Linear Interpolation with a Nonparametric Accelerated Failure-Time Model

Assuming a nonparametric accelerated failure-time model, a method is proposed for extrapolating low stress-response probabilities on negative-sloping line segments in the stress-failure-time plane. The method (analogous to linear interpolation in dose-response studies) results in simultaneous extrapolation ahead in time and down in stress. In the two-sample (two stress level) setting, a goodness-of-fit test of the nonparametric model is considered and two hypothesis tests are proposed, each pertaining to a condition that (in addition to the assumed model) must be met for the method to be valid. The method is illustrated with censored strand-life data, and some simulated examples are used to consider power and robustness. The failure-time model is a scale family with curvature constraints. The failure probability is assumed to be a sigmoid function of the product of time and a sigmoid scaling function of stress. Time percentiles as functions of stress are assumed to be convex. The goodness-of-fit test is of the Kolmogorov-Smirnov type. Given data at stresses x 1 and x 2, the linear interpolation procedure requires a negative-sloping line such that (a) the failure probability increases along the line as the stress increases from x 1 to x 2, and (b) at the point on the line at x 1 the ratio of the failure probability to time increases in time. A Kolmogorov-Smirnov hypothesis test is used to support (a), and a binomial probability test is used for (b).