An Additive Metric From All the Principal Components of a Perfect Scale 1

A new type of metric is established for perfect scales that is different from the three previously known types of metrics. Given a perfect scale of m types of dichotomous items, each rank of persons is regarded as a point in an m ‐dimensional space defined by the m non‐constant principal components. A non‐Euclidean distance function is defined for this space. It is proved that the resulting metric is additive: the non‐Euclidean distance between any two ranks i and k is the sum of the distances from i to j and from j to k whenever i ≤ j ≤ k. Treating the principal component space as non‐Euclidean may also be useful for the study of non‐scale structures and of quasi‐scales.