Number and Continuous Magnitude Processing Depends on Task Goals and Numerosity Ratio

A large body of evidence shows that when comparing non-symbolic numerosities, performance is influenced by irrelevant continuous magnitudes, such as total surface area, density, etc. In the current work, we ask whether the weights given to numerosity and continuous magnitudes are modulated by top-down and bottom-up factors. With that aim in mind, we asked adult participants to compare two groups of dots. To manipulate task demands, participants reported after every trial either (1) how accurate their response was (emphasizing accuracy) or (2) how fast their response was (emphasizing speed). To manipulate bottom-up factors, the stimuli were presented for 50 ms, 100 ms or 200 ms. Our results revealed (a) that the weights given to numerosity and continuous magnitude ratios were affected by the interaction of top-down and bottom-up manipulations and (b) that under some conditions, using numerosity ratio can reduce efficiency. Accordingly, we suggest that processing magnitudes is not rigid and static but a flexible and adaptive process that allows us to deal with the ever-changing demands of the environment. We also argue that there is not just one answer to the question ‘what do we process when we process magnitudes?’, and future studies should take this flexibility under consideration.

Since duration affected RTs, we conducted the same regression but separately for each duration. For durations of 50 ms, convex hull ratio was the first to enter, F (1, 122) = 9.95, p = .002. The correlation coefficient was .28, indicating approximately 7.5% of the variance in RT could be accounted for by convex-hull ratio. In the next and final step, in addition to convex hull ratio, numerical ratio was entered into the equation. In this final regression step, the correlation coefficient was .35, indicating 12% of the variance of the RT could be accounted for by the mentioned continuous ratios. For durations of 100 ms, numerosity was the first to enter into the regression, F (1, 122) = 33.85, p < .001. The correlation coefficient was .47, indicating approximately 22% of the variance in RT could be accounted for by the numerosity ratio at an exposure duration of 100 ms. In the second and final step, convex hull ratio was entered into the equation. The correlation coefficient was .58, indicating approximately 34% of the variance in RT could be accounted for by numerosity and convex hull ratios at exposure durations of 100 ms (see Table S3). For durations of 200 ms, numerosity ratio was the first to enter into the regression, F (1, 122) = 35.23, p < .001. The correlation coefficient was .48, indicating approximately 22% of the variance in RT could be accounted for by numerosity ratio at exposure durations of 200 ms. In the final step, convex hull and average diameter ratios were entered into the equation. The correlation coefficient was .55, indicating approximately 31% of the variance in RT could be accounted for by numerosity and convex hull ratios at exposure durations of 100 ms (see Table S3).

Accuracy as a dependent measure
In step 1 of the analysis, numerosity ratio was entered into the regression equation and was significantly related to accuracy, F (1, 500) = 334.34, p < .001. The correlation coefficient was -.63, indicating approximately 40% NUMBER AND CONTINUOUS MAGNITUDE S2 of the variance in accuracy rates could be accounted for by the numerosity ratio. In the next steps, in addition to numerosity ratio, density, total circumference, convex hull and total surface area ratios were entered into the regression equation. In this final regression step, the correlation coefficient was .69, indicating 47% of the variance of the accuracy rates could be accounted for by the mentioned numerosity and continuous ratios (see Table S4). Note that duration was not entered into the regression.

Speed emphasis condition RT as a dependent measure
In step 1 of the analysis, total circumference ratio was entered into the regression equation and was significantly related to RT, F (1, 371) = 45.9, p < .001. The correlation coefficient was .33, indicating approximately 11% of the variance in RT could be accounted for by numerosity ratio. In the next steps, in addition to total circumference ratio, density and total surface ratio were entered into the regression equation. In the final regression step, the correlation coefficient was .38, indicating 15% of the variance of the RT could be accounted for by the mentioned continuous ratios (see Table S5). Note that neither numerosity ratio nor duration were entered into the regression equation.

Accuracy as a dependent measure
In step 1 of the analysis, numerosity ratio was entered into the regression equation and was significantly related to accuracy, F (1, 500) = 234.68, p < .001. The correlation coefficient was -.57, indicating approximately 32% of the variance in accuracy could be accounted for by numerosity ratio. In the next steps, in addition to numerosity ratio, density, total circumference and total surface ratios were entered into the regression equation. In the final regression step, the correlation coefficient was .66, indicating 44% of the variance of the accuracy rates could be accounted for by the mentioned magnitude ratios (see Table S6).

RT -across durations
Step and variable B ± SE β R 2 Change in R 2 Step

Summary of Multiple Stepwise Regression Analysis for Variables Predicting RT in the Accuracy Emphasis Condition by Duration
Step and variable B ± SE β R 2 Change in R 2

RT -for duration of 50 ms
Step 1

Summary of Multiple Stepwise Regression Analysis for Variables Predicting Accuracy in the Accuracy Emphasis Condition
Accuracy -across durations Step and variable B ± SE β R 2 Change in R 2 Step

RT -across durations
Step and variable B ± SE β R 2 Change in R 2 Step

Summary of Multiple Stepwise Regression Analysis for Variables Predicting Accuracy in the Speed Emphasis Condition
Accuracy -across durations Step and variable B ± SE β R 2 Change in R 2 Step