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

the individual approach may be more suited to investigate developmental trajectories of the
spatial dimension of number magnitude representation.
Addressing the problem of possible model overfitting
Finally, a general issue in science regards the question of model overfitting in the way
of evaluating whether it is signal which is modeled by a given model or rather signal plus
noise. In the latter case, one runs the risk of capitalizing on measurement error due to a more
complex model with a higher number of degrees of freedom (see Myung & Pitt, 1997; Pitt,
1999, Pitt, Myung, & Zhang, 2002 for a discussion of this point). Transferred to the current
study this taps on the different number of free parameters between the logarithmic and the
two-linear model. While the logarithmic model [based on a function such as y = a log(x) + b]
has the two free parameters a and b, a two-linear model without a fixed breakpoint has four
free parameters (a 1 , a 2 and b 1 , b 2 ) two for each linear part of the model (i.e., y = a 1 x +b 1 for
the first part and y = a 2 x +b 2 for the second part). Yet, when using a fixed breakpoint (as
assumed for the current two-linear model) the number of free parameters comes down to
three. When the first linear segment is described by the function y = a 1 x +b 1 with two free
parameters and the constraint of ending at a fixed breakpoint (e.g., 10) it defines a fixed point
at this endpoint which the second linear part of the model has to cut. Thereby, the second part
of the two-linear model with a fixed breakpoint is described sufficiently by one further free
parameter, be it either constant or slope. As the second linear element has to cut a fixed point
at the breakpoint its slope is fixed when the constant is estimated while on the other hand the
constant is fixed when the slope is estimated. Taken together, the proposed two-linear model
with fixed breakpoint at 10 has exactly one free parameter more than the logarithmic model.
Nevertheless, this could make the difference. To account for the higher number of free
parameters in our model as compared to both the simple linear as well as the logarithmic
model R 2 values were adjusted by the number of free parameters of either model prior to any
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