3. Time-to-event Models#

Lecture errata:#

There is a typo that says:

\[ \text{Bayesian inference about} \longrightarrow L(\theta \mid y_1, \ldots, y_n) + \pi(\theta) \]

Which should instead be:

\[ \text{Bayesian inference about} \longrightarrow L(\theta \mid y_1, \ldots, y_n) \cdot \pi(\theta) \]

Survival models#

Survival analysis studies the time \(T\) until an event occurs (\(T \ge 0\)). Key functions:

  • Cumulative Distribution Function (CDF): \(F(t) = P(T \le t)\), probability event occurs by time \(t\).

  • Survival Function: \(S(t) = P(T > t) = 1 - F(t)\), probability event has not occurred by time \(t\).

  • Probability Density Function (PDF): \(f(t) = \frac{d}{dt}F(t) = -\frac{d}{dt}S(t)\), instantaneous rate of event at time \(t\).

  • Hazard Function (Hazard Rate): \(h(t) = \lim_{\Delta t \to 0} \frac{P(t \le T < t+\Delta t | T \ge t)}{\Delta t}\). Instantaneous potential for event at \(t\), given survival up to \(t\).

\[ h(t) = \frac{f(t)}{S(t)} = -\frac{d}{dt} \ln(S(t)) \]

  • Cumulative Hazard Function: \(H(t) = \int_0^t h(u) du\). Related to survival by:

\[ S(t) = e^{-H(t)} = e^{-\int_0^t h(u) du} \]

Probability of event between \(a\) and \(b\) (\(a<b\)) is \(P(a < T \le b) = F(b) - F(a) = S(a) - S(b)\).

Exponential Distribution#

Has a constant hazard rate \(\lambda\).

  • Parameter: Rate \(\lambda > 0\).

  • PDF: \(f(t|\lambda) = \lambda e^{-\lambda t}\), for \(t \ge 0\).

  • CDF: \(F(t|\lambda) = 1 - e^{-\lambda t}\).

  • Survival Function: \(S(t|\lambda) = e^{-\lambda t}\).

  • Hazard Function: \(h(t|\lambda) = \frac{\lambda e^{-\lambda t}}{e^{-\lambda t}} = \lambda\).

  • Cumulative Hazard Function: \(H(t|\lambda) = \lambda t\).

Weibull Distribution#

Allows hazard rates to increase (\(\alpha > 1\)), decrease (\(\alpha < 1\)), or be constant (\(\alpha = 1\), Exponential case). Two common parameterizations exist.

Rate model (BUGS)#

  • Parameters: Shape \(r > 0\), Rate \(\lambda > 0\).

  • PDF: \(f(t|r, \lambda) = \lambda r t^{r-1} e^{-\lambda t^r}\), for \(t > 0\).

  • CDF: \(F(t|r, \lambda) = 1 - e^{-\lambda t^r}\).

  • Survival Function: \(S(t|r, \lambda) = e^{-\lambda t^r}\).

  • Hazard Function: \(h(t|r, \lambda) = \frac{\lambda r t^{r-1} e^{-\lambda t^r}}{e^{-\lambda t^r}} = \lambda r t^{r-1}\).

  • Cumulative Hazard Function: \(H(t|r, \lambda) = \lambda t^r\).

Scale model (PyMC)#

  • Parameters: Shape \(\alpha > 0\), Scale \(\beta > 0\).

  • PDF: \(f(t|\alpha, \beta) = \frac{\alpha}{\beta} \left(\frac{t}{\beta}\right)^{\alpha - 1} e^{-(t/\beta)^{\alpha}} = \frac{\alpha}{\beta^\alpha} t^{\alpha - 1} e^{-(t/\beta)^{\alpha}}\), for \(t > 0\).

  • CDF: \(F(t|\alpha, \beta) = 1 - e^{-(t/\beta)^\alpha}\).

  • Survival Function: \(S(t|\alpha, \beta) = e^{-(t/\beta)^\alpha}\).

  • Hazard Function: \(h(t|\alpha, \beta) = \frac{\frac{\alpha}{\beta^\alpha} t^{\alpha - 1} e^{-(t/\beta)^{\alpha}}}{e^{-(t/\beta)^\alpha}} = \frac{\alpha}{\beta^\alpha} t^{\alpha - 1} = \frac{\alpha}{\beta} \left(\frac{t}{\beta}\right)^{\alpha-1}\).

  • Cumulative Hazard Function: \(H(t|\alpha, \beta) = (t/\beta)^\alpha\).

Converting from BUGS to PyMC parameterization#

Let \(X\) be Weibull distributed.

BUGS parameterization: shape \(r > 0\), rate \(\lambda > 0\).

\[ f_{BUGS}(x|r, \lambda) = \lambda r x^{r-1} e^{-\lambda x^r}, \quad \text{for } x > 0 \]

PyMC parameterization: shape \(\alpha > 0\), scale \(\beta > 0\).

\[ f_{PyMC}(x|\alpha, \beta) = \frac{\alpha}{\beta^\alpha} x^{\alpha - 1} e^{-(x/\beta)^{\alpha}}, \quad \text{for } x > 0 \]

Relationship:

  1. \(\alpha = r\)

  2. \(\beta = \lambda^{-1/r} = \lambda^{-1/\alpha}\)

From (2), \(\lambda = 1/\beta^\alpha\).

Substituting BUGS parameters with PyMC equivalents:

\[\begin{split} \begin{align*} f_{BUGS}(x|r, \lambda) &= \lambda r x^{r-1} e^{-\lambda x^r} \\ &= \left(\frac{1}{\beta^\alpha}\right) \alpha x^{\alpha-1} e^{-\left(\frac{1}{\beta^\alpha}\right) x^\alpha} \\ &= \frac{\alpha}{\beta^\alpha} x^{\alpha-1} e^{-\frac{x^\alpha}{\beta^\alpha}} \\ &= \frac{\alpha}{\beta^\alpha} x^{\alpha-1} e^{-\left(\frac{x}{\beta}\right)^\alpha} \\ &= f_{PyMC}(x|\alpha, \beta) \end{align*} \end{split}\]

The parameterizations are equivalent if \(\alpha = r\) and \(\beta = \lambda^{-1/\alpha}\).

Censoring and Truncation#

Event time \(T\) may not be fully observed.

Right-Censoring: Most common. Study ends or subject lost before event. True event time \(T_i >\) observed censoring time \(C_i\). Data is \((y_i, \delta_i)\), where \(y_i = \min(T_i, C_i)\) and \(\delta_i\) indicates censoring:

  • \(\delta_i = 0\) if event observed (\(T_i \le C_i\)).

  • \(\delta_i = 1\) if censored (\(T_i > C_i\)).

Left-Censoring: Event occurred before time \(C_i\), exact time unknown (\(T_i < C_i\)).

Truncation: Differs from censoring. Subjects included only if event time is within a range (e.g., left-truncation: only subjects with \(T_i > T_{start}\) are observed).

Likelihood with Right-Censoring:

For \(n\) subjects (\(k\) observed events, \(n-k\) censored), likelihood \(L\) for parameters \(\theta\):

  • Observed event \(y_i\) (\(\delta_i=0\)): contribution is PDF \(f(y_i|\theta)\).

  • Censored time \(y_i\) (\(\delta_i=1\)): contribution is Survival \(S(y_i|\theta) = P(T_i > y_i | \theta)\).

Total likelihood:

\[ L(\theta | \mathbf{y}, \boldsymbol{\delta}) = \prod_{i=1}^n [f(y_i|\theta)]^{1-\delta_i} [S(y_i|\theta)]^{\delta_i} \]

Used for inference (MLE, Bayesian).

Types of Censoring:

  • Type I Censoring: Study ends at fixed time \(C\). Number of events is random.

  • Type II Censoring: Study ends after fixed number of events \(k\). Time \(C\) is random.