This document summarizes the mathematical formulations of 11 hazard-based and odds-based regression models as presented in the thesis “Bayesian and Frequentist Approaches for Flexible Parametric Hazard-Based Regression Models with Generalized Log-logistic Baseline Distribution”.
For each model, the following probabilistic functions are defined:
Notation:
The PH model assumes that covariates act multiplicatively on the hazard rate function.
Hazard Rate Function: \[h_{PH}(t; x) = h_0(t) e^{x'\beta}\]
Cumulative Hazard Function: \[H_{PH}(t; x) = H_0(t) e^{x'\beta}\]
Survival Function: \[S_{PH}(t; x) = [S_0(t)]^{e^{x'\beta}}\]
Odds Function: \[R_{PH}(t; x) = [S_0(t)]^{-e^{x'\beta}} - 1\] (Alternatively expressed via cumulative hazard: \(\exp(H_0(t)e^{x'\beta}) - 1\))
Derivative of Odds Function: \[r_{PH}(t; x) = h_{PH}(t; x) [S_{PH}(t; x)]^{-1} = h_0(t)e^{x'\beta} [S_0(t)]^{-e^{x'\beta}}\]
The AH model assumes that covariates scale the time axis of the hazard rate function (time-scale change in hazard progression).
Hazard Rate Function: \[h_{AH}(t; x) = h_0(t e^{x'\beta})\]
Cumulative Hazard Function: \[H_{AH}(t; x) = H_0(t e^{x'\beta}) e^{-x'\beta}\]
Survival Function: \[S_{AH}(t; x) = \exp \left( -H_0(t e^{x'\beta}) e^{-x'\beta} \right) = [S_0(t e^{x'\beta})]^{e^{-x'\beta}}\]
Odds Function: \[R_{AH}(t; x) = [S_0(t e^{x'\beta})]^{-e^{-x'\beta}} - 1\]
Derivative of Odds Function: \[r_{AH}(t; x) = \frac{h_{AH}(t; x)}{S_{AH}(t; x)} = h_0(t e^{x'\beta}) [S_0(t e^{x'\beta})]^{-e^{-x'\beta}}\]
The AFT model assumes that covariates act multiplicatively on the time scale, affecting the rate at which an individual proceeds along the time axis.
Hazard Rate Function: \[h_{AFT}(t; x) = h_0(t e^{x'\beta}) e^{x'\beta}\]
Cumulative Hazard Function: \[H_{AFT}(t; x) = H_0(t e^{x'\beta})\]
Survival Function: \[S_{AFT}(t; x) = S_0(t e^{x'\beta})\]
Odds Function: \[R_{AFT}(t; x) = R_0(t e^{x'\beta})\]
Derivative of Odds Function: \[r_{AFT}(t; x) = r_0(t e^{x'\beta}) e^{x'\beta}\]
The GH model nests the PH, AH, and AFT models. It separates the covariate effects into a time-scale change (\(\beta_1\)) and a relative hazard ratio (\(\beta_2\)).
Hazard Rate Function: \[h_{GH}(t; x) = h_0(t e^{x'\beta_1}) e^{x'\beta_2}\]
Cumulative Hazard Function: \[H_{GH}(t; x) = H_0(t e^{x'\beta_1}) e^{x'(\beta_2 - \beta_1)}\]
Survival Function: \[S_{GH}(t; x) = [S_0(t e^{x'\beta_1})]^{e^{x'(\beta_2 - \beta_1)}}\]
Odds Function: \[R_{GH}(t; x) = [S_0(t e^{x'\beta_1})]^{-e^{x'(\beta_2 - \beta_1)}} - 1\]
Derivative of Odds Function: \[r_{GH}(t; x) = h_0(t e^{x'\beta_1}) e^{x'\beta_2} [S_0(t e^{x'\beta_1})]^{-e^{x'(\beta_2 - \beta_1)}}\]
The PO model assumes that the odds of the event occurring are proportional across different covariate levels.
Odds Function: \[R_{PO}(t; x) = R_0(t) e^{x'\beta}\]
Derivative of Odds Function: \[r_{PO}(t; x) = r_0(t) e^{x'\beta}\]
Survival Function: \[S_{PO}(t; x) = \frac{1}{1 + R_0(t) e^{x'\beta}}\]
Hazard Rate Function: \[h_{PO}(t; x) = \frac{r_0(t) e^{x'\beta}}{1 + R_0(t) e^{x'\beta}}\]
Cumulative Hazard Function: \[H_{PO}(t; x) = \ln(1 + R_0(t) e^{x'\beta})\]
A novel model introduced in the thesis where covariates accelerate the odds function.
Odds Function: \[R_{AO}(t; x) = R_0(t e^{x'\beta}) e^{-x'\beta}\]
Derivative of Odds Function: \[r_{AO}(t; x) = r_0(t e^{x'\beta})\]
Survival Function: \[S_{AO}(t; x) = \left[ 1 + R_0(t e^{x'\beta}) e^{-x'\beta} \right]^{-1}\]
Hazard Rate Function: \[h_{AO}(t; x) = \frac{r_0(t e^{x'\beta})}{1 + R_0(t e^{x'\beta}) e^{-x'\beta}}\]
Cumulative Hazard Function: \[H_{AO}(t; x) = \ln \left( 1 + R_0(t e^{x'\beta}) e^{-x'\beta} \right)\]
The GO model nests the PO, AFT, and AO models.
Odds Function: \[R_{GO}(t; x) = R_0(t e^{x'\beta_1}) e^{x'(\beta_2 - \beta_1)}\]
Derivative of Odds Function: \[r_{GO}(t; x) = r_0(t e^{x'\beta_1}) e^{x'\beta_2}\]
Survival Function: \[S_{GO}(t; x) = \left[ 1 + R_0(t e^{x'\beta_1}) e^{x'(\beta_2 - \beta_1)} \right]^{-1}\]
Hazard Rate Function: \[h_{GO}(t; x) = \frac{r_0(t e^{x'\beta_1}) e^{x'\beta_2}}{1 + R_0(t e^{x'\beta_1}) e^{x'(\beta_2 - \beta_1)}}\]
Cumulative Hazard Function: \[H_{GO}(t; x) = \ln \left( 1 + R_0(t e^{x'\beta_1}) e^{x'(\beta_2 - \beta_1)} \right)\]
A semi-parametric model allowing for crossing survival curves, including PO and PH as special cases.
Odds Function: \[R_{YP}(t; \beta_1, \beta_2, x) = e^{x'(\beta_1 - \beta_2)} R_0(t)^{e^{x'\beta_2}}\]
Derivative of Odds Function: \[r_{YP}(t; \beta_1, \beta_2, x) = r_0(t) e^{x'\beta_1} R_0(t)^{e^{x'\beta_2} - 1}\]
Hazard Rate Function: \[h_{YP}(t; \beta_1, \beta_2, x) = \frac{e^{x'(\beta_1+\beta_2)} h_0(t)}{F_0(t)e^{x'\beta_1} + S_0(t)e^{x'\beta_2}}\]
Survival Function: \[S_{YP}(t; \beta_1, \beta_2, x) = \left[ 1 + \left( \frac{F_0(t)}{S_0(t)} \right)^{e^{x'\beta_2}} e^{x'(\beta_1 - \beta_2)} \right]^{-1}\]
Cumulative Hazard Function: \[H_{YP}(t; \beta_1, \beta_2, x) = \ln \left( 1 + R_{YP}(t; \beta_1, \beta_2, x) \right)\]
A model containing PH, AFT, and PO as special cases.
Survival Function: \[S_{GOH}(t; \theta, \beta, x) = \{ 1 + \theta \exp(\varphi_0(t) + x'\beta) \}^{-1/\theta}\] (Where \(\theta > 0\) is the GORH parameter)
Hazard Rate Function: \[h_{GOH}(t; \theta, \beta, x) = \frac{\varphi'_0(t) \exp(\varphi_0(t) + x'\beta)}{1 + \theta \exp(\varphi_0(t) + x'\beta)}\]
Cumulative Hazard Function: \[H_{GOH}(t; \theta, \beta, x) = \frac{1}{\theta} \ln \{ 1 + \theta \exp(\varphi_0(t) + x'\beta) \}\]
Odds Function: \[R_{GOH}(t; \theta, \beta, x) = \{ 1 + \theta \exp(\varphi_0(t) + x'\beta) \}^{1/\theta} - 1\]
Derivative of Odds Function: \[r_{GOH}(t; \theta, \beta, x) = h_{GOH}(t) \cdot (R_{GOH}(t) + 1)\]
A semi-parametric model containing PH, PO, AFT, AH, YP, and GH models.
Hazard Rate Function: \[h_{SM}(t; \beta_1, \beta_2, \beta_3, x) = \frac{e^{x'(\beta_2+\beta_3+\beta_1)} h_0(te^{\beta_1 x'})}{e^{x'(\beta_2+\beta_1)} F_0(te^{\beta_1 x'}) + e^{\beta_3 x'} S_0(te^{\beta_1 x'})}\]
Odds Function: \[R_{SM}(t; \beta_1, \beta_2, \beta_3, x) = e^{x'(\beta_2 - \beta_3 + \beta_1)} R_0(te^{\beta_1 x'})^{e^{(\beta_3 - \beta_1)'x}}\]
Survival Function: \[S_{SM}(t; \beta_1, \beta_2, \beta_3, x) = \left[ 1 + e^{x'(\beta_2 - \beta_3 + \beta_1)} \left( \frac{F_0(te^{\beta_1 x'})}{S_0(te^{\beta_1 x'})} \right)^{e^{(\beta_3 - \beta_1)'x}} \right]^{-1}\]
Cumulative Hazard Function: \[H_{SM}(t) = \ln(1 + R_{SM}(t))\]
Derivative of Odds Function: \[r_{SM}(t) = \frac{d}{dt} R_{SM}(t)\]
A universal class proposed in the thesis that encompasses all hazard-based and odds-based regression models (PH, PO, AH, AO, AFT, GH, GO).
Odds Function: \[R_{AM}(t; \beta_1, \beta_2, \beta_3, x) = e^{x'(\beta_2 - \beta_1)} R_0(te^{x'\beta_1})^{e^{x'(\beta_3 - \beta_1)}}\]
Survival Function: \[S_{AM}(t; \beta_1, \beta_2, \beta_3, x) = \left[ 1 + e^{x'(\beta_2 - \beta_1)} R_0(te^{x'\beta_1})^{e^{x'(\beta_3 - \beta_1)}} \right]^{-1}\]
Hazard Rate Function: \[h_{AM}(t; \beta_1, \beta_2, \beta_3, x) = \frac{e^{x'(\beta_2 + \beta_3 - \beta_1)} h_0(te^{x'\beta_1})}{e^{x'(\beta_2 - \beta_1)} F_0(te^{x'\beta_1}) + S_0(te^{x'\beta_1})}\]
Derivative of Odds Function: \[r_{AM}(t; \beta_1, \beta_2, \beta_3, x) = r_0(te^{x'\beta_1}) e^{x'(\beta_2 + \beta_1)} R_0(te^{x'\beta_1})^{e^{x'(\beta_3 - \beta_1)} - 1}\]
Cumulative Hazard Function: \[H_{AM}(t; \beta_1, \beta_2, \beta_3, x) = \ln \left( 1 + e^{x'(\beta_2 - \beta_1)} R_0(te^{x'\beta_1})^{e^{x'(\beta_3 - \beta_1)}} \right)\]
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