Independent Hazards

Most of this comes from Kalbfleisch (1978), with an excellent technical outline by Ibrahim (2001).

Recall that the cumulative baseline hazard \(H_0(t) = \int_0^t \lambda_0(t) dt\) where the integral is the Riemann-Stieltjes integral. The central idea is to develop a prior for the cumulative hazard \(H_0(t)\), which will then admit a prior for the hazard, \(\lambda_0(t)\).

The Gamma Process is such a prior. Each realization of a Gamma Process is a cumulative hazard function that is centered around some prior cumulative hazard function, \(H^*\), with a sort of dispersion/concentration parameter, \(\beta\) that controls how tightly the realizations are distributed around the prior \(H^*\).

Okay, now the math. Let \(\mathcal{G}(\alpha, \beta)\) denote the Gamma distribution with shape parameter \(\alpha\) and rate parameter \(\beta\). Let \(H^*(t)\) for \(t\geq 0\) be our prior cumulative hazard function. For example we could choose \(H^*\) to be the exponential cumulative hazard, \(H^*(t)= \eta\cdot t\), where \(\eta\) is a fixed hyperparameter. By definition \(H^*(0)=0\). The Gamma Process is defined as having the following properties:

• \(H_0(0) = 0\)
• \(\lambda_0(t) = H_0(t) – H_0(s) \sim \mathcal G \Big(\ \beta\big(H^*(t) – H^*(s)\big)\ , \ \beta \ \Big)\), for \(t>s\)

The increments in the cumulative hazard is the hazard function. The gamma process has the property that these increments are independent and Gamma-distributed. For a set of time increments \(t\geq0\), we can use the properties above to generate one realization of hazards \(\{\lambda_0(t) \}_{t\geq0}\). Equivaltently, one realization of the cumulative hazard function is \(\{H_0(t)\}_{t\geq0}\), where \(H_0(t) = \sum_{k=0}^t \lambda_0(k)\). We denote the Gamma Process just described as \[H_0(t) \sim \mathcal{GP}\Big(\ \beta H^*(t), \ \beta \Big), \ \ t\geq0\]

Below in Panel A are some prior realizations of \(H_0(t)\) with a Weibull \(H^*\) prior for various concentration parameters, \(\beta\). Notice for low \(\beta\) the realizations are widely dispersed around the mean cumulative hazard. Higher \(\beta\) yields to tighter dispersion around \(H^*\).

Since there’s a correspondence between the \(H_0(t)\), \(\lambda_0(t)\), and \(S_0(t)\), we could also plot prior realizations of the baseline survival function \(S_0(t) = \exp\big\{- H_0(t) \big\}\) using the realization \(\{H_0(t)\}_{t\geq0}\). This is shown in Panel B with the Weibull survival function \(S^*\) corresponding to \(H^*\).

Correlated Hazards

In the previous section, the hazards \(\lambda_0(t)\) between increments were a priori independent – a naive prior belief perhaps. Instead, we might expect that the hazard at the current time point is centered around the hazard in the previous time point. We’d also expect that a higher hazard at the previous time point likely means a higher hazard at the current time point (positive correlation across time).

Nieto‐Barajas et al (2002) came up with a correlated Gamma Process that expresses exactly this prior belief. The basic idea is to introduce a latent stochastic process \(\{u_t\}_{t\geq0}\) that links \(\lambda_0(t)\) with \(\lambda_0(t-1)\). Here is the correlated Gamma Process,

• Draw a hazard rate for the first time interval, \(I_1=[0, t_1)\), \(\lambda_1 \sim \mathcal G(\beta H^*(t_1), \beta)\),
• Draw a latent variable \(u_1 | \lambda_1 \sim Pois(c \cdot \lambda_1)\)
• Draw a hazard rate for second time interval \(I_2 = [t_1, t_2)\), \(\lambda_2 \sim \mathcal G(\beta( H^*(t_2) – H^*(t_1) ) + u_1, \beta + c )\)

• In general for \(k\geq1\), define \(\alpha_k = \beta( H^*(t_k) – H^*(t_{k-1}) )\)

  • \(u_k | \lambda_k \sim Pois(c\cdot \lambda_k)\)
  • \(\lambda_{k+1} | u_k \sim \mathcal G( \alpha_k + u_k, \beta + c )\)

Notice that if \(c=0\), then \(u=0\) with probability \(1\) and the process reduces to the independent Gamma Process in the previous section. Now consider \(c=1\). Then, the hazard rate in the next interval has mean \(E[\lambda_{k+1} | u_k] = \frac{ \alpha_k + u_k }{\beta+c}\). Now \(u_k \sim Pois(\lambda_k)\) is centered around the current \(\lambda_k\) – allowinng \(\lambda_k\) to influence \(\lambda_{k+1}\) through the latent variable \(u_k\). The higher the current hazard, \(\lambda_k\), the higher \(u_k\), and the higher the mean of the next hazard, \(\lambda_{k+1}\).

Below are some realizations of a correlated and independent Gamma processes centered around the \(Weibull(2,1.5)\) hazard shown in red. One realization is higlighted in blue to make it easier to see the differences between correlated and independent realizations

Notice the correlated gamma process looks very snake-y. This is because of the autoregressive structure on \(\{\lambda_0(t)\}_{t\geq0}\) induced by the latent process\(\{ u_t\}_{t\geq0}\).