Background
Methods
The Bayesian method to estimate the biomarker optimal threshold and its credible interval
Determination of the optimal threshold using the decision theory
Bayesian optimal-threshold estimation
Heterogeneity in the variance of biomarker measurements
Optimal threshold with Student-t distribution
Simulations
Marker values truly distributed according to a normal law (Design 1)
Marker values truly distributed according to a Student-t law (Design 2)
Marker distributed according to a mixture of two normal laws in the diseased subjects (Design 3)
Application
Heterogeneity in the mean and the variance of biomarker measurements
Optimal threshold using a mixture of Dirichlet processes
Simulations (Design 4)
Application
Results
Heterogeneity in the variance of biomarker measurements
Simulations
Marker values truly distributed according to a normal law (Design 1)
Relative bias*
| Coverage probability†
| CI mean width†
| |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mode | Median | Mean | Quantile | HPD | Quantile | HPD | |||||||||||
N
0
|
N
1
|
σ
0
|
σ
1
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
|
100 | 100 | 0.07 | 0.07 | −0.00022 | −0.00014 | −0.00003 | −0.00016 | −0.00003 | −0.00018 | 0.945 | 0.948 | 0.943 | 0.944 | 0.022 | 0.021 | 0.022 | 0.021 |
100 | 100 | 0.05 | 0.05 | 0.00006 | 0.00015 | −0.00003 | 0.00021 | −0.00003 | 0.00021 | 0.947 | 0.951 | 0.945 | 0.944 | 0.014 | 0.015 | 0.014 | 0.015 |
100 | 100 | 0.03 | 0.03 | −0.00019 | 0.00010 | −0.00020 | 0.00016 | −0.00020 | 0.00011 | 0.948 | 0.956 | 0.945 | 0.951 | 0.010 | 0.011 | 0.010 | 0.011 |
50 | 50 | 0.07 | 0.07 | 0.00124 | 0.00114 | 0.00114 | 0.00106 | 0.00115 | 0.00104 | 0.949 | 0.953 | 0.948 | 0.950 | 0.032 | 0.031 | 0.032 | 0.031 |
50 | 50 | 0.05 | 0.05 | 0.00004 | 0.00030 | 0.00002 | 0.00031 | 0.00002 | 0.00030 | 0.950 | 0.951 | 0.949 | 0.948 | 0.020 | 0.021 | 0.020 | 0.022 |
50 | 50 | 0.03 | 0.03 | −0.00038 | 0.00071 | −0.00042 | 0.00072 | −0.00043 | 0.00070 | 0.944 | 0.941 | 0.940 | 0.940 | 0.015 | 0.016 | 0.015 | 0.016 |
100 | 100 | 0.07 | 0.03 | 0.00047 | 0.00567 | −0.00047 | 0.00518 | −0.00015 | 0.00530 | 0.949 | 0.941 | 0.950 | 0.934 | 0.012 | 0.013 | 0.012 | 0.013 |
100 | 100 | 0.03 | 0.07 | −0.00067 | −0.00544 | 0.00025 | −0.00492 | −0.00006 | −0.00510 | 0.946 | 0.938 | 0.946 | 0.935 | 0.012 | 0.013 | 0.012 | 0.013 |
50 | 50 | 0.07 | 0.03 | 0.00057 | 0.00727 | −0.00120 | 0.00608 | −0.00055 | 0.00651 | 0.948 | 0.944 | 0.946 | 0.940 | 0.018 | 0.019 | 0.018 | 0.019 |
50 | 50 | 0.03 | 0.07 | −0.00114 | −0.00719 | 0.00063 | −0.00610 | 0.00000 | −0.00651 | 0.953 | 0.948 | 0.949 | 0.942 | 0.018 | 0.019 | 0.018 | 0.019 |
Marker values truly distributed according to a Student-t law (Design 2)
Relative bias*
| Coverage probability†
| CI mean width†
| ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mode | Median | Mean | Quantile | HPD | Quantile | HPD | ||||||||
ν
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
|
1 | 0.2905 | 0.0055 | 0.2944 | 0.0103 | 0.2929 | 0.0082 | 0.000 | 0.937 | 0.000 | 0.932 | 0.033 | 0.029 | 0.034 | 0.030 |
4 | 0.0431 | 0.0029 | 0.0437 | 0.0040 | 0.0435 | 0.0036 | 0.530 | 0.953 | 0.532 | 0.955 | 0.024 | 0.022 | 0.024 | 0.022 |
8 | 0.0147 | 0.0014 | 0.0149 | 0.0019 | 0.0148 | 0.0017 | 0.863 | 0.951 | 0.868 | 0.952 | 0.023 | 0.021 | 0.023 | 0.021 |
12 | 0.0086 | 0.0009 | 0.0086 | 0.0013 | 0.0086 | 0.0011 | 0.918 | 0.951 | 0.920 | 0.955 | 0.022 | 0.020 | 0.022 | 0.020 |
Marker distributed according to a mixture of two normal laws in the diseased subjects (Design 3)
Relative bias*
| Coverage probability†
| CI mean width†
| |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mode | Median | Mean | Quantile | HPD | Quantile | HPD | |||||||||
σ
2
|
p
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
| Gauss |
t
|
0.10 | 0.3 | 0.0280 | −0.0011 | 0.0287 | 0.0001 | 0.0285 | −0.0004 | 0.736 | 0.956 | 0.734 | 0.951 | 0.025 | 0.024 | 0.024 | 0.023 |
0.10 | 0.2 | 0.0211 | −0.0011 | 0.0215 | −0.0001 | 0.0214 | −0.0004 | 0.804 | 0.957 | 0.804 | 0.954 | 0.024 | 0.022 | 0.024 | 0.022 |
0.10 | 0.1 | 0.0123 | 0.0005 | 0.0125 | 0.0010 | 0.0124 | 0.0007 | 0.889 | 0.959 | 0.887 | 0.955 | 0.023 | 0.021 | 0.023 | 0.021 |
0.075 | 0.3 | 0.0084 | −0.0019 | 0.0085 | −0.0015 | 0.0085 | −0.0017 | 0.924 | 0.953 | 0.923 | 0.953 | 0.023 | 0.022 | 0.023 | 0.021 |
0.075 | 0.2 | 0.0062 | −0.0010 | 0.0063 | −0.0007 | 0.0063 | −0.0008 | 0.935 | 0.952 | 0.936 | 0.952 | 0.023 | 0.021 | 0.023 | 0.021 |
0.075 | 0.1 | 0.0029 | −0.0008 | 0.0030 | −0.0007 | 0.0030 | −0.0007 | 0.947 | 0.954 | 0.942 | 0.953 | 0.022 | 0.020 | 0.022 | 0.020 |
Application: PSA nadir threshold
Heterogeneity in the mean and the variance of biomarker measurements
Simulations (Design 4)
Relative bias*
| Coverage probability†
| CI mean width†
| ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mode | Median | Mean | Quantile | HPD | Quantile | HPD | ||||||||||
N
|
σ
1
|
σ
2
| Gauss | Dirichlet | Gauss | Dirichlet | Gauss | Dirichlet | Gauss | Dirichlet | Gauss | Dirichlet | Gauss | Dirichlet | Gauss | Dirichlet |
200 | 0.07 | 0.07 | 0.0392 | −0.0294 | 0.0405 | −0.0087 | 0.0400 | −0.0177 | 0.623 | 0.947 | 0.631 | 0.926 | 0.020 | 0.066 | 0.020 | 0.069 |
200 | 0.08 | 0.05 | 0.1657 | −0.0064 | 0.1668 | 0.0281 | 0.1664 | 0.0082 | 0.000 | 0.956 | 0.000 | 0.943 | 0.020 | 0.080 | 0.020 | 0.087 |
200 | 0.10 | 0.05 | 0.1717 | −0.0107 | 0.1728 | 0.0167 | 0.1724 | 0.0023 | 0.000 | 0.959 | 0.000 | 0.945 | 0.020 | 0.069 | 0.021 | 0.074 |
100 | 0.07 | 0.07 | 0.0386 | −0.0394 | 0.0412 | −0.0093 | 0.0403 | −0.0223 | 0.792 | 0.944 | 0.797 | 0.922 | 0.029 | 0.082 | 0.029 | 0.086 |
100 | 0.08 | 0.05 | 0.1644 | −0.0074 | 0.1666 | 0.0340 | 0.1658 | 0.0112 | 0.000 | 0.960 | 0.000 | 0.947 | 0.029 | 0.093 | 0.029 | 0.100 |
100 | 0.10 | 0.05 | 0.1713 | −0.0118 | 0.1731 | 0.0206 | 0.1725 | 0.0039 | 0.000 | 0.961 | 0.000 | 0.948 | 0.025 | 0.077 | 0.026 | 0.082 |
50 | 0.07 | 0.07 | 0.0373 | −0.0413 | 0.0425 | −0.0063 | 0.0406 | −0.0215 | 0.879 | 0.971 | 0.884 | 0.952 | 0.042 | 0.100 | 0.041 | 0.096 |
50 | 0.08 | 0.05 | 0.1623 | 0.0007 | 0.1668 | 0.0483 | 0.1651 | 0.0243 | 0.017 | 0.969 | 0.020 | 0.960 | 0.041 | 0.114 | 0.041 | 0.106 |
50 | 0.10 | 0.05 | 0.1689 | 0.0032 | 0.1736 | 0.0431 | 0.1719 | 0.0244 | 0.012 | 0.963 | 0.015 | 0.962 | 0.042 | 0.102 | 0.042 | 0.096 |
30 | 0.07 | 0.07 | 0.0333 | −0.0219 | 0.0426 | 0.0015 | 0.0392 | −0.0081 | 0.923 | 0.982 | 0.929 | 0.974 | 0.055 | 0.102 | 0.054 | 0.098 |
30 | 0.08 | 0.05 | 0.1576 | 0.0394 | 0.1652 | 0.0705 | 0.1624 | 0.0589 | 0.111 | 0.961 | 0.127 | 0.962 | 0.054 | 0.117 | 0.053 | 0.111 |
30 | 0.10 | 0.05 | 0.1643 | 0.0498 | 0.1723 | 0.0679 | 0.1693 | 0.0629 | 0.091 | 0.944 | 0.104 | 0.951 | 0.055 | 0.104 | 0.054 | 0.100 |