Exploring the randomness of mentally generated head–tail sequences

It is well known that people deviate from randomness as they attempt to mentally generate head–tail sequences as randomly as possible. This deviation from randomness is quantified by an excess of repetitions or alternations between successive responses more than would be expected by chance. We conducted an experiment in which a sample of students was asked to mentally simulate a sequence as if it is produced by a fair coin. We propose several models based on Markov chains for analysing the dynamic of head–tail outcomes in these sequences. First, we explore observed Markov chains and suggest some practical solutions to reduce the number of parameters. However, there is a need for more sophisticated models, and in this case, we propose latent Markov models and mixture of Markov chains to analyse these head–tail sequences. A generalization of the so-called mixture transition distribution (MTD) model is also considered.

[1]  C. Morris.  Parametric Empirical Bayes Inference: Theory and Applications , 1983 .

[2]  Yuriy S. Kharin,et al.  Markov Chain of Conditional Order: Properties and Statistical Analysis , 2014 .

[3]  Zhiwei Zhang,et al.  A TWO-STATE MIXED HIDDEN MARKOV MODEL FOR RISKY TEENAGE DRIVING BEHAVIOR. , 2015, The annals of applied statistics.

[4]  P. Brugger,et al.  Random number generation in dementia of the Alzheimer type: A test of frontal executive functions , 1996, Neuropsychologia.

[5]  Francesco Bartolucci,et al.  Longitudinal analysis of self‐reported health status by mixture latent auto‐regressive models , 2014 .

[6]  A. Raftery.  A model for high-order Markov chains , 1985 .

[7]  Lola L. Lopes,et al.  Distinguishing between random and nonrandom events. , 1987 .

[8]  R. Katz.  On Some Criteria for Estimating the Order of a Markov Chain , 1981 .

[9]  Francesco Bartolucci,et al.  A Class of Latent Markov Models for Capture–Recapture Data Allowing for Time, Heterogeneity, and Behavior Effects , 2007, Biometrics.

[10]  T. Saaty,et al.  Why the magic number seven plus or minus two , 2003 .

[11]  W. Wagenaar,et al.  The perception of randomness , 1991 .

[12]  A Baddeley,et al.  Random Generation and the Executive Control of Working Memory , 1998, The Quarterly journal of experimental psychology. A, Human experimental psychology.

[13]  Donald Hedeker,et al.  A Random-Effects Probit Model for Predicting Medical Malpractice Claims , 1994 .

[14]  A. Raftery,et al.  The Mixture Transition Distribution Model for High-Order Markov Chains and Non-Gaussian Time Series , 2002 .

[15]  J. Danion,et al.  Inhibition of inappropriate responses is preserved in the think-no-think and impaired in the random number generation tasks in schizophrenia , 2007, Journal of the International Neuropsychological Society.

[16]  M. Puterman,et al.  Maximum-penalized-likelihood estimation for independent and Markov-dependent mixture models. , 1992, Biometrics.

[17]  M. Tanner,et al.  Mixtures of marginal models , 2000 .

[18]  P S Albert,et al.  A two-state Markov mixture model for a time series of epileptic seizure counts. , 1991, Biometrics.

[19]  A. Rapoport,et al.  Randomization in individual choice behavior. , 1997 .

[20]  David V. Budescu,et al.  A Markov model for generation of random binary sequences. , 1987 .

[21]  R. Altman.  Mixed Hidden Markov Models , 2007 .

[22]  George B. Macready,et al.  Concomitant-Variable Latent-Class Models , 1988 .

[23]  A. Farcomeni,et al.  A note on the mixture transition distribution and hidden Markov models , 2010 .

[24]  Francesco Bartolucci,et al.  Latent Markov model for longitudinal binary data: An application to the performance evaluation of nursing homes , 2009, 0908.2300.

[25]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[26]  J. Vermunt,et al.  Discrete-Time Discrete-State Latent Markov Models with Time-Constant and Time-Varying Covariates , 1999 .

[27]  T. Petrie.  Probabilistic functions of finite-state markov chains. , 1967, Proceedings of the National Academy of Sciences of the United States of America.

[28]  B. Kedem,et al.  Regression Theory for Categorical Time Series , 2003 .

[29]  Zhiwei Zhang,et al.  Ordinal latent variable models and their application in the study of newly licensed teenage drivers. , 2013, Journal of the Royal Statistical Society. Series C, Applied statistics.

[30]  Jeroen K. Vermunt,et al.  Longitudinal Research Using Mixture Models , 2010 .

[31]  Mauro Pesenti,et al.  Age-Related Differences in Random Generation , 1998, Brain and Cognition.

[32]  Yuriy S. Kharin,et al.  A Markov chain of order s with r partial connections and statistical inference on its parameters , 2007 .

[33]  Clifford Konold,et al.  Making Sense of Randomness " Implicit Encoding as a Basis for Judgment , 1997 .

[34]  Francesco Bartolucci,et al.  Latent Markov models: a review of a general framework for the analysis of longitudinal data with covariates , 2014 .

[35]  R. Nickerson,et al.  The production and perception of randomness. , 2002, Psychological review.

[36]  D. Budescu.  Analysis of dichotomous variables in the presence of serial dependence. , 1985 .

[37]  F. V. D. Pol,et al.  MIXED MARKOV LATENT CLASS MODELS , 1990 .

[38]  S. Zeger,et al.  Markov regression models for time series: a quasi-likelihood approach. , 1988, Biometrics.

[39]  J. Towse.  On random generation and the central executive of working memory. , 1998, British journal of psychology.

[40]  N. Breslow,et al.  Approximate inference in generalized linear mixed models , 1993 .

[41]  Estimation for mixtures of Markov processes , 2002 .

[42]  Geert Molenberghs,et al.  Strategies to fit pattern-mixture models. , 2002, Biostatistics.

[43]  S. Rabe-Hesketh,et al.  Prediction in multilevel generalized linear models , 2009 .

[44]  Pedro Puig,et al.  Parsimonious higher order Markov models for rating transitions , 2018 .

[45]  D. Bates,et al.  Approximations to the Log-Likelihood Function in the Nonlinear Mixed-Effects Model , 1995 .

[46]  A. Neuringer.  Can people behave "randomly?": The role of feedback. , 1986 .

[47]  G. Goldenberg,et al.  Components of Random Generation by Normal Subjects and Patients with Dysexecutive Syndrome , 1993, Brain and Cognition.