 0.540 | 0.541 | 0.542 | 0.602 | ** |
0.603
| * | * | * | * |
M | 0.804 | * | * | * | * | 0.792 | 0.804 | 0.800 | 0.804 | 0.693 | ** | ** |
0.804
| * | * | * |
N | 0.781 | * | * | * | * | 0.759 | 0.781 | 0.775 | 0.781 | 0.692 | ** | ** | * |
0.781
| * | * |
O
| 0.795 | * | * | * | * | 0.782 | 0.795 | 0.790 | 0.795 | 0.686 | ** | ** | * | * |
0.795
| * |
P | 0.810 | * | * | * | * | 0.802 | 0.810 | 0.806 | 0.810 | 0.695 | ** | ** | * | * | * |
0.810
|
Validated in population | Developed in population | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
A | B | C | D | E | F | G | H | I | ||
A
|
0.770
| 0.768 | 0.767 | 0.770 | 0.767 | 0.767 | 0.753 | 0.754 | 0.762 | |
B
| 0.764 |
0.765
| 0.759 | 0.763 | 0.759 | 0.764 | 0.745 | 0.746 | 0.761 | |
C
| 0.783 | 0.777 |
0.785
| 0.785 | 0.784 | 0.773 | 0.774 | 0.774 | 0.763 | |
D
| 0.777 | 0.773 | 0.776 |
0.777
| 0.776 | 0.771 | 0.763 | 0.764 | 0.763 | |
E
| 0.790 | 0.781 | 0.795 | 0.793 |
0.795
| 0.777 | 0.785 | 0.786 | 0.763 | |
F
| 0.855 | 0.858 | 0.845 | 0.852 | 0.845 |
0.858
| 0.819 | 0.821 | 0.857 | |
G
| 0.664 | 0.655 | 0.672 | 0.667 | 0.671 | 0.651 |
0.676
| 0.676 | 0.642 | |
H
| 0.696 | 0.690 | 0.702 | 0.700 | 0.703 | 0.689 | 0.705 |
0.705
| 0.682 | |
I
| 0.896 | 0.914 | 0.864 | 0.885 | 0.863 | 0.917 | 0.811 | 0.814 |
0.922
|