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Development of Numerical Estimation: Data and Models

Young, Christopher J.
http://rave.ohiolink.edu/etdc/view?acc_num=osu1316454592

Abstract Details

2011, Doctor of Philosophy, Ohio State University, Psychology.
Numerical estimation—a process of mapping a non-numerical quantity to an approximate numerical one (and vice-versa)—improves greatly with age and experience. What this change entails and how it occurs has engendered a lively and provocative debate in the literature on numerical cognition. At the center of this debate lies a range of potential cognitive processes that are theoretically capable of generating observed changes in estimation performance. Choice of the most likely model of these cognitive processes is a difficult problem: models vary greatly in complexity, leading to a trade-off between their ability to fit any one data set and their ability to predict new data. In this dissertation, I argue that conventional attempts to evaluate models have not been adequately sensitive to this trade-off. As a result, conventional model selection techniques run the risk of favoring complex models that fail to predict the likelihood of future responses on the same laboratory task, on similar laboratory tasks, and on real-world analogues. To address this issue, I (1) applied conventional model selection procedures to analyze the performance of 740 subjects in 9 experiments using one numerical estimation task (number line estimation), (2) contrasted conventional model fits with a hierarchical Bayesian analysis of the same data set in order to identify the models most likely to have generated the data, and (3) evaluated the model parameters and fits of competing models to predict performance in other numerical estimation tasks (i.e., measurement estimation, numerosity estimation, and number categorization) as well as real-world numerical performance (i.e., math achievement scores and memory for numbers). Results indicated that conventional model fits differed based on whether the models were fit to individual children’s estimates or to fictional estimates obtained by averaging over them. Analyses of individual and averaged data supported the logarithmic, linear and segmental models as contenders but not power models. Cross-validation and Bayesian model selection techniques tended to support conventional model selection procedures, though simulations indicated the more complex segmental linear models data generating functions were often over-fit by other functions, and also fit other data generating functions well. Consistent with these simulations of over-fitting, I found that the most likely parameters of segmental linear models failed to predict parameter values on other numerical estimation tasks, whereas the parameters of the logarithmic and linear models generalized better to other numerical estimation tasks as well as being the most predictive of real-world performance.
John Opfer (Advisor)
Mark Pitt (Committee Member)
Vladimir Sloutsky (Committee Member)
124 p.
Cognitive Psychology
Numerical Estimation; Model Selection; Bayesian Analysis

Recommended Citations

Citations

  • Young, C. J. (2011). Development of Numerical Estimation: Data and Models [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1316454592

    APA Style (7th edition)

  • Young, Christopher. Development of Numerical Estimation: Data and Models. 2011. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1316454592.

    MLA Style (8th edition)

  • Young, Christopher. "Development of Numerical Estimation: Data and Models." Doctoral dissertation, Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1316454592

    Chicago Manual of Style (17th edition)

osu1316454592
603
© 2011, some rights reserved.Creative Commons License Development of Numerical Estimation: Data and Models by Christopher J. Young is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by The Ohio State University and OhioLINK.