'normalized inter-element distances' (power transformed Hartmann distances).

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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 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.

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

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distanceNormalized(x, rep=100, quant=c(0.05, 0.5, 0.95), significant=FALSE,
    trim=10, indexcol=FALSE, prob, digits=2, output=1,
    progress=TRUE, upper=TRUE)

Arguments

x

repgrid object.

rep

Number of random grids to generate to produce sample distribution for Hartmann distances (default is 100). Note that a lot of samples may take a while to calculate. Set progress = TRUE to monitor progress for large samples.

quant

The propabities of the quantiles from the power transformed Hartmann distance distribution that will be returned. The default is c(.05, .5, .95). This corresponds to the lower 5 %, the mean and the upper 5 % of the distribution.

significant

Whether to only show values that are outside the quantiles defined in quant, i.e. onsidered as 'significant' (default is FALSE.) The first and last value of quant is used to determine the indifference region. This options only applies when output == 1 is used.

trim

The number of characters a element names are trimmed to (default is 10). If NA no trimming is done. Trimming simply saves space when displaying the output.

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 FALSE). Note that the index column is the first matrix column.

prob

The probability of each rating value to occur. If NULL (default) the distribution is uniform. The number of values must match the length of the rating scale.

digits

Numeric. Number of digits to round to (default is 2).

output

The output type. The default (output=1) will print the power transformed Hartmann distances to the console. output=0 will suppress the printing to the console. In all cases a list containig the results of the calculations is returned invisibly. See value for details.

progress

Whether to show a progress bar (default is TRUE). May be useful when the distribution is estimated on the basis of many quasis.

upper

Logical. Whether to display only upper part of the distance matrix (default TRUE).

Details

The 'power tranformed or normalized Hartmann distance' are calulated as follows: The simulated Hartmann distribution is added a constant as the Box-Cox transformation can only be applied to positive values. Then a range of values for lambda in the Box-Cox 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 z-transformed 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), 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. London: Wiley.

See Also

distanceHartmann and distanceSlater.

Examples

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## 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)

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