| Abstract: | In an earlier note, a new metric for bounded response scales (MBR) was introduced which resembles the city-block metric but is bounded above. It was suggested the MBR may be more appropriate than minkowski metrics for data obtained with bounded response scales. In this article, some formal properties of the MBR are investigated and it is shown that it is indeed a metric. Empirical predictions are then derived from the MBR and contrasted with those of a “monotonicity hypothesis,” which holds that dissimilarity judgements tend to be biased towards overestimation of larger distances, and with the predictions of the minkowski metrics, which imply additivity of collinear segments. Some empirical results are presented which contradict the monotonicity hypothesis and the minkowski metrics, and favor the MBR. Finally, the logic used to motivate the MBR is invoked to define a subadditive concatenation for bounded norms in the one-dimensional case, which may be useful in psychophysical work where the upper bounds are often real, rather than due to the response scale. This concatenation predicts understimation for doubling and overestimation for halving and middling tasks. |
