pwexploglik: Profile Log-Likelihood Function for Change Points in...

In lrstat: Power and Sample Size Calculation for Non-Proportional Hazards and Beyond

Profile Log-Likelihood Function for Change Points in Piecewise Exponential Approximation

Description

Obtains the profile log-likelihood function for change points in the piecewise exponential approximation to a survival function.

Usage

pwexploglik(tau, S, ...)

Arguments

tau	The numeric vector of change points.
S	The survival function of a univariate survival time.
...	Additional arguments to be passed to S.

Details

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.

Value

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.

Author(s)

Kaifeng Lu, [e-mail]

Examples

pwexploglik(tau = c(0.5, 1.2, 2.8), pweibull,
            shape = 1.37, scale = 1/0.818, lower.tail = FALSE)

lrstat documentation built on Aug. 8, 2025, 7:36 p.m.