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

analyses. However, in the following, the results of two additional analyses will be presented
addressing the problem of possible overfitting of the data by the more complex two-linear
model.
In a first approach it was evaluated in how far the results observed depended on the
inclusion of one particular data point (i.e., to-be-estimated number) to examine over-fitting of
the data. When iteratively omitting one item from the analysis results in considerable changes
of each of the models’ performance this indicates that the overall result is highly dependable
on the inclusion of particular data points. On the other hand, when results do not change
considerably this argues for a high stability of the overall result indicating that it is rather
invariant to the exclusion of specific items. A respective analysis for the current data showed
that the results for both models were very stable (logarithmic: mean adjusted R 2 = .68, SEM <
0.001; two-linear: mean adjusted R 2 = .82, SEM < 0.001; mean number of participants with an
optimal break-point around 10: 62.72, SEM = 0.87; mean slope of single digit segment: 3.93,
SEM < 0.01; mean slope of two-digit segment: 0.32, SEM < 0.001). More interestingly, the
two-linear model outperformed the logarithmic one reliably in terms of adjusted R 2 no matter
which data point was omitted from the analysis (all t > 10.23, all p < .001). Note that this
stability is true for a logarithmic model and for a two-linear model with a fixed break point. A
model with a variable break point could be potentially more flexible and thus more variable
when single data points are omitted.
Another way to address the point of possible overfitting of the two-linear model is to
evaluate in how far a two-linear model accounts for data produced by a logarithmic model and
vice versa. Generally, data produced by logarithmic model (including some random noise)
must not be accounted for better by two-linear fitting as compared to logarithmic fitting.
When data produced by a logarithmic model is accounted for better by two-linear fitting than
by logarithmic fitting this would suggest overfitting of the data by the two-linear model.
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