Some Statistical Considerations in Multidimensional Scaling

ABSTRACT This study had three purposes: First, to discuss the multidimensional scaling model with respect to its statistical problems; second, to describe a set of statistical assumptions which seem reasonable in light of the nature of the scaling model and to derive from them a set of estimates and predictions for a scaling experiment; and third, to investigate the usefulness and success of these estimates in a particular scheme for gathering scaling data and to determine whether this scheme led to reasonable conclusions about color space. The ways in which stimuli are perceived can often be assessed by obtaining scales for the stimuli with respect to their attributes. The problem of obtaining these scales has been approached more and more by using information about perceived differences. This information is transformed, through multidimensional scaling, into a representation of the stimuli as points in a space, usually Euclidean, of specified dimensionality. Unfortunately, if one plots the difference measures used against the corresponding interpoint distances, the resulting curve may not pass through the origin and may even at times be nonlinear. The source of these difficulties may be the traditional assumption of a symmetric distribution, often the normal or Gaussian, for differences. This assumption occurs, for example, whenever the mean is used as a direct estimate of the desired parameter. In order to illustrate these difficulties and to provide an alternate hypothesis which may be more acceptable, a statistical model is derived for perceived difference starting from assumptions first made by Thurstone in his unidimensional scaling models. The model is extended to include the distribution of ratios of differences. It is shown that the model can provide a solution to both the so‐called additive constant problem and to some of the nonlinearity problems in the relationship of perceived difference to scaled interpoint distance. One frequent use of multidimensional scaling is in the construction of a uniform color scale. Here, some problems have arisen with the assigning of measures to difference. In order to investigate these problems, an experiment was conducted using 21 colors falling in a plane of constant lightness. Each pair was compared to a standard pair of colors by asking subjects to specify the ratio of the difference between the colors in the variable pair to that between those in the standard pair. The data for 20 subjects with normal color vision were analyzed for each individual and for the total group, using four types of estimators: the mean ratio judgments, two types of unbiased estimators for population differences according to the statistical model, and a maximum likelihood estimator for these differences, also according to the model. It was found that this method of collecting difference data, which is somewhat more economical than many other methods available, gave reasonable solutions for individual color spaces. It was found also that the estimators provided by the model and the means worked equally well for the scaling of these differences for the group. The relationship between mean differences and scaled interpoint distances was linear and passed through the origin. It appears that difference data collected in this fashion do not suffer from the problems indicated in other data. A generalization of the statistical model which accommodates this fact is offered, in which it is postulated that judgments for ratios of small differences are not as variable as those for larger differences and hence that the nonlinearity predicted with the assumption of constant variability does not occur.