Normalized distances

Slater and Hartmann distances are approaches to standardize the inter-element distances. The function explained here are a further approach to eliminate the shortcomings of Slater's (1977) and Hartmann's (1992) approach.

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) and Hartmann (1992).

Hartmann (1992) suggested a transformation of Slater (1977) distances to make them independent from the size of a grid. Hartmann distances are supposed to yield stable cutoff values used to determine 'significance' of inter-element distances. It can be shown that Hartmann distances are still affected by grid parameters like size and the range of the rating scale used. The function distanceNormalize applies a Box-Cox (1964) transformation to the Hartmann distances in order to remove the skew of the Hartmann distance distribution. The normalized values show to have more stable cutoffs (quantiles) and better properties for comparison across grids of different size and scale range.

R-Code
The function distanceNormalize will return Slater, Hartmann or power transfpormed Hartmann distances if prompted. It is also possible to return the quantiles of the sample distribution and only the element distances consideres 'significant' according to the quantiles defined.

> distanceNormalized(boeker)

Power transformed Hartmann distances

1    2     3    4     5     6     7     8     9    10    11    12    13    14    15 1 self             -0.43  0.94 1.55     0 -1.36  0.53 -0.97  0.15  1.84 -5.18  1.84  2.14  1.55  2.47 2 ideal self            -0.97 1.37 -0.43 -3.66 -1.04 -0.43 -0.84  0.61 -5.81 -1.36 -1.68 -2.29 -1.23 3 mother                      0.85  2.04 -0.14  1.94  0.61  1.84  1.28 -2.17  1.46  0.69  1.02  1.65 4 father                            1.94 -1.55  1.65 -0.22  0.85  1.19 -4.13  0.69   0.3  -0.7  0.38 5 kurt                                    0.45  1.74  0.45  2.04  1.11 -1.43  1.28  0.22  0.69  0.77 6 karl                                         -0.07  0.94  2.04 -0.43  1.37  -0.7 -1.04 -0.22 -0.14 7 george                                             -0.77  1.74  0.22 -1.99  1.74   0.3 -0.57  1.37 8 martin                                                    1.46  1.28  -1.3 -0.97 -0.77 -0.36 -0.43 9 elizabeth                                                       0.61  0.38  0.15  -0.9 -0.36  0.61 10 therapist                                                           -3.95  1.65  2.14  1.37  1.65 11 irene                                                                     -4.31 -5.54 -4.01 -3.83 12 childhood                                                                        3.08  2.58  3.52 13 self befor                                                                             2.95  3.37 14 self with                                                                                    3.08 15 self as dr

The 'Power transformed Hartmann distance' sample distribution has the following quantiles: 5%  50%   95% -1.63  0.00  1.65

Standard deviation: 1

To only output inter-element distances that are judged to 'significant' of the basis of the provided quantiles use the argument significant=TRUE.

> distanceNormalized(boeker, sign=T)

Power transformed Hartmann distances

1   2    3    4    5     6    7    8    9   10    11    12    13    14    15 1 self                                                      1.81 -5.21  1.81  2.11        2.43 2 ideal self                          -3.65                     -5.88       -1.66 -2.28 3 mother                         2.01       1.91      1.81      -2.16 4 father                         1.91                           -4.12 5 kurt                                      1.71      2.01 6 karl                                                2.01 7 george                                              1.71      -1.97  1.71 8 martin 9 elizabeth 10 therapist                                                    -3.94        2.11 11 irene                                                              -4.31 -5.58    -4 -3.82 12 childhood                                                                 3.03  2.54  3.45 13 self befor                                                                       2.9  3.31 14 self with                                                                             3.03 15 self as dr

The 'Power transformed Hartmann distance' sample distribution has the following quantiles: 5%  50%   95% -1.65  0.01  1.63

Standard deviation: 1

Calculation
The form of normalization applied by Hartmann (1992) does not account for skewness or kurtosis. Here, a form of normalization - a power transformation - is explored that takes into account these higher moments of the distribution. For this purpose Hartmann values are transformed using the Box-Cox family of transformations (Box & Cox, 1964). The transformation is defined as

$$ Y_i^{\lambda}= \left\{ \begin{matrix} \frac{(Y_i + c)^\lambda - 1}{\lambda} & \mbox{for }\lambda \neq 0 \\ ln(Y_i + c) & \mbox{for }\lambda = 0 \end{matrix} \right. $$

As the transformation requires values $$\ge 0$$ a constant $$c$$ is added to derive positive values only. For the present transformation $$c$$ is defined as the minimum Hartmann distances from the quasis distribution. In order to derive at a transformation that resembles the normal distribution as close as possible, an optimal $$\lambda$$ is searched by selecting a $$\lambda$$ that maximizes the correlation between the quantiles of the transformed values $$Y_i^\lambda$$ and the standard normal distribution. As a last step, the power transformed values $$Y_i^\lambda$$ are z-transformed to remove the arbitrary scaling resulting from the Box-Cox transformation yielding $$Y_i^P$$.

$$ Y_{i}^P = \frac{Y^{\lambda}_i - \overline Y^{\lambda}}{\sigma_{Y^{\lambda}}} $$

Literature

 * Box, G. E. P., & Cox, D. R. (1964). An analysis of transformations. Journal of the Royal Statistical Society. Series B (Methodological), 26(2), 211-252.
 * 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. (1977). The measurement of intrapersonal space by Grid technique. Vol II. London: Wiley.