Analysis of variance procedures based on a proximity measure between subjects.

Abstract

Analysis of variance techniques are presented for the problem of ^-independent and AT-dependent samples when the data provided by each subject are used to generate a numerical proximity measure between subjects. The term proximity refers to any index of similarity or correspondence a researcher wishes to define, usually from the responses that subjects have provided to a set of objects or items. Differences between the ^-independent or AT-dependent samples are tested by evaluating the patterning of these proximities in relation to an appropriate randomization model. The usual univariate analysis of -variance methods based on randomization can be interpreted as special cases of the more general procedures. Finally, to illustrate the discussion we present a numerical example that deals with the subjective evaluation of criminal activity over a set of California cities.

Behavioral Geography and the Theoretical/Quantitative Revolution

“…In addition, Tobler's (1978) suggestions for developing error‐ellipses [variations of the Tissot Indicatrix] proved to be important in displaying and analyzing variability in subjective location errors. Probabilistic Multidimensional Scaling. This was developed by Zinnes and MacKay (1983) (Geography, Marketing, and Psychology) and was a Probabilistic Scaling Model (PROSCAL) of perceptions and/or cognitions of people's preferences for places. Spatial Autocorrelation (Hubert and Golledge 1981a, b, 1982; Hubert, Golledge, and Costanzo 1981, 1982): Emphasis was placed on metric and non‐metric measures of spatial association (i.e., spatial autocorrelation) and methods for analyzing square and rectangular data matrices derived from different explanatory models. Made necessary by the need to evaluate alternative predictive models—such as variations of the Spatial Interaction Model—predictions of shopping center choices, and by comparing objective and subjective models of behavior—models such as CONGRU (Olivier 1970), PROFIT (Carroll and Chang (1970), and Quadratic Assignment Procedures (QAP: Hubert and Golledge 1981a, b) were developed and tested. Interaction Models: Different versions of the Social Gravity/Spatial Interactance Model were developed to include subjective distance, travel time, and attitudinal and preference factors (Huff 1963; Cadwallader 1973, 1976; Fotheringham 1981, 1983, 1984a, b, 1986).…”

“…Spatial Autocorrelation (Hubert and Golledge 1981a, b, 1982; Hubert, Golledge, and Costanzo 1981, 1982): Emphasis was placed on metric and non‐metric measures of spatial association (i.e., spatial autocorrelation) and methods for analyzing square and rectangular data matrices derived from different explanatory models. Made necessary by the need to evaluate alternative predictive models—such as variations of the Spatial Interaction Model—predictions of shopping center choices, and by comparing objective and subjective models of behavior—models such as CONGRU (Olivier 1970), PROFIT (Carroll and Chang (1970), and Quadratic Assignment Procedures (QAP: Hubert and Golledge 1981a, b) were developed and tested.…”

Methods for defining user groups and user-adjusted information structures

“…The best strategies to make confirmatory analysis in this situation appear to be based on the test of Mantel (Arabie and Hubert 1992, Hubert et al 1982, Legendre and Fortin 1989, Legendre and Vaudor 1991. This test has the advantage of avoiding the problem of the non-independence between correlations (Rounds et al 1992) that other methods that have been used in published literature do not really control Schvaneveldt 1988, Gillan et al 1992).…”

“…In our case the matrix is formed by ones in the positions corresponding to the correlation's between pairs of the same groups and zeros of the pairs of different groups. This permits us to calculate the equivalent of an analysis of variance over the matrix of proximity (Hubert et al 1982).…”

A note on the use of directional statistics in weighted euclidean distances multidimensional scaling models

“…Standard linear statistical methods may be used with such data if it is borne in mind that the observations are not independent of one another. Another possibility might be to use the analysis of variance for proximities data, recently developed by Hubert, Golledge, and Costanzo [1982]. As before, it must be remembered that the WMDS weights are not independent of one another, unless an externally supplied group space is used in estimation procedure.…”