Slater Distance

Desription of Slater distance
As a similarity measure in grids different types of Minkowski metrics, especially the euclidean and city-block metric are frequently used. The euclidean distance is the sum of squared differences between the ratings on two different elements. This measure depends on the range of the rating scale and the number of constructs used, that is, on the size of the grid. An approach to standardize the euclidean distance to make it independent from size and range of ratings of the grid and thus comparable between grids of different size was proposed by Slater (1977, pp. 94).

$$D$$ is a the grid matrix $$G$$ centered around the construct means.

$$ d_{ij} =g_{..} - g_{ij} $$

Where $$g_{..}$$ is the mean of the construct.

$$ P=D^TD $$

$$ S= trace(P) $$

Euclidean distances results in:

$$ (\sum{ (d_{ij} - d_{ik} )^2})^{1/2} $$

$$ \Leftrightarrow (\sum{ (d_{ij}^2 + d_{ik}^2 - 2d_{ij}d_{ik})})^{1/2} $$

$$ \Leftrightarrow (\sum{ d_{ij}^2 } + \sum{d_{ik}^2} - 2\sum{d_{ij}d_{ik} })^{1/2} $$

$$ \Leftrightarrow (S_j + S_k - 2P_{jk})^{1/2} $$

For the standardization, Slater proposes to use the expected euclidean distance between a random pair of elements taken from the grid. The average for $$S_j$$ and $$S_k$$ would then be $$S_{avg} = S/m$$ where $$m$$ is the number of elements in the grid. The average of the off-line diagonals of $$P$$ is -S/m(m-1) (see Slater, 1951, for a proof). Inserted into the formula above it gives the following expected average euclidean distance $$U$$ which is outputted as unit of expected distance in Slater's INGRID program.

 U = (2S/(m-1))^{1/2} 

The calculated euclidean distances are then divided by the unit of expected distance to form the matrix of standardized element distances E_{std}

 E_{std} = E/U 

Distances calculated to be bigger than 1 are greater than expected, smaller than 1 are smaller than expected. These distances can be used to compare element distances between different grids, where the grid do not need to have the same constructs or elements.

Literature

 * Slater, P. (1951). NEEDS TO BE ADDED.
 * Slater, P. (1977). The measurement of intrapersonal space by Grid technique. Vol II. London: Wiley.