Calculate Slater distance.
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The euclidean distance is often used as a measure of similarity between
distance. A drawback of this measure is that it
depends on the range of the rating scale and the number of constructs used,
i. e. on the size of a grid.
An approach to standardize the euclidean distance to make it independent from size and range of ratings and was proposed by Slater (1977, pp. 94). The 'Slater distance' is the Euclidean distance divided by the expected distance. Slater distances bigger than 1 are greater than expected, lesser than 1 are smaller than expected. The minimum value is 0 and values bigger than 2 are rarely found. Slater distances have been be used to compare inter-element distances between different grids, where the grids do not need to have the same constructs or elements. Hartmann (1992) showed that Slater distance is not independent of grid size. Also the distribution of the Slater distances is asymmetric. Hence, the upper and lower limit to infer 'significance' of distance is not symmetric. The practical relevance of Hartmann's findings have been demonstrated by Schoeneich and Klapp (1998). To calculate Hartmann's version of the standardized distances see
A matrix with Slater distances.
The Slater distance is calculated as follows.
For a derivation see Slater (1977, p.94).
Let matrix D contain the row centered ratings. Then
P = D^TD
S = tr(P)
The expected 'unit of expected distance' results as
U = (2S/(m-1))^.5
where m denotes the number of elements of the grid. The standardized Slater distances is the matrix of Euclidean distances E devided by the expected distance U.
Hartmann, A. (1992). Element comparisons in repertory grid technique: Results and consequences of a Monte Carlo study. International Journal of Personal Construct Psychology, 5(1), 41-56.
Schoeneich, F., & Klapp, B. F. (1998). Standardization of interelement distances in repertory grid technique and its consequences for psychological interpretation of self-identity plots: An empirical study. Journal of Constructivist Psychology, 11(1), 49-58.
Slater, P. (1977). The measurement of intrapersonal space by Grid technique. Vol. II. London: Wiley.
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