Statistical consistency and hypothesis testing for nonmetric multidimensional scaling (Q1819882)

Nonmetric multidimensional scaling (NMDS) methods deal generally with data \(r_{im}\) from a group of subjects \((i=1,...,I)\) for a set of objects on different comparisons \((m=1,...,M)\). For some given functions \(v_ m(\lambda)\), the aim is to find \(\lambda\) (unknown vector of values representing the objects) such that the rank order of the \(v_ m(\lambda)\) is similar to the rank order of some function of the \(r_{im}.\) Statistical properties are explored assuming that \(r_{im}\) are realizations of a random variable \(r_ m\) and using Monte-Carlo studies. In the case taken into account here (Shepard-Kruskal method), the ''true'' \(\lambda^ 0\) belongs to an unknown set \(\Lambda^ 0:\) the classical concept of (point) consistency has to be extended to a set-valued one. A variety of sufficient conditions are stated and applied to different models. Hypothesis testing for NMDS is also provided in a maximum- likelihood framework.