In celehs/DPH: Discrete Proportional Hazards Models

Notation

• $t$-year event status: $D_t = I(T \leq t)$

• True Positive Rate: $\text{TPR}_t(z) = P(Z \geq z | D_t = 1)$

• False Positive Rate: $\text{FPR}_t(z) = P(Z \geq z | D_t = 0)$

• Receiver Operating Characteristic: $\text{ROC}_t(u) = \text{TPR}_t{\text{FPR}_t^{-1}(u)}$

• Area Under Curve: $\text{AUC}_t = \int \text{ROC}_t(u) du$

Estimation

• $\widehat{\Lambda}0(t) = \sum{m=1}^t \widehat{\lambda}_{0m}$

• $\widehat{S}(t | Z_i) = \exp[-\hat{\Lambda}_0(t) \exp{{\widehat{\beta} ^\top \psi(Z_i)}}]$

• $\widehat{\text{TPR}}t(z) = \dfrac{\sum{i=1}^n {1 - \widehat{S}(t | Z_i)} I(Z_i \geq z)}{\sum_{i=1}^n {1 - \widehat{S}(t | Z_i)}}$

• $\widehat{\text{FPR}}t(z) = \dfrac{\sum{i=1}^n \widehat{S}(t | Z_i) I(Z_i \geq z)}{\sum_{i=1}^n \widehat{S}(t | Z_i)}$

• $\widehat{\text{AUC}}_t = \int \widehat{\text{TPR}}_t{\widehat{\text{FPR}}_t^{-1}(u)} du$

References

• T. Cai, S. Cheng, Robust combination of multiple diagnostic tests for classifying censored event times, Biostatistics, Volume 9, Issue 2, April 2008, Pages 216–233, https://doi.org/10.1093/biostatistics/kxm037

• Hajime Uno, Tianxi Cai, Lu Tian & L. J Wei (2007) Evaluating Prediction Rules for t-Year Survivors With Censored Regression Models, Journal of the American Statistical Association, 102:478, 527-537, https://doi.org/10.1198/016214507000000149

celehs/DPH documentation built on April 2, 2021, 7:27 p.m.