Electronic supplementary description of the
Structural Dimensional Analysis – Motorics (SDA-M)
Abstract
Because rating and sorting methods do not allow a psychometric analysis of the
representational structure, a methodical procedure called the structural dimensional analysis
of mental representation (SDA-M) is applied. The SDA-M contains four steps: (1) A special
split procedure involving a multiple sorting task is used to create a distance scaling among
the basic action concepts (BACs) of a suitably predetermined set. (2) A hierarchical cluster
analysis is used to transform the set of BACs into a hierarchical structure. (3) A factor
analysis is used to reveal the dimensions in this structured set of BACs. (4) The cluster
solutions are tested for invariance within and between individuals and groups. A practical
implication of the measurement of mental representations with SDA-M arises from the fact
that the measured representation structures can be analyzed not only on a group level but also
on an individual level.
Introduction
In descriptions of mental representations, the representational units and the structural
composition of these cognitive representations in LTM are of main interest. In general, BACs
are not represented in isolation, but are part of a hierarchical representation system. The
structure of this knowledge representation is understood as the internal grouping or clustering
of conceptual units (BACs) in individual sub domains. This approach views the relationships
among conceptual units as being feature based. They can be characterized according to the
type (feature classes), number, and weighting (relevance) of the features of a conceptual
representation system. This assignment of features is called dimensioning. Dimensioning is
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characterized in object concepts through the shared features of objects (e.g., color, size,
purpose) or in BACs through the shared properties of movements (e.g., temporal control,
amplitude, purpose). Hence, it is not just the structural design of a concept system but also
the dimensioning (feature binding and feature weighting) that is of interest. The relationship
between these two aspects of a mental representation is also of interest (Lander, 1991;
Schack, 2010b). Methodical approaches for measuring mental representations must consider
and analyze the structure and feature dimensions of representation. For this reason the
method is termed structural dimensional analysis of mental representation (SDA-M).
Method
Step 1: Splitting technique
A primary assumption of the SDA-M (and other methodological approaches to measuring
mental representations) is that the structure of movement representations can be explicated
only to a limited degree. Thus a splitting technique is used in a first step for this purpose.
Each concept is offered as an anchor (i.e., reference object, at the top of the list) to which the
remaining N − 1 concepts are either classified or declassified according to an individually
chosen similarity criterion (e.g., similarity or dissimilarity between concepts). This procedure
continues with the emerging (positive or negative) partial quantities by retaining the reference
concept (anchor) until an individual discontinuance criterion is reached. By this procedure N
decision trees are established, as each concept occupies an anchor (reference) position (at the
top of the list). Subsequently, the algebraic branch sums (Σ) are determined on the partial
quantities per decision tree and are submitted to a Z-transformation for standardization and
finally combined into a Z-matrix. This matrix forms the starting point of all further analyses.
If the Z-values are distributed normally, N(0, 1), then the classification probabilities of the
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concepts of the reference concept (which are found in the last joint of the decision tree) can
be established.
In case of our investigation (Stoeckel et al. this issue) the children judged the similarity of all
pictures (BACs) with one another (described in full in the paper), which produced positive or
negative partial quantities between each amount of pictures (BAC) (family Wusel or family
Sauber in relation to the anchoring picture). Because this anchor role of standard was
assigned to each picture (BAC) in succession, we ended up with a total of 9 decision trees
whose nodes contained the resulting subsets and whose borders took either a negative or
positive sign depending on whether the element was judged as belonging to or not belonging
to the standard. To obtain a measure of the distance between the successively judged
elements and the standard (with interval scaling), algebraic sums were computed over the
subsets located on one branch of the decision tree. These sum scores were then z transformed.
Step 2: Determination of the hierarchical structure of representation.
First, the Z-matrix was transformed into a Euclidean distance matrix for structural analysis,
and formed the basis for later hierarchical cluster analysis. As a result of the cluster analysis
an individual cluster solution on the N concepts was formed, and depicted as a dendogram.
Each cluster solution was established by determining the incidental Euclidean distance (dcrit).
All junctures that lay below the incidental value formed the apical pole of an underlying
concept cluster. The incidental distance was defined as
𝑑𝑐𝑟𝑖𝑡 = √2𝑁 √1 − 𝑟𝑐𝑟𝑖𝑡 (𝛼, 𝐹𝐺 / H0 : r = 1)
where rcrit denotes the incidental value of the correlation of two line vectors of the Z-matrix,
provided that H0 is valid. The value for rcrit results from the t-distribution for α (to be
determined), with FG = N − 2 degrees of freedom.
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Step 3: Determination of the feature dimensions of representations.
The Z-matrix was transformed into a correlation matrix, and served as the basis for
orthogonal factor analysis with a subsequent cluster-oriented rotation procedure. This
procedure yielded a factor matrix classified according to the concept cluster, the elements of
which were factor charges (c), or property values. A cluster within an individual cluster
solution (dendogram) stood out in that its elements (concepts) are at least equally and highly
charged. The incidental value for the factor charges based on the factor analysis was defined
by
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𝑐𝑐𝑟𝑖𝑡 = √ 𝑟𝑐𝑟𝑖𝑡 (𝛼, 𝐹𝐺 / 𝐻0 ∶ 𝑟 = 0)
𝑚
where m refers to the number of factors that are to be extracted (estimated a priori). Here rcrit
refers to the incidental value of the correlation of the two line vectors from the Z-matrix
(assuming that H0 is valid). As in step two, rcrit results from the t-distribution for α, with FG =
N − 2 degrees of freedom. This factor matrix formed the final solution of the individual SDAM as a concept quantity.
Step 4: Measurement of individual differences between the representations within each
group, and determination of the structural invariance (λ) between cluster solutions.
The number of made-up clusters of the pair-wise cluster solutions (r, s) was compared with
one another. The number of concepts within the made-up clusters (partial quantities nj, nk),
and the average quantities of the made up clusters (nik) were used for the structural invariance
measure of two cluster solutions, which was determined by
𝜆𝑖𝑘 = √𝑘𝑖𝑘 𝐺𝐴𝑀 (𝑝𝑖𝑘 )
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in which 𝑘𝑖𝑘 =
𝑀𝑖𝑛(𝑟,𝑠)
𝑀𝑎𝑥(𝑟,𝑠)
, with k0 = 2/3 as a limiting value, 𝑝𝑖𝑘 =
𝑛𝑖𝑘
√𝑛𝑖 𝑛𝑘
, 0 ≤ 𝑝𝑖𝑘 ≤ 1, with p0
= 0.7, and GAM(pik) comprises the weighted arithmetic measures of all relative average
quantities (pik). The differential threshold was determined by using the top two interval
criteria with λ0 = 0.683. Based on the invariance analysis, an invariance matrix (λ) was
established. The clusters represent subgroups of homogenous cluster solutions of the applied
concept quantities. The subgroup-specific cluster solutions were determined by summing the
Z-matrices of the individuals belonging to the subgroup-specific cluster (following a restandardization of the Z-summary values). Differences between groups were established in a
similar fashion. The established structures for each age group were determined by the
invariance measure (λ), which can be regarded as sufficient differential criterion if the
solutions are found to be of pairwise invariance (Schack, 2004, 2011; Schack & Mechsner,
2006; Schack & Ritter, 2009).
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Schack, T. (2011). A method for Measuring Mental Representation. Handbook of
Measurement in Sport, pp 203-214. Champaign, IL: Human Kinetics, in press.
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