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

APPENDIX A
A remark on how to evaluate the logarithmic vs. two-linear issue
There are different ways of evaluating whether the logarithmic or the two-linear model
provide a better fit of the empirical data. In the following, these different approaches shall be
summarized. Generally, (i) approaches averaging data across participants can be distinguished
from (ii) approaches contrasting individual performance measures.
Averaging across participants (fixed effects analysis):
(a) After averaging the estimates for each to-be-located number across all participants
a comparison may be of interest which of the two models (logarithmic vs. two-linear)
provides a better fit of the empirical data (e.g., in terms of adjusted R 2 ). This can be done for
instance by a stepwise multiple regression analysis in which both a linear as well as a
logarithmic predictor is entered. By the logic of the stepwise regression analysis the predictor
which accounts for the largest part of the variance is included in the model first. Any further
predictor will only be incorporated into the model when it adds significantly to the variance
explained by the final regression model. Thereby, it would be possible to identify which of the
two predictors (i.e., linear or logarithmic) is incorporated into the regression model first and to
evaluate whether the inclusion of the other predictor would add reliably to the explanatory
power of the model or not. When running this stepwise regression analysis on mean estimates
of all Italian-speaking children it could be observed that only the two-linear predictor was
considered in the final regression model while the logarithmic predictor was not incorporated
[R = .99, adj. R 2 = .97, F(1, 16) = 553.59, p < .001, b two-lin = .99, p < .001, b log = - .35, p =
.18]. This observation was substantiated by the results of two forward regression analyses.
When entering the logarithmic and the two-linear predictor successively the logarithmic
predictor on its own was found to be a reliable predictor of estimation performance [b = .96, p
< .001]. However, when entering the two-linear predictor the logarithmic predictor was no
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