We propose a non-parametric estimate of the scale change parameter characterizing the difference between two survival functions under the accelerated failure time model using an estimating equation based on restricted means. Our estimator is different from the rank-based estimator of Louis (1981), which is obtained from a modification of the efficient score of Cox’s (1972) proportional hazards model. We derive the asymptotic properties of the proposed estimator. In a simulation study, we investigate the finite sample properties of our estimator and compare its performance with Louis’s estimator and parametric competitors in terms of bias, efficiency, and accuracy of coverage probabilities. We found that with sample sizes ranging from 50 to 500 per group, our estimator was unbiased with reasonably accurate type I error rates and coverage probabilities. Efficiency relative to fitting the correct parametric model ranged from 83% to 93%. Point estimates and confidence intervals based on our estimator were similar to those obtained based on Louis’s estimator. An example is presented to illustrate an application of the methodology in practice.