distanceNormalized: 'normalized inter-element distances' (power transformed...
In OpenRepGrid: Tools to analyse repertory grid data

Description Usage Arguments Details Value Author(s) References See Also Examples

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.

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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)

`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`).

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.

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

Mark Heckmann

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.

distanceHartmann and distanceSlater.

## 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)