Performs nonparametric hazard rate estimation using kernel functions. Easy‑to‑use version of HAZRD.

Routine HAZEZ is an implementation of the methods discussed by Tanner and Wong (1984) for estimating the hazard rate in survival or reliability data with right censoring. It uses the biweight kernel,

and a modified likelihood to obtain data‑based estimates of the smoothing parameters α, β, and k needed in the estimation of the hazard rate. For kernel K(x), define the “smoothed” kernel
Ks(x – x(j)) as follows:

where djk is the distance to the k‑th nearest failure from x(j), and x(j)is the j‑th ordered observation (from smallest to largest). For given α and β, the hazard at point x is given by:

where N = NOBS, δi is the censor code (0 = failed, 1 = censored) for the i‑th ordered observation, and wi is the weight of the i‑th ordered observation (given by 1/(N – i + 1)). The hazard may be estimated via routine after the smoothing parameters have been obtained

Let

The likelihood is given by:

where Π denotes product. Since the likelihood, as specified, will lead to degenerate estimates, Tanner and Wong (1984) suggest the use of a modified likelihood. The modification consists of deleting the observation xi in the calculation of h(xi) and H(xi) when the likelihood term for xi is computed. For a given k, α and β can then be estimated via the usual optimization techniques.

Estimates for α and β are computed as follows. For a given β, a closed form solution is available for α. The problem is thus reduced to the estimation of β. To estimate α and β, a grid search is first performed. Experience indicates that if the initial estimate of β from this grid search is greater than exp(6), then the modified likelihood is degenerate because the hazard rate does not change with time. In this situation, β should be taken to be infinite, and an estimate of α corresponding to infinite β is computed directly. When the estimate of β from the grid search is less than exp(6) (approximately 400), a secant algorithm is used to optimize the modified likelihood. The secant algorithm is said to have converged when the change in β from one iteration to the next is less than 0.00001. Additionally, convergence is assumed when the value of β becomes greater than exp(6). This corresponds to an infinite β with a degenerate likelihood.

A grid of k‑values is used to find the optimum value of the likelihood with respect to k. The grid is determined by HAZEZ and consists of at most 10 points. The starting value in the grid is the smallest possible value of k. An increment of 2 is then used to obtain the remaining grid points.

For each grid value, the modified likelihood is optimized with respect to α and β. That grid point, which leads to the smallest likelihood, is taken to be the optimal k.

The following example is illustrated in Tanner and Wong (1984), and the data are taken from Stablein, Carter, and Novak (1981). It involves the survival times of individuals with nonresectable gastric carcinoma. Only those individuals treated with radiation and chemotherapy are used.