English or languish - Probing the ramifications
of Hong Kong's language policy
Multidimensional Scaling
project index | statistical modelling (diagnostics | prognostics) | cluster analysis

Key Features

  • Input data - non-metric proximity data

  • Output data - metric data that lends itself to useful statistical analysis

  • Sample Size - As multidimensional scaling measures the perceptual space of individuals, a single respondent is sufficient for measurement. As the perceptual space of an entire population is likely to be complex, large sample sizes are recommended.

  • Objective - Transform simple nonmetric data into easily manipulated metric data for the purpose of understanding the subjective perceptions.

  • General Uses -

    1. The identification of individual and shared perceptual criteria utilized by subjects for the purpose of ordering the world. cluster analysis (key features)
    2. Modfication of non-metric data for use in metric-analytical procedures. cluster analysis (proximity measures)
    3. The sorting of objects, people, or notions into perceptual spaces.
    4. The comparison of subjective perceptual space with more objective empirical measures.

  • Statistical Procedure - Minimize the difference between dij - dij while constraining dij to be monotonic. See Kruskall's stress formula below for explanation of dij and dij

Key Terms

  • Ideal Element
    mds (research design issues)

    An ideal element is an imagined element of a stimulus set whose attributes are assigned by the respondent. As other elements of the stimulus set are compared with the imagined ideal element, the ideal element is treated statistically as it it were just another element of the stimulus set. As the ideal element appears on the perceptual map, it is a useful indicator for understanding the preference patterns of the respondent.

    Obviously ideal elements are only useful when comparisons are made for which respondents are queried with regard to their own preferences or pre-conceived notions about the way thing should be. more...

  • Kruskall's Stress Formula
    mds (key features)

    Stress = [Σ(dij - dij)2 / Σ(dij - dbar)2]1/2

    where dij = a monotonic transformation of the input proximity measure

    where dij = the estimated distance between the ith and jth observations

    where dbar = the mean of dij for all i and j such that i ≠ j.

  • Perceptual dimension or scale (or simply dimension)
    mds (number and interpretation of dimensions)

    Perceptual dimensions (scales) are statistically generated constructs representing the probable perceptual framework used by respondents to compare the elements of a stimulus set.

    Each dimension (scale) represents a separate bipolar standard of comparison. The dimension can be either implicit or explicit in the mind of the respondent.

  • Perceptual map
    mds (research design issues)

    A two-dimensional space on which the elements of a stimulus set are plotted according to their relative positions as defined by the dimensions of the respondents perceptual space. A three-dimensional space would require three two-dimensional maps in order to show the exact location of each of the elements relative to each pair of dimensions.

    The perceptual map is a key feature of multidimensional scaling, because it allows the researcher to visualize the distance between the elements of the stimulus set. more...

  • Proximity Data
    key features (input data)

    Nonmetric data obtained from pair-wise comparisons or rank orderings of single elements of a stimulus set that are qualitatively different and for which the explicit standards of comparison are simple measures of inequality (proximity). more...

  • Stimulus pair
    mds (research design issues)

    The stimulus pair is a key element of the nonmetric data set.

    In order to obtain nonmetric input data that can be transformed into appropriate metric data output one either rank orders individual stimuli or compares pairs of stimuli belonging to the stimulus set

  • Stimulus set
    mds (metric data output | research design issues | key terms (perceptual map | proximity data)

    he elements of a set of objects, people, or ideas that are ranked by respondents and form the basis for collecting proximity data.

Procedural Topics

Research Design Issues


Reference List

Greenberg, Marshall G. 1969. A variety of approaches to nonmetric multidimensional scaling. A monograph presented at the 16th International Meeting of the Institute of Management Sciences, New York (March). mds (metric data output)

Kruskal, Joseph B. 1964a. Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika 20 (March) pp - 27. mds (number and interpretation of dimensions)

------------. 1964b. Nonmetric multidimensional scaling: a numerical method. Psychometrika 29 (June) pp. 115-129. mds (number and interpretation of dimensions)

Neidell, Lester A. 1969. The use of nonmetric multidimenional scaling in marketing analysis. Journal of Marketing 33 (October) pp 37-43.