Bayesian estimates of autocorrelations in single-case designs

Researchers in the single-case design tradition have debated the size and importance of the observed autocorrelations in those designs. All of the past estimates of the autocorrelation in that literature have taken the observed autocorrelation estimates as the data to be used in the debate. However, estimates of the autocorrelation are subject to great sampling error when the design has a small number of time points, as is typically the situation in single-case designs. Thus, a given observed autocorrelation may greatly over- or underestimate the corresponding population parameter. This article presents Bayesian estimates of the autocorrelation that greatly reduce the role of sampling error, as compared to past estimators. Simpler empirical Bayes estimates are presented first, in order to illustrate the fundamental notions of autocorrelation sampling error and shrinkage, followed by fully Bayesian estimates, and the difference between the two is explained. Scripts to do the analyses are available as supplemental materials. The analyses are illustrated using two examples from the single-case design literature. Bayesian estimation warrants wider use, not only in debates about the size of autocorrelations, but also in statistical methods that require an independent estimate of the autocorrelation to analyze the data.     

Memory for spatial location in two-dimensional arrays

Memory for spatial location in a two-dimensional display was studied in two experiments by means of a recognition task. The first experiment investigated the effects of presenting multiple reference points at the same time as sequentially shown to-be-remembered (TBR) locations. In the second experiment, the TBR locations were presented either one by one or four at a time with and without multiple reference points. The reference points tended to decrease the false alarm rate during acquisition, rather than to increase the hit rate, which suggests that they affected the precision rather than the amount of systematic bias in memory for the locations. Similar effects were obtained when the TBR locations were presented four at a time. The reference points were also found to serve as effective retrieval cues in a final recognition test. Reference points may thus facilitate both acquisition and retrieval of information about spatial location.     

Role of vision and kinesthesis in location and distance estimates

Three experiments were conducted to investigate the visual/kinesthetic coding characteristics of two movement informations, i.e., location and distance. The first experiments used matching task procedures, the second experiment used visual augmentation and reduction procedures and the third experiment made use of an ischemic nerve block procedure. The findings' of the first experiment indicated a clear kinesthetic coding for distance information. Results suggest a kinesthetic coding for location cue, with the presence of vision (alone) on both phases of movements improving its accuracy. The second experiment ruled out a visual coding of distance cue and indicated a kinesthetic representation of location cue. Again, simultaneous presence of vision (only at the time of criterion movement) enhanced its performance. The third experiment confirmed a kinesthetic signalling of distance cue. Location cue, in conditions of the third experiment did not get influenced. Probably, muscle receptors in relation with central commands can handle location information. A multiple coding for location information is suggested.     

Induction of combination rules in two-dimensional function learning

Previous studies have typically found that when people learn to combine two dimensions of a stimulus to select a response, they learn additive combination rules more easily than nonadditive (e.g., multiplicative) ones. The present experiments demonstrate that in some situations people can learn multiplicative rules more easily than other (e.g., additive) rules. Subjects learned to produce specified response durations when presented with stimulus lines varying in length and angle of orientation. When stimuliand correct responses were related by a multiplicative combination of power functions, learning was relatively easy (Experiment 1). In contrast, systematic response biases occurred during the early phases of learning an additive combination of linear functions (Experiment 2) and a more complex (nonadditive and nonmultiplicative) combination of linear functions (Experiment3), suggesting that people have a tendency to induce a multiplicative combination of power functions. However, the initial biases decreased with practice. These results are explained in terms of a revised adaptive regression model of function learning originally proposed by Koh and Meyer (1991). Differences between the present results and previous results in the literature are discussed.     

Representational pseudoneglect and reference points both influence geographic location estimates

Our mental representation of the world is far from objective. For example, western Canadians estimate the locations of North American cities to be too far to the west. This bias could be due to a reference point effect, in which people estimate more space between places close to them than far from them, or to representational pseudoneglect, in which neurologically intact individuals favor the left side of space when asked to image a scene. We tested whether either or both of these biases influence the geographic world representations of neurologically intact young adults from Edmonton and Ottawa, which are in western and eastern Canada, respectively. Individuals were asked to locate North American cities on a two-dimensional grid. Both groups revealed effects of representational pseudoneglect in this novel paradigm, but they also each exhibited reference point effects. These results inform theories in both cognitive psychology and neuroscience.     

Global-scale location and distance estimates: common representations and strategies in absolute and relative judgments

The authors examined whether absolute and relative judgments about global-scale locations and distances were generated from common representations. At the end of a 10-week class on the regional geography of the United States, participants estimated the latitudes of 16 North American cities and all possible pairwise distances between them. Although participants were relative experts, their latitude estimates revealed the presence of psychologically based regions with large gaps between them and a tendency to stretch North America southward toward the equator. The distance estimates revealed the same properties in the representation recovered via multidimensional scaling. Though the aggregated within- and between-regions distance estimates were fitted by Stevens's law (S. S. Stevens, 1957), this was an averaging artifact: The appropriateness of a power function to describe distance estimates depended on the regional membership of the cities. The authors conclude that plausible reasoning strategies, combined with regionalized representations and beliefs about the location of these relative to global landmarks, underlie global-scale latitude and distance judgments.     

Bivariate estimates of giftedness

The bivariate normal probability distribution is presented as a method for estimating the prevalence of giftedness. The bivariate model requires the specification of two cutoff values for two selection variables and information regarding the size of the correlation between variables. The use and interpretation of bivariate estimates of the prevalence of giftedness is discussed.     

Biserial estimates of correlation

Brogden's coefficient of selective efficiency and Clemans' are more efficient than biserialr when the correlation is high. Some empirical sampling studies are reported comparing the standard errors of these statistics.     

Bayesian chance

This paper explores how the Bayesian program benefits from allowing for objective chance as well as subjective degree of belief. It applies David Lewis??s Principal Principle and David Christensen??s principle of informed preference to defend Howard Raiffa??s appeal to preferences between reference lotteries and scaling lotteries to represent degrees of belief. It goes on to outline the role of objective lotteries in an application of rationality axioms equivalent to the existence of a utility assignment to represent preferences in Savage??s famous omelet example of a rational choice problem. An example motivating causal decision theory illustrates the need for representing subjunctive dependencies to do justice to intuitive examples where epistemic and causal independence come apart. We argue to extend Lewis??s account of chance as a guide to epistemic probability to include De Finetti??s convergence results. We explore Diachronic Dutch book arguments as illustrating commitments for treating transitions as learning experiences. Finally, we explore implications for Martingale convergence results for motivating commitment to objective chances.     

Bayesian probability

Bayesian decision theory is here construed as explicating a particular concept of rational choice and Bayesian probability is taken to be the concept of probability used in that theory. Bayesian probability is usually identified with the agent’s degrees of belief but that interpretation makes Bayesian decision theory a poor explication of the relevant concept of rational choice. A satisfactory conception of Bayesian decision theory is obtained by taking Bayesian probability to be an explicatum for inductive probability given the agent’s evidence.     

Statistical properties of staircase estimates

The bias and variability of staircase estimators were studied by means of repeated computer simulations of staircase runs. Both forced-choice and yes-no staircases were simulated. The influence of the shape of the psychometric function, the location and spacing of the stimuli, the number of trials in a run, and the method of deriving the estimate from the data are discussed. The forced-choice staircase is compared to the yes-no staircase, and the limitations of the simulation procedure are outlined.     

Bayesian rules of updating

This paper discusses the Bayesian updating rules of ordinary and Jeffrey conditionalisation. Their justification has been a topic of interest for the last quarter century, and several strategies proposed. None has been accepted as conclusive, and it is argued here that this is for a good reason; for by extending the domain of the probability function to include propositions describing the agent's present and future degrees of belief one can systematically generate a class of counterexamples to the rules. Dynamic Dutch Book and other arguments for them are examined critically. A concluding discussion attempts to put these results in perspective within the Bayesian approach.This paper was completed with assistance from a Mind Association Fellowship. I would like to thank the Mind Association, and also Paul Castell for his patient advice on previous drafts.     

Overconfidence in interval estimates

Judges were asked to make numerical estimates (e.g., "In what year was the first flight of a hot air balloon?"). Judges provided high and low estimates such that they were X% sure that the correct answer lay between them. They exhibited substantial overconfidence: The correct answer fell inside their intervals much less than X% of the time. This contrasts with choices between 2 possible answers to a question, which showed much less overconfidence. The authors show that overconfidence in interval estimates can result from variability in setting interval widths. However, the main cause is that subjective intervals are systematically too narrow given the accuracy of one's information-sometimes only 40% as large as necessary to be well calibrated. The degree of overconfidence varies greatly depending on how intervals are elicited. There are also substantial differences among domains and between male and female judges. The authors discuss the possible psychological mechanisms underlying this pattern of findings.     

The guttman-harris uniqueness estimates

It is shown that, under very general conditions, uniqueness estimates proposed independently by Guttman [1957] and by Harris [1963] provide tighter upper bounds on the unknown uniqueness values of factor analysis than do existing estimates.The Illinois Department of Mental Health Drug Abuse Program is co-sponsored by the State of Illinois and the Department of Psychiatry, Division of the Biological Sciences and Pritzker School of Medicine, University of Chicago. Send reprint requests to Drug Abuse Programs—Research, East Pavilion, Museum of Science and Industry, 57th Street and South Lake Shore Drive, Chicago, Illinois 60637.     

We are All Bayesian,Everyone is Not a Bayesian

Medical research makes intensive use of statistics in order to support its claims. In this paper we make explicit an epistemological tension between the conduct of clinical trials and their interpretation: statistical evidence is sometimes discarded on the basis of an (often) underlined Bayesian reasoning. We suggest that acknowledging the potentiality of Bayesian statistics might contribute to clarify and improve comprehension of medical research. Nevertheless, despite Bayesianism may provide a better account for scientific inference with respect to the standard frequentist approach, Bayesian statistics is rarely adopted in clinical research. The main reason lies in the supposed subjective elements characterizing this perspective. Hence, we discuss this objection presenting the so-called Reference analysis, a formal method which has been developed in the context of objective Bayesian statistics in order to define priors which have a minimal or null impact on posterior probabilities. Furthermore, according to this method only available data are relevant sources of information, so that it resists the most common criticisms against Bayesianism.

     

Bayesian contour integration

The process by which the human visual system parses an image into contours, surfaces, and objects—perceptual grouping—has proven difficult to capture in a rigorous and general theory. A natural candidate for such a theory is Bayesian probability theory, which provides optimal interpretations of data under conditions of uncertainty. But the fit of Bayesian theory to human grouping judgments has never been tested, in part because methods for expressing grouping hypotheses probabilistically have not been available. This paper presents such methods for the case ofcontour integration—that is, the aggregation of a sequence of visual items into a “virtual curve.” Two experiments are reported in which human subjects were asked to group ambiguous configurations of dots (in Experiment 1, a sequence of five dots could be judged to contain a “corner” or not; in Experiment 2, an arrangement of six dots could be judged to fall into two disjoint contours or one smooth contour). The Bayesian theory accounts extremely well for subjects’ judgments, explaining more than 75% of the variance in both tasks. The theory thus provides a far more quantitatively precise account of human contour integration than has been previously possible, allowing a very precise calculation of the subjective goodness of a virtual chain of dots. Because Bayesian theory is inferentially optimal, this finding suggests a “rational justification,” and hence possibly an evolutionary rationale, for some of the rules of perceptual grouping.     

Weak Bayesian coherentism

Recent results in probability theory have cast doubt on coherentism, purportedly showing (a) that coherence among a set of beliefs cannot raise their probability unless individual beliefs have some independent credibility, and (b) that no possible measure of coherence makes coherence generally probability-enhancing. I argue that coherentists can reject assumptions on which these theorems depend, and I derive a general condition under which the concurrence of two information sources lacking individual credibility can raise the probability of what they report.