Yaakov Kareev

Seven (indeed, plus or minus two) and the detection of correlations.

Capacity limitations of working memory force people to rely on samples consisting of 7 +/- 2 items. The implications of these limitations for the early detection of correlations between binary variables were explored in a theoretical analysis of the sampling distribution of phi, the contingency coefficient. The analysis indicated that, for strong correlations (phi > .50), sample sizes of 7 +/- 2 are most likely to produce a sample correlation that is more extreme than that of the population. Another analysis then revealed that there is a similar cutoff point at which useful correlations (i.e., for which each variable is a valid predictor of the other) first outnumber correlations for which this is not the case. Capacity limitations are thus shown to maximize the chances for the early detection of strong and useful relations.

Through a narrow window: working memory capacity and the detection of covariation.

The samples of data that people use in their attempts to detect relationships in the environment are limited in size by working memory capacity. The implications of that constraint are explored by analyzing the sampling distribution of the most common measure of relationship--the product moment correlation (Pearsons rxy). This distribution is skewed when the population correlation differs from zero (i.e., when a correlation exists), and the more so, the smaller the sample. As both the median and the mode of the sampling distribution are more extreme than the population value, it follows that samples likely to be encountered indicate a correlation stronger than that in the population. Thus, the limited capacity of working memory may serve as an amplifier that helps people to avoid missing strong relationships. As the distribution is more skewed the smaller the sample size, the effect suggests an explanation for the fact that young children detect meaningful covariation fairly rapidly.

Not that bad after all : generation of random sequences

People simulating a random generator fail in the rate of each event, which is too close to the theoretical rate, and in overalternation between events. It is suggested that both failures stem from attempts to produce within short-term memory (STM) limitations a typical sequence in the standard task, 398 Ss of 3 age groups performed 3 coin-tossing tasks: standard, guessing, and guessing with feedback. The proportion of events was more variable and alternation rate was higher in the guessing than in the standard task. High alternation rates are shown to be byproducts of typical sequences. An estimate of the window size within which people operate highly correlated with age (corresponding to changes in STM capacity), further supporting the typicality assumption. Peoples grasp of randomness is therefore concluded to be better than hitherto believed

On the Misperception of Variability

It has long been claimed that people perceive the world as less variable and more regular than it actually is. Such misperception, if shown to exist, could explain some perplexing behaviors. However, evidence supporting the claim is indirect, and there is no explanation of its cause. As a possible cause, the authors suggest that people use sample variability as an estimate of population variability. This is so because the sampling distribution of sample variance is downward attenuated, the attenuation being substantial for sample sizes that people consider. Results of 5 experiments show that people use sample variability, uncorrected for sample size, in tasks in which a correction is normatively called for, and indeed perceive variability as smaller than it is.

Does decision quality (always) increase with the size of information samples? Some vicissitudes in applying the law of large numbers.

Adaptive decision making requires that contingencies between decision options and their relative assets be assessed accurately and quickly. The present research addresses the challenging notion that contingencies may be more visible from small than from large samples of observations. An algorithmic account for such a seemingly paradoxical effect is offered within a satisficing-choice framework. Accordingly, a choice is only made when the sample contingency describing the relative evaluation of the 2 options exceeds a critical threshold. Small samples, because of the high dispersion of their sampling distribution, facilitate above-threshold contingencies. Across a broad range of parameters, the resulting small-sample advantage in terms of hits is stronger than their disadvantage in false alarms. Computer simulations and experiments support the model predictions. The relative advantage of small samples is most apparent when information loss is low, when the threshold is high relative to the ecological contingency, and when the sampling process is self-truncated.

Do the Weak Stand a Chance? Distribution of Resources in a Competitive Environment

When two agents of unequal strength compete, the stronger one is expected to always win the competition. This expectation is based on the assumption that evaluation of performance is complete, hence flawless. If, however, the agents are evaluated on the basis of only a small sample of their performance, the weaker agent still stands a chance of winning occasionally. A theoretical analysis indicates that, to increase the chance of this happening the weaker agent ought to give up on enough occasions so that he or she can match the stronger agent on the remaining ones. We model such a competition in a game, present its game-theoretic solution, and report an experiment, involving 144 individuals, in which we tested whether players (both weak and strong) are actually sensitive to their relative strengths and know how to allocate their resources accordingly. Our results indicate that they do.

Teaching by Examples: Implications for the Process of Category Acquisition:

A new paradigm, the “teaching-by-examples” paradigm, was used to shed new light on the process of category acquisition. In four experiments (n = 90, 90, 115, 117), manipulating the variables of category structure, status of non-target category, learning mode, and teaching mode, participants first learned a category and then taught it to someone else. High agreement between participants on the teaching sequences was found across conditions, and a typical sequence was identified for each category structure. The typical participant-produced sequences started with several ideal positive cases, followed by an ideal negative case and then borderline cases. The efficiency of such sequences for teaching was tested in another experiment (n = 60), in which they were compared with sequences emphasizing category borders and sequences emphasizing each dimension separately. The typical participant-produced sequences induced the most efficient learning. It is proposed that the pattern of performance may provide a rich source of data for testing and fine-tuning models of category acquisition.

Improving performance and increasing the use of non-positive testing in a rule-discovery task

A well-documented characteristic of rule discovery behaviour is subjects’ infrequent use of negative testing. Previous attempts at increasing the use of negative testing have met with little success. In an evaluation task, we found that subjects appreciate the benefits of negative testing and disconfirmation (Kareev & Halberstadt, this issue). Further, when given the choice, subjects prefer to begin their inquiry by employing a reception mode of inquiry, and only later switch to a generative strategy (Halberstadt & Kareev, 1992). In the present study we had subjects solve two rule-discovery problems. For the training problem, 180 subjects were assigned either to the traditional generation mode, in which subjects had to generate number triplets, or to a reception mode, in which subjects were presented with number triplets by the experimenter. For the subsequent test problem both groups used the traditional generation mode. Results revealed that subjects trained by the reception mode were more likely to use non-positive tests and more likely to solve the second problem. Apparently, training under the less demanding reception mode enabled subjects to realize the potential relevance of nonpositive testing.

Choosing Between Adaptive Agents: Some Unexpected Implications of Level of Scrutiny

Even with ample time and opportunity to use extensive data, people often make do with small samples, which increases their risk of making the wrong decision. A theoretical analysis indicates, however, that when the decision involves continually selecting among competing, adaptive agents who are eager to be selected, an error-prone evaluation may be beneficial to the decision maker. In this case, the chance of an error can motivate competitors to exert greater effort, improving their level of performance—which is the prime concern of the decision maker. This theoretical argument was tested empirically by comparing the effects of two levels of scrutiny of performance. Results show that minimal scrutiny can indeed lead to better performance than full scrutiny, and that the effect is conditional on a bridgeable difference between the competitors. We conclude by pointing out that small-sample-based, error-prone decisions may also maintain competition and diversity in the environment.

Nonproportional Sampling and the Amplification of Correlations

A theoretical analysis shows that sample correlations between two binary variables will be inflated when the frequency distributions of the two variables are flatter (i.e., closer to equal frequencies for the two values) in the sample than in the population. A correlation-assessment study in which participants were free to choose their own sample revealed an overwhelming preference for samples that included roughly the same number of observations for the two values of dichotomous variables, irrespective of their actual distribution in the population. Subjective estimates of observed correlations followed the sample correlations—which were inflated, as predicted—more closely than the true correlations. Peoples sampling behavior thus resembles that of a research designer who maximizes the chance of detecting a relationship, at the cost of diminished accuracy in estimating its strength.