'normalized interelement distances' (power transformed Hartmann distances).
Description
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 interelement
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 BoxCox (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.
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.
Usage
1 2 3 
Arguments
x 

rep 
Number of random grids to generate to produce
sample distribution for Hartmann distances
(default is 
quant 
The propabities of the quantiles from the
power transformed Hartmann distance distribution that will be returned.
The default is 
significant 
Whether to only show values that are outside the quantiles
defined in 
trim 
The number of characters a element names are trimmed to (default is

indexcol 
Logical. Whether to add an extra index column so the
column names are indexes instead of element names. This option
renders a neater output as long element names will stretch
the output (default is 
prob 
The probability of each rating value to occur.
If 
digits 
Numeric. Number of digits to round to (default is

output 
The output type. The default ( 
progress 
Whether to show a progress bar (default is 
upper 
Logical. Whether to display only upper part of the distance matrix
(default 
Details
The 'power tranformed or normalized Hartmann distance' are calulated as follows: The simulated Hartmann distribution is added a constant as the BoxCox transformation can only be applied to positive values. Then a range of values for lambda in the BoxCox transformation (Box & Cox, 1964) are tried out. The best lambda is the one maximizing the correlation of the quantiles with the standard normal distribution. The lambda value maximizing normality is used to transform Hartmann distances. As the resulting scale of the power transformation depends on lambda, the resulting values are ztransformed to derive a common scaling.
The code for the calculation of the optimal lambda was written by Ioannis Kosmidis.
Value
A matrix containing Hartmann distances (output=1
and output=2
)
or a list (output=3
) containing:
hartmann 
matrix of Hartmann distances 
h.quantiles 
quantiles for Hartmann distances 
h.vals 
random values of Hartmann 
h.sd 
standard deviation of distribution of Hartmann values 
slater 
matrix of Slater distances 
sl.quantiles 
quantiles for Slater distances 
sl.vals 
vector of all Slater distances 
ls.sd 
standard deviation of random Slater distances 
normalized 
matrix of power transformed Hartmann distances 
n.quantiles 
quantiles for power transformed Hartmann distances 
n.vals 
vector of all power transformed Hartmann distances 
n.sd 
standard deviation of random power transformed Hartmann distances 
Author(s)
Mark Heckmann
References
Box, G. E. P., & Cox, D. R. (1964). An Analysis of Transformations. Journal of the Royal Statistical Society. Series B (Methodological), 26(2), 211252.
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), 4156.
Slater, P. (1977). The measurement of intrapersonal space by Grid technique. London: Wiley.
See Also
distanceHartmann
and distanceSlater
.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  ## Not run:
distanceNormalized(bell2010)
distanceNormalized(bell2010, trim=40, index=T, sig=T)
### histogram of power transformed Hartmann distances indifference region
d < distanceNormalized(bell2010, out=0)
hist(d$n.vals, breaks=100)
abline(v=d$n.quant, col="red")
### histogram of Hartmann distances and indifference region
hist(d$h.vals, breaks=100)
abline(v=d$h.quant, col="red")
## End(Not run)
