Parameter Dependencies in an Accumulation-to-Threshold Model of Simple Perceptual Decisions

234 Pages·2015·7.36 MB·English

Parameter Dependencies in an Accumulation-to-Threshold
Model of Simple Perceptual Decisions
A Thesis
Presented in Partial Fulfillment of the Requirements for the Degree
Master of Arts in the Graduate School of The Ohio State University
By
Vyacheslav Y. Nikitin, B.S.
Graduate Program in Psychology
The Ohio State University
2015
Master’s Examination Committee:
Patricia Van Zandt, Advisor
Michael Edwards
Jay Myung
Paul De Boeck
(cid:13)c Copyright by
Vyacheslav Y. Nikitin
2015
Abstract
It is a common assumption in sequential sampling models of simple perceptual
decisions that parameters are statistically independent across trials. This thesis ad-
dresses theoretical and empirical implications of assuming statistically dependent
parameters. Three questions are answered: how to formulate flexible multivariate
distributions of parameters of sequential sampling models, what are the predictive
consequences of parameter dependencies for mean sample paths and joint distribu-
tion of responses and response times, and what correlation matrix is consistent with
a benchmark dataset collected from a brightness discrimination task without explicit
correlation manipulations.
The key to studying dependent parameters is a flexible framework of copulas
that allow arbitrary combinations of dependence structures with marginal distribu-
tions. Adding correlations to a widely-used diffusion model shows that initial points
and absorption times of mean sample paths can be strongly affected by correlations.
Whereas the impact of correlation on the joint distribution of behavior is potentially
strong adjustment of asymmetry in reaction time distributions of the two responses.
Finally, in an experiment without explicit manipulation of correlations, the posterior
distribution is consistent with small to moderate correlations between parameters.
ii
Thus, under typical experimental conditions, the usual assumption of statistical in-
dependenceisanadequatesimplificationofhowparametersofsimpledecisionmaking
vary across trials.
iii
This is dedicated to the crucible of science
iv
Acknowledgments
I want to thank my adviser Patricia Van Zandt for advice, proofreading and
invaluable computing resources that made this thesis possible. I am also grateful to
my committee members - Jay Myung, Michael Edwards and Paul De Boeck - for
provoking questions and insightful comments during the development of this thesis.
v
Vita
2010 ........................................B.S. Psychology
2011-2012 .................................. University Fellow,
The Ohio State University
2012-present ................................Graduate Teaching Associate,
The Ohio State University
Fields of Study
Major Field: Psychology
vi
Table of Contents
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Formal Modeling Approach . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Data-Generating Experimental Tasks . . . . . . . . . . . . . . . . . 18
1.3 Benchmark Behavioral Data . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Cognitive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Bayesian Statistical Framework . . . . . . . . . . . . . . . . . . . . 27
1.5.1 Bayesian Models . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.2 MCMC Background . . . . . . . . . . . . . . . . . . . . . . 30
1.5.3 Metropolis-Hastings Algorithm . . . . . . . . . . . . . . . . 34
2. Motivating Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1 Modeling Parameter Dependencies . . . . . . . . . . . . . . . . . . 38
2.2 Predictions of Dependent Parameters . . . . . . . . . . . . . . . . . 41
2.3 Correlation Structure in a Benchmark Dataset . . . . . . . . . . . . 43
vii
3. Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Ratcliff Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Copula-based Multivariate Distributions . . . . . . . . . . . . . . . 59
3.3 Generalized Decision Models . . . . . . . . . . . . . . . . . . . . . 68
4. Theoretical and Empirical Studies . . . . . . . . . . . . . . . . . . . . . . 74
4.1 Design and Parameter Settings . . . . . . . . . . . . . . . . . . . . 75
4.2 Study A1 - Mean Sample Paths . . . . . . . . . . . . . . . . . . . . 79
4.2.1 Mean Sample Paths Calculation . . . . . . . . . . . . . . . . 82
4.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3 Study A2 - Response times and Responses . . . . . . . . . . . . . . 112
4.3.1 Joint Distribution Calculation . . . . . . . . . . . . . . . . . 113
4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.4 Study B - Benchmark Dataset Analysis . . . . . . . . . . . . . . . 132
4.4.1 Outlier filtering . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.4.2 Bayesian Models . . . . . . . . . . . . . . . . . . . . . . . . 135
4.4.3 MCMC Sampler . . . . . . . . . . . . . . . . . . . . . . . . 139
4.4.4 Convergence Diagnostics . . . . . . . . . . . . . . . . . . . . 143
4.4.5 Characterizing Parameter Dependencies . . . . . . . . . . . 145
4.4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Appendix A. Study A1 - Mean Sample Paths . . . . . . . . . . . . . . . . . 174
Appendix B. Study A2 - Response Times and Responses . . . . . . . . . . . 191
Appendix C. Study B - Benchmark Dataset Analysis . . . . . . . . . . . . . 200
viii
List of Tables
Table Page
4.1 WeibullfunctionparametersusedtoobtainpredictionsinstudyA.The
values were reported by Vandekerckhove, Tuerlinckx, and Lee (2011). 76
4.2 Non-correlation parameters used to obtain predictions in study A. The
values were reported by Vandekerckhove et al. (2011). Starting point
values were transformed to decision bias values. . . . . . . . . . . . . 77
4.3 Correlation parameters used to obtain predictions in study A. The val-
ues were picked to range from low to high and represent three possible
correlation patterns. For sample paths only column one matters, espe-
cially rows 4 - 6. Predicting response and response times depends on
all the values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Summary of the posterior distribution of the non-correlation parame-
ters of the normal copula model for subject “kr”. Each parameter is
summarized by a mean and lower/upper boundaries of its HDI. Note:
ACC is accuracy condition and SPD is speed condition. . . . . . . . . 154
4.5 Summary of the posterior distribution of the correlation parameters
of the normal copula model for all three subjects. Each parameter is
summarized by a mean and lower/upper boundaries of its HDI. Note:
ACC is accuracy condition and SPD is speed condition. . . . . . . . . 155
ix