Mapping numerical magnitudes along the right lines: differentiating between scale and bias.

Previous investigations on the subjective scale of numerical representations assumed that the scale type can be inferred directly from stimulus-response mapping. This is not a valid assumption, as mapping from the subjective scale into behavior may be nonlinear and/or distorted by response bias. Here we present a method for differentiating between logarithmic and linear hypotheses robust to the effect of distorting processes. The method exploits the idea that a scale is defined by transformational rules and that combinatorial operations with stimulus magnitudes should be closed under admissible transformations on the subjective scale. The method was implemented with novel variants of the number line task. In the line-marking task, participants marked the position of an Arabic numeral within an interval defined by various starting numbers and lengths. In the line construction task, participants constructed an interval given its part. Two alternative approaches to the data analysis, numerical and analytical, were used to evaluate the linear and log components. Our results are consistent with the linear hypothesis about the subjective scale with responses affected by a bias to overestimate small magnitudes and underestimate large magnitudes. We also observed that in the line-marking task, participants tended to overestimate as the interval start increased, and in the line construction task, they tended to overconstruct as the interval length increased. This finding suggests that magnitudes were encoded differently in the 2 tasks: in terms of their absolute magnitudes in the line-marking task and in terms of numerical differences in the line construction task.