Revisiting the Langlie procedure

TL;DR

This paper critically examines the Langlie procedure for sensitivity testing, revealing that its input sequence does not converge when used, thus challenging its applicability beyond median estimation and questioning its potential for nonparametric adaptation.

Contribution

It proves that the Langlie procedure's input sequence fails to converge, demonstrating limitations in its use for nonparametric sensitivity estimation.

Findings

The input sequence in the Langlie procedure does not converge with probability one.

The procedure is only suitable for median estimation, not for other quantiles.

The parametric model assumption is essential for the procedure's convergence.

Abstract

Introduced in 1962, the Langlie procedure is one of the most popular approaches to sensitivity testing. It aims to estimate an unknown sensitivity distribution based on the outcomes of binary trials. Officially recognized by the U.S. Department of Defense, the procedure is widely used both in civil and military industry. It first provides an experimental design for how the binary trials should be conducted, and then estimates the sensitivity distribution via maximum likelihood under a simple parametric model like logistic or probit regression. Despite its popularity and longevity, little is known about the statistical properties of the Langlie procedure, but it is well-established that the sequence of inputs tend to narrow in on the median of the sensitivity distribution. For this reason, the U.S. Department of Defense's protocol dictates that the procedure is only appropriate for estimating the median of the distribution, and no other quantiles. This begs the question of whether the parametric model assumption can be disposed of altogether, potentially making the Langlie procedure entirely nonparametric, much like the Robbins-Monro procedure. In this paper we answer this question in the negative by proving that when the Langlie procedure is employed, the sequence of inputs converges with probability zero.