The influence of the place-value structure of the Arabic number ...

Splitting the overall interval in two should do nothing to the validity of the logarithmic model.
This may be of particular interest for the segment representing two-digit numbers as the
proposition of a linear representation of all two-digit numbers seems to be a strong claim
against the background of the problem size effect, which should be tested prior to any further
analysis. Given the case there is no problem size effect for the two-digit numbers (or e.g., the
larger half of the items), interpreting the relatively good fit of a logarithmic function to
indicate a logarithmic magnitude representation may be premature as there might actually be
no explicit representation of the magnitude of these items. Instead, this might reflect the case
of intact magnitude representation up to a given quantity (e.g., 10) with each magnitude above
this point corresponding to a rather fuzzy representation of “many”. Please note that the
assumption of an overall magnitude representation as claimed by the logarithmic model is
violated by such a data pattern. On the other hand, a two-linear model as proposed by
Ebersbach et al. (2008) would be able to account for the data pattern in a more appropriate
way. Possible confounds to a logarithmic interpretation of this kind may for instance be
inherent in the data pattern reported by Opfer, Thompson, & Furlong (in press, Figure 5, p. 8)
interpreted to indicate a logarithmic magnitude representation of even preschoolers (see also
Brysbeart, 1995; Muldoon, Simms, Towse, Burns, & Yue, this issue for comparable data
patterns). Second, by running linear and logarithmic analyses separate for one- and two-digit
numbers as testing specific model predictions of the two-linear model with a breakpoint at 10
no special adjustment for the degrees of freedom of the two models is necessary as these are
identical (i.e., linear: y = a·x + b; logarithmic: y = a·log(x) + b).
Fitting individual participants (random effects analysis):
(a) When intending to evaluate model adequacy on a more individual basis a measure
of model fit (e.g., adjusted R 2 ) can also be calculated for both the logarithmic as well as the
two-linear model for each participant individually. Afterwards the two matrices are directly
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