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\begin{document}

\section{Smooth accelerated failure time models}

\subsection{Time-constant acceleration factors}

Let survival to time $t$ for covariates $x$ be modelled as an accelerated failure time model using
\begin{align*}
S(t|x) &= S_0(t \exp(-\eta(x;\beta)))
\end{align*}
where $\eta$ is a linear predictor. Moreover, let the baseline survival function be modelled as
\begin{align*}
S_0(t) &= \exp(-\exp(\eta_0(\log(t);\beta_0)))
\end{align*}
where $\eta_0$ is a linear predictor. Then the combined regression model is
\begin{align*}
S(t|x) &= \exp(-\exp(\eta_0(\log(t) -\eta(x; \beta);\beta_0)))
\end{align*}
We can calculate the hazard, such that
\begin{align*}
h(t|x) &= \frac{\partial}{\partial t} \left(-\log(S(t|x))\right) \\
&= \exp(\eta_0(\log(t) -\eta(x; \beta);\beta_0)) \eta_0'(\log(t) -\eta(x; \beta);\beta_0)/t
\end{align*}

\subsection{Time-dependent acceleration factors}

We can model survival as an accelerated failure time model with time-dependent effects as
\begin{align*}
S(t|x) &= S_0\left(\int_0^t\exp(-\eta_1(x,u;\beta)) \mathrm{d}u\right) = S_0\left(t\exp(-\eta(x,t;\beta))\right)
\end{align*}
for a time-specific linear predictor $\eta_1$ and where $\eta$ now models for cumulative time-dependent effects.
By differentiation with respect to time $t$, we have that
\begin{align*}
\exp(-\eta_1(x,t;\beta)) &= \frac{\partial}{\partial t} t\exp(-\eta(x,t;\beta)) \\
&= \exp(-\eta(x,t;\beta)) \left(1- t \frac{\partial}{\partial t} \eta(x,t;\beta)\right)
\end{align*}
so that $\eta_1(x,t;\beta)=\eta(x,t;\beta) - \log\left(1- t \frac{\partial}{\partial t} \eta(x,t;\beta)\right)$. This shows that we can recover the time-specific acceleration factors from $\eta$.

The combined regression model is then
\begin{align*}
S(t|x) &= \exp(-\exp(\eta_0(\log(t) -\eta(x,t; \beta);\beta_0)))
\end{align*}

The hazard is then
\begin{align*}
h(t|x) &= \frac{\partial}{\partial t} -\log(S(t|x)) \\
&= \exp(\eta_0(\log(t) -\eta(x,t; \beta);\beta_0))\, \eta_0'(\log(t) -\eta(x,t; \beta);\beta_0)\times \\
\end{align*}

\subsection{R implementation}

The linear predictor $\eta_0$ is modelled using natural splines. The linear predictor $\eta$ can be modelled freely provided the linear predictor is a smooth function of time.

Initial values for the time-constant log acceleration factors were calculated from a Weibull regression. The time-varying log acceleration factors were assumed to be zero.  Cox regression and the Breslow estimator was used to calculate baseline survival; parameters for the baseline were estimated using linear regression with natural splines fitted to the log-times at the event times.

We included quadratic penalties to ensure that $\eta_0'(\log(t) -\eta(x,t; \beta);\beta_0)$ and $1/t-\frac{\partial}{\partial t}\eta(x,t; \beta)$ were positive. For speed, the model has been implemented in C++, with optimisation using the Nelder-Mead algorithm.

\end{document}

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