Slater and Hartmann distance

Slater and Hartmann distances are approaches to standardize the inter-element distances. A standardization is desireable as the distribution of euclidean distance between elements depends on the size of a grid. To be able to compare distances across grids a standardization is needed.

Desription
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, p. 94).

R-Code
> distanceSlater(boeker)

Slater distances

1   2    3    4    5    6    7    8    9   10   11   12   13   14   15 1 self           1.06 0.84 0.75 0.99 1.19 0.91 1.14 0.97  0.7 1.68  0.7 0.65 0.75  0.6 2 ideal self         1.14 0.78 1.06 1.51 1.15 1.06 1.12  0.9 1.74 1.19 1.24 1.33 1.18 3 mother                  0.86 0.67 1.01 0.68  0.9  0.7 0.79 1.31 0.76 0.88 0.83 0.73 4 father                       0.68 1.22 0.73 1.02 0.86  0.8 1.57 0.88 0.94  1.1 0.93 5 kurt                              0.92 0.72 0.92 0.67 0.82  1.2 0.79 0.96 0.88 0.87 6 karl                                      1 0.84 0.67 1.06 0.78  1.1 1.15 1.02 1.01 7 george                                      1.11 0.72 0.96 1.28 0.72 0.94 1.08 0.78 8 martin                                           0.76 0.79 1.19 1.14 1.11 1.05 1.06 9 elizabeth                                              0.9 0.93 0.97 1.13 1.05  0.9 10 therapist                                                 1.54 0.73 0.65 0.78 0.73 11 irene                                                          1.59 1.72 1.55 1.53 12 childhood                                                            0.5 0.58 0.42 13 self befor                                                               0.52 0.45 14 self with                                                                      0.5 15 self as dr

Note that Slater distances cannot be compared across grids with a different number of constructs.

Another possibility to prompt Slater distances is using the function distanceHartmann and chose argument output = 2. This function will also return quantile of the distribution of Slazer distances for random grids. This allows to determine 'significance' intervals in which Slater distances.

> distanceHartmann(boeker, output=2)

Slater distances

1   2    3    4    5    6    7    8    9   10   11   12   13   14   15 1 self             1.06 0.84 0.75 0.99 1.19 0.91 1.14 0.97  0.7 1.68  0.7 0.65 0.75  0.6 2 ideal self           1.14 0.78 1.06 1.51 1.15 1.06 1.12  0.9 1.74 1.19 1.24 1.33 1.18 3 mother                    0.86 0.67 1.01 0.68  0.9  0.7 0.79 1.31 0.76 0.88 0.83 0.73 4 father                         0.68 1.22 0.73 1.02 0.86  0.8 1.57 0.88 0.94  1.1 0.93 5 kurt                                0.92 0.72 0.92 0.67 0.82  1.2 0.79 0.96 0.88 0.87 6 karl                                        1 0.84 0.67 1.06 0.78  1.1 1.15 1.02 1.01 7 george                                        1.11 0.72 0.96 1.28 0.72 0.94 1.08 0.78 8 martin                                             0.76 0.79 1.19 1.14 1.11 1.05 1.06 9 elizabeth                                                0.9 0.93 0.97 1.13 1.05  0.9 10 therapist                                                   1.54 0.73 0.65 0.78 0.73 11 irene                                                            1.59 1.72 1.55 1.53 12 childhood                                                              0.5 0.58 0.42 13 self befor                                                                 0.52 0.45 14 self with                                                                        0.5 15 self as dr

The 'Slater distance' sample distribution has the following quantiles: 5% 50%  95% 0.72 0.99 1.24

Standard deviation: 0.16

Note that Slater distances cannot be compared across grids of different size. Another option is to output only values that can be considered 'significant', smaller or bigger than expected. This means they lay outside the upper and lower quantile of the distribution of Slater distance (significant=TRUE). The default setting is to use a the 5% and 95% quantile. You may change this using the argument  quant.

> distanceHartmann(boeker, significant=TRUE, output=2)

Slater distances

1   2    3    4    5    6    7    8    9   10   11   12   13   14   15 1 self                                                      0.7 1.68  0.7 0.65       0.6 2 ideal self                          1.51                     1.74      1.24 1.33 3 mother                         0.67      0.68       0.7      1.31 4 father                         0.68                          1.57 5 kurt                                     0.72      0.67 6 karl                                               0.67 7 george                                             0.72      1.28 0.72 8 martin 9 elizabeth 10 therapist                                                   1.54      0.65 11 irene                                                            1.59 1.72 1.55 1.53 12 childhood                                                              0.5 0.58 0.42 13 self befor                                                                 0.52 0.45 14 self with                                                                        0.5 15 self as dr

The 'Slater distance' sample distribution has the following quantiles: 5% 50%  95% 0.73 0.99 1.24

Standard deviation: 0.15

Note that Slater distances cannot be compared across grids of different size.

Calculation
$$D$$ 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.

Hartmann distance
Hartmann (1992) showed in a Monte Carlo study that Slater distances (see above) based on random grids, for which Slater coined the expression quasis, have a skewed distribution, a mean and a standard deviation depending on the number of constructs elicited. Hence, the distances cannot be compared across grids with a diffenrent number of constructs. As a remedy he suggested a linear transformation (z-transformation) of the Slater distance values which take into account their estimated (or expected) mean and their standard deviation to standardize them. Hartmann distances represent a more accurate version of Slater distances. Note that Hartmann distances are multiplied by -1 to allow an interpretation similar to correlation coefficients: negative Hartmann values represent an above average dissimilarity, i.e. a big Slater distance and positive values represent an above average similarity (i.e. a a small Slater distance).

The function distanceHartmann conducts a small Monte Carlo simulation for the supplied grid. I. e. a number of quasis of the same size and with the same scale range as the grid under investigation are generated. A distrubution of Slater distances derived from the quasis is calculated and used for the Hartmann standardization.

R-Code
Hartmann distances can be prompted as follows. As a default they are based on 100 random grids. > distanceHartmann(boeker)

Hartmann distances

1    2     3    4     5     6     7     8     9    10    11    12    13    14    15 1 self             -0.36  0.99 1.62  0.06 -1.24  0.58 -0.88  0.21  1.91 -4.36  1.91  2.23  1.62  2.57 2 ideal self            -0.88 1.43 -0.36 -3.25 -0.94 -0.36 -0.75  0.66  -4.7 -1.24 -1.54 -2.09 -1.12 3 mother                      0.91  2.12 -0.08  2.02  0.66  1.91  1.34 -1.98  1.52  0.74  1.08  1.71 4 father                            2.02 -1.42  1.71 -0.15  0.91  1.25 -3.62  0.74  0.35 -0.62  0.43 5 kurt                                    0.51  1.81  0.51  2.12  1.16  -1.3  1.34  0.28  0.74  0.83 6 karl                                         -0.01  0.99  2.12 -0.36  1.43 -0.62 -0.94 -0.15 -0.08 7 george                                             -0.69  1.81  0.28 -1.82  1.81  0.35 -0.49  1.43 8 martin                                                    1.52  1.34 -1.18 -0.88 -0.69 -0.29 -0.36 9 elizabeth                                                       0.66  0.43  0.21 -0.81 -0.29  0.66 10 therapist                                                           -3.48  1.71  2.23  1.43  1.71 11 irene                                                                     -3.75 -4.57 -3.53 -3.39 12 childhood                                                                        3.22  2.69  3.69 13 self befor                                                                             3.08  3.53 14 self with                                                                                    3.22 15 self as dr

The 'Hartmann distance' sample distribution has the following quantiles: 5%  50%   95% -1.52  0.06  1.76

Standard deviation: 1

It is also possible to return the quantiles of the sample distribution and only the element distances consideres 'significant' according to the quantiles defined. > distanceHartmann(boeker, significant=TRUE)

Hartmann distances

1   2    3    4    5     6    7    8    9   10    11   12    13    14    15 1 self                                                      1.94 -4.42 1.94  2.26         2.6 2 ideal self                          -3.29                     -4.76      -1.56 -2.12 3 mother                         2.15       2.04      1.94      -2.01 4 father                         2.04                           -3.66 5 kurt                                      1.84      2.15 6 karl                                                2.15 7 george                                              1.84      -1.84 1.84 8 martin 9 elizabeth 10 therapist                                                    -3.52       2.26 11 irene                                                              -3.8 -4.63 -3.57 -3.43 12 childhood                                                                3.27  2.73  3.74 13 self befor                                                                     3.12  3.57 14 self with                                                                            3.27 15 self as dr

The 'Hartmann distance' sample distribution has the following quantiles: 5%  50%   95% -1.54  0.07  1.74

Standard deviation: 1

Calculation
The Hartmann distance is calculated as follows (Hartmann 1992, p. 49).

$$D = -1 \frac{D_{slater} - M_c}{sd_c}$$

Where $$D_{slater}$$ denotes the Slater distances of the grid, $$M_c$$ the sample distribution's mean value and $$sd_c$$ the sample distributions's standard deviation.

Literature

 * 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.
 * Slater, P. (1951). The transformation of a matrix of negative correlations. British Journal of Statistical Psychology, 6, 101-106.
 * Slater, P. (1977). The measurement of intrapersonal space by Grid technique. Vol II. London: Wiley.