Applications of Scaling to Developmental Research

Abstract Many theories of development and change involve strong assumptions of individual differences in growth and change not found in many traditional statistical methodologies. Nonmetric multidimensional scaling techniques are proposed as one family of techniques currently underutilized by developmental psychologists. Nonmetric scaling solutions generate maps of the underlying stimulus patterns, which make fewer assumptions of the data than is often the case in statistical model construction. Additionally, some models construct graphic representations of systematic subject differences. The basic graphic analogy underlying metric scaling solutions is outlined, along with the three fundamental assumptions of distance in scaling. Stress measures for determining the relative fit of candidate nonmetric solutions to the data are presented. Two applications from developmental psychology are presented that illustrate the application of nonmetric scaling solutions. The first, a weighted Euclidean analysis of data on children's conservation of volume, incorporates a specific model of interindividual differences as well as a model of intraindividual variation. The second example concerns a nonmetric analysis of developmental change in ratings of children's behavior, and determines optimal levels of environmental characteristics for individuals. This example has particular relevance to developmental researchers analyzing behavior from a system theoretic point of view. In this external nonmetric analysis, ideal optimal models are easily computable, given common regression analysis software. Common problems generally encountered in conducting nonmetric scaling analysis, local minima, degenerate solutions, and lack of convergence are defined, and courses of action to prevent their occurrence are proposed. Scaling methods may be broadly defined as statistical techniques developed to study relationships between stimuli presented to subjects at the inter- and intraindividual level. In this respect factor analysis, cluster analysis, and discriminant analysis qualify as scaling techniques under this definition. Although recent developments in comparison of scaling and its more parametric counterparts such as factor analysis reveal many close parallels, which will be discussed, the concentration in this chapter will be on models that either make less restrictive assumptions of the data or that incorporate models that more closely conform to certain assumptions of a developmental perspective of growth and change. In recent years, scaling techniques have been applied to a wide variety of areas, including counseling psychology (Davison, Richards and Rounds, 1986), personality (Rosenberg and Jones, 1972), and speech development (Shepard, 1972). In spite of this range of applications, metric and nonmetric multidimensional scaling has remained a relatively unused technique in developmental assessment. This is odd, given the fact that several metric models allow for simultaneous estimation of intra- and interindividual change processes. Complaints about the theoretical restrictions and lack of applicability of many standard linear models that assume the normal distribution are common in the developmental literature (Nesselroade and Ford, 1987; Labouvie and Nesselroade, in press; Nesselroade and Cattell, 1989). Scaling alternatives are attractive because several of them do not require the multivariate normality of tests frequently employed. This chapter presents the techniques of scaling as an often overlooked set of techniques that can be applied to data analyzable by other techniques mentioned here, as well as by types of data not amenable to the common methods of analysis due to limitations imposed by the specific design of an experiment, or theoretical assumptions about the nature of the phenomenon under study. In one sense, attempting to write a single chapter on scaling applications is as difficult as writing a single chapter on the possible developmental applications of the linear model. To complicate matters, scaling models have been articulated by several researchers who independently developed their models. For this reason, early scaling models often used very different terminology to describe similar processes—a confusing development that hides hierarchical relationships between models and confuses the beginning student. For this reason, this chapter will use the terminology of Young (1987) and Davison (1983), because these model designations highlight the explicit measurement assumptions of each model by reference to its geometric representation. The scaling techniques discussed here produce a geometric or spatial representation of relationships between stimuli or individuals. The end goal of the analyses is to produce a geometric representation of the underlying individual and/or stimulus dimensions that fit the data reasonably well and that can be plotted. Central to all scaling techniques discussed here is the notion of spatial proximity between pairs of objects as a measure of their relatedness. Some measures of proximity found in developmental research include correlation coefficients, joint probabilities, conditional probabilities, and transitional probabilities. Data gathering techniques currently underutilized can also be analyzed by scaling models. These include direct ratings of perceived (dis)similarity between stimuli that come from presenting subjects with sets of pairwise comparisons of a defined stimulus set or from asking parents or children the percent of times that a given strategy of behavior results in a criterion behavior. In many respects the development of a proximity measure is limited only by the ingenuity of the researcher. A necessary correlate of this statement is that the researcher must be able to defend his or her choice of a proximity measure as reasonable and conforming to the assumptions of the scaling technique employed. These assumptions relate to the precise manner in which proximity (or, to be more precise, lack of proximity) conforms to geometric analogy underlying the model. The explicit choice of the type of distance (Euclidean or non-Euclidean) assumed in the analysis will be understood within the generalized distance function. The way in which models that make increasingly less restrictive assumptions regarding how observed data relate to the underlying latent dimensions will then be presented. The best way to appreciate scaling alternatives in developmental psychology is to go through examples of analysis. To this end, two analyses of developmental data will be presented. The first example will discuss an individual differences model of perception in a developmental study. The second will involve assessment of a developmental analysis of behavior from a systems perspective.