Performs nonparametric hazard rate estimation using kernel functions and quasi‑likelihoods.

Routine HAZRD 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 then

where N = NOBS, δi is the i‑th observation’s censor code (1 = censored, 0 = failed), and wi is the i‑th ordered observation’s weight, which may be chosen as either 1/(N ‑ i + 1), or ln(1 + 1/(N ‑ i + 1)). After the smoothing parameters have been obtained, the hazard may be estimated via HAZST.

Let

The likelihood is given by

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

Estimates for α and β are computed as follows: for given β, a closed form solution is available for α. The problem is thus reduced to the estimation of β. A grid search for β is first performed. Experience indicates that if the initial estimate of β from this grid search is greater than, say, e6, 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 should be directly computed. When the estimate of β from the grid search is less than e6, a secant algorithm is used to optimize the modified likelihood. The secant algorithm iteration stops when the change in β from one iteration to the next is less than 10−5. Alternatively, the iterations may cease when the value of β becomes greater than e6, at which point an infinite β with a degenerate likelihood is assumed.

To find the optimum value of the likelihood with respect to k, a user‑specified grid of k‑values is used. 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 taken from Tanner and Wong (1984). The data are from Stablein, Carter, and Novak (1981) and involve the survival times of individuals with nonresectable gastric carcinoma. Individuals treated with radiation and chemotherapy are used. For each value of k from 18 to 22 with increment of 2, the default grid search for β is performed. Using the optimal value of β in the grid, the optimal parameter estimates of α and β are computed for each value of k. The final solution is the parameter estimates for the value of k which optimizes the modified likelihood (VML). Because the IPRINT = 1 option is in effect, HAZRD prints all of the results in the output.