Obtains the profile log-likelihood function for change points in the piecewise exponential approximation to a survival function.
pwexploglik(tau, S, ...)A list with the following three components:
piecewiseSurvivalTime: A vector that specifies the starting
time of piecewise exponential survival time intervals.
lambda: A vector of hazard rates for the event. One for
each analysis time interval.
loglik: The value of the profile log-likelihood.
The numeric vector of change points.
The survival function of a univariate survival time.
Additional arguments to be passed to S.
Kaifeng Lu, kaifenglu@gmail.com
This function computes the profile log-likelihood for change points in a piecewise exponential survival model.
Let \(S(t)\) denote the survival function of a univariate survival time, and \(\tau\) be a vector of \(J-1\) change points. The piecewise exponential survival model divides the time axis into \(J\) intervals defined by the change points \(\tau\), where each interval \([t_j, t_{j+1})\) has a constant hazard rate \(\lambda_j\). The time intervals are specified as: $$[t_1, t_2), [t_2, t_3), \ldots, [t_{J}, t_{J+1})$$ where \(t_1 = 0\), \(t_{J+1} = \infty\), and \(t_j = \tau_{j-1}\) for \(j = 2, \ldots, J\).
For each subject, the expected number of events occurring in the \(j\)-th interval is $$d_j = E\{I(t_j < Y \leq t_{j+1})\} = S(t_j) - S(t_{j+1})$$ The expected exposure in the \(j\)-th interval is: $$e_j = E\{(Y-t_j)I(t_j < Y \leq t_{j+1}) + (t_{j+1} - t_j)I(Y > t_{j+1})\}$$ which can be shown to be equivalent to $$e_j = \int_{t_j}^{t_{j+1}} S(t) dt$$
The log-likelihood for the piecewise exponential model is: $$\ell(\tau,\lambda) = \sum_{j=1}^J \{d_j \log(\lambda_j) - e_j \lambda_j\}$$ The profile log-likelihood for \(\tau\) is obtained by maximizing \(\ell(\tau,\lambda)\) with respect to \(\lambda\) for fixed \(\tau\). The maximum likelihood estimate of the hazard rate in the \(j\)-th interval is $$\lambda_j = \frac{d_j}{e_j}$$ Substituting back, the profile log-likelihood is $$\ell(\tau) = \sum_{j=1}^J d_j \log(d_j/e_j) - 1$$ where we use the fact that \(\sum_{j=1}^J d_j = 1\).
pwexploglik(tau = c(0.5, 1.2, 2.8), pweibull,
shape = 1.37, scale = 1/0.818, lower.tail = FALSE)
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