Effect of Rancher’s Management Philosophy, Grazing Practices, and Personal Characteristics on Sustainability Indices for North Central Texas Rangeland Page: 69
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Mair and de Leeuw (2010) describe a general framework for multivariate analysis with
optimal scaling whereby, multivariate data are collected into a multivariable "aspect". This
aspect of a multivariable is a function that is used to measure how well the multivariable satisfies
some criterion. Two different families of aspects exist: non-correlational and correlational. It is
quite possible that the two will give the same transformations. If the goal is to achieve
bilinearization, then the non-correlational aspect, LINEALS, should be chosen. "A function of
two variables is bilinear if it is linear with respect to each of its variables. The simplest example
is f(x, y) = x y" (Weisstein, 2011).
Many parametric statistical routines, like regression assume that the effects of each
variable on the target are linear; if the effects are non-linear, transformation of data are
recommended (Field, 2005). Therefore, because binearization was desirable, the R package
'aspect' and the non-correlational aspect, LINEALS was chosen for use in optimal scaling of
data obtained during the study.
When using the R package 'aspect', the LINEALS function makes transformations of the
variables such that all bivariate regressions are exactly linear. This is a function combining
elements of two vector spaces (consisting of an ordered collection of numbers) to yield an
element of a third vector space that is linear in each of its arguments. In the case of LINEALS
this is rjl of the Pearson correlation matrix and the Pearson correlation ratio. A matrix of these
quantifications is available in the R output. After optimization, asymptotic tests and estimates
can still be used on the transformed data (de Leeuw and Mair, 2010)
To assess the relative success of the imputation, part of the output is characterized as
"loss". Loss of information occurs when discretizing (grouping) variables and then performing
optimal scaling (Mair and de Leeuw, 2010). They do not quantify loss, but simply state, "the
lower, the better". They explain further by reporting that eventually, the value of the LINEALS
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Becker, Wayne. Effect of Rancher’s Management Philosophy, Grazing Practices, and Personal Characteristics on Sustainability Indices for North Central Texas Rangeland, dissertation, December 2011; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc103289/m1/80/: accessed July 27, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .