Richard B. Anderson

The power law as an emergent property

Recent work has shown that the power function, a ubiquitous characteristic of learning, memory, and sensation, can emerge from the arithmetic averaging of exponential curves. In the present study, the forgetting process was simulated via computer to determine whether power curves can result from the averaging of other types of component curves. Each of several simulations contained 100 memory traces that were made to decay at different rates. The resulting component curves were then arithmetically averaged to produce an aggregate curve for each simulation. The simulations varied with respect to the forms of the component curves: exponential, range-limited linear, range-limited logarithmic, or power. The goodness of the aggregate curve’s fit to a power function relative to other functions increased as the amount of intercomponent slope variability increased, irrespective of componentcurve type. Thus, the power law’s ubiquity may reflect the pervasiveness of slope variability across component functions. Moreover, power-curve emergence may constitute a methodological artifact, an explanatory construct, or both, depending on the locus of the effect. John Wixted, and Steven Sloman, for their helpful comments on the manuscript.

Need probability affects retention: A direct demonstration

Recent memory theory has emphasized the concept ofneed probability—that is, the probability that a given piece of learned information will be tested at some point in the future. It has been proposed that, in real-world situations, need probability declines over time and that the memory-loss rate is calibrated to match the progressive reduction in need probability (J. R. Anderson & Schooler, 1991). The present experiments were designed to examine the influence of the slope of the need-probability curve on the slope of the retention curve. On each of several trials, subjects memorized a list of digits, then retained the digits in memory for 1, 2, 4, 8, or 16 sec. Some trials ended with a recall test; other trials ended with the message, “no test.” In Experiment 1, the likelihood of encountering a memory test (i.e., the need probability) was made to either increase or decrease as the retention interval increased; in Experiment 2, need probability either was flat (invariant across retention intervals) or decreased as the retention interval increased. The results indicated that the shape of the need-probability curve influenced the slope of the retention curve (Experiment 1) and that the effect became larger as the experimental session progressed (Experiment 2). The findings support the notion that memory adapts to need probabilities and that the rate of forgetting is influenced by the slope of the need-probability curve. In addition, all of the forgetting curves approximated a power function, suggesting that need probability influences the slope but not the form of forgetting.

The perception of scatterplots.

Four experiments investigated the perception of correlations from scatterplots. All graphic properties, other than error variance, that have been shown to affect subjective but not objective correlation(r) were held constant. Participants in Experiment 1 ranked 21 scatterplots according to the magnitude ofr. In Experiments 2 and 3, participants made yes/no judgments to indicate whether a scatterplot was high (signal) or low (noise). Values ofr for signal and noise scatterplots varied across participants. Differences between correlations for signal and for noise scatterplots were constant inr in Experiment 2, and constant inr2 in Experiment 3. Standard deviations of the ranks in Experiment 1 and ď values in Experiments 2 and 3 showed that discriminability increased with the magnitude ofr. In Experiment 4, faculty and graduate students in psychology and sociology made point estimates ofr for single scatterplots. Estimates were negatively accelerated functions of objective correlation.

Sample size and the detection of correlation--a signal detection account: comment on Kareev (2000) and Juslin and Olsson (2005).

Simulations examined the hypothesis that small samples can provide better grounds for inferring the existence of a population correlation, p, than can large samples. Samples of 5, 7, 10, 15, or 30 data pairs were drawn either from a population with p=0 or from one with p>0. When decision accuracy was assessed independently for each level of the decision criterion, there was a criterion-specific small-sample advantage. For liberal criteria, accuracy was greater for large than for small samples, but for conservative criteria, the opposite result occurred. There was no small-sample advantage when accuracy was measured as the area under a receiver operating characteristic curve or as the posterior probability of a hit. The results show that small-sample advantages can occur, but under limited conditions.

Sample Size and Correlational Inference

In 4 studies, the authors examined the hypothesis that the structure of the informational environment makes small samples more informative than large ones for drawing inferences about population correlations. The specific purpose of the studies was to test predictions arising from the signal detection simulations of R. B. Anderson, M. E. Doherty, N. D. Berg, and J. C. Friedrich (2005). The results of a simulation study in the present article confirmed and extended previous theoretical claims (R. B. Anderson et al., 2005) that in a yes/no correlation detection task, small-sample advantages should occur but should be restricted to particular decision conditions. In 3 behavioral studies, participants viewed larger or smaller samples of data pairs and judged whether each sample had been drawn from a population characterized by a zero correlation or from one characterized by a greater-than-zero correlation. Consistent with traditional statistical theory, accuracy tended to be greater for larger than for smaller samples, though there was a small-sample advantage in 1 experimental condition. The results are discussed in relation to alternative theoretical and behavioral paradigms such as those of Y. Kareev (e.g., 2005) and K. Fiedler and Y. Kareev (2006).

Variation in scatterplot displays.

Scatterplots are typically constructed for the purpose of showing the association between two variables. We argue that such scatterplots should not vary in ways that are unrelated to the degree of association. Noting that the publication standards for the preparation of scatterplots are minimal, we review empirical literature that shows that a number of arbitrary features of scatterplots influence the inferences drawn by users. We discuss nine such features. We also review the literature that recommends how scatterplots should be prepared, and we examine 221 scatterplots published in recent journals. There was considerable variation among those published scatterplots. This article concludes with a call for further standardization by way of flexible guidelines.

Sample Size and the Detection of Means: A Signal Detection Account

Using statistical theory as a basis, Kareev (e.g., 1995) claimed that people’s ability to correctly infer the existence of a population correlation should be greater for small than for large samples. Simulations by R. B. Anderson, Doherty, Berg, and Friedrich (2005) identified conditions favoring small samples but could not determine whether such an advantage was due to sampling skew, variance, or central tendency displacement. In the present study, we investigated theoretical effects of sample size(n) on the detection of population means under circumstances in which sampling variance is unconfounded with skew or central tendency displacement. The results demonstrate an extremely limited, criterion-specific, small-sample advantage that was attributable ton-related sampling variance and that occurred only with highly conservative, suboptimal criterion placement.

Exploring unexplained variation

In a 1974 commencement address, Richard Feynman described scientific integrity as a kind of utter honesty, a kind of leaning over backwards to tell the whole truth. We argue that investigators could tell more of the truth and increase the value of their papers by highlighting and discussing unexplained variation, a major source of which is individual differences. An argument that unexplained individual differences must have many sources is presented, and means of representing that variation are illustrated. We believe that such a change in reporting of research results is likely to advance the progress of scientific psychology, but perhaps the most compelling argument for what we propose is simply that telling the whole story as fully as possible is good scientific practice. The Appendix provides two examples of what we are urging, taken from recent psychological literature.

Belief bias in the perception of sample size adequacy

In two experiments participants were instructed to set aside their own, complete knowledge of a statistical population parameter and to take the perspective of an agent whose knowledge was limited to a random sample. Participants rated the appropriateness of the agents conclusion about the adequacy of the sample size (which, objectively, was more than adequate). They also rated the agents intelligence. Whereas previous work suggests that unbelievable statistical conclusions impact reasoning by provoking critical thought which enhances the detection of research flaws, the present studies presented participants an unflawed scenario designed to assess effects of believability on bias. The results included the finding that participants’ complete knowledge did indeed bias their perceptions not only of the adequacy of the sample size, but also of the rationality of the agent drawing the conclusion from the sample. The findings were interpreted in the context of research on belief bias, social attribution, and Theory of Mind.

Inhibitory consequences of memory selection

When subjects select a prime from a visual display while leaving a distractor prime unselected, response time (RT) or response accuracy to a subsequent probe may be impeded if the distractor prime and probe are identical, or if they are related to one another. This phenomenon, negative priming (NP), has obvious implications for understanding perceptual selection. However, it is not known whether NP results from other kinds of selection processes. The present studies were designed to investigate whether NP occurs when primes are selected from working memory rather than from a visual display. In the two experiments, the subjects memorized two primes, selected one prime for further processing, and classified the contents of a probe display. Significant NP occurred in both Experiments. In Experiment 2, however, NP occurred only under easy-selection conditions; the effect was reversed under difficult-selection conditions. The findings indicate a role for NP in memory processing, but contrast with the results from perceptual selection studies showing greater NP under difficult-selection than under easy-selection conditions. The present finding suggests a complex and perhaps strategy-dependent relationship between memory selection difficulty and NP.