Adding possibilities can reduce the Gambler's Fallacy: A naïve-probability paradox

This paper reports a novel paradox of intuitive probabilistic reasoning detected in naïve reasoners’ responses in two separate experiments where we manipulated the number of sets (or possibilities) of the problem keeping constant the probability of the critical set. Experiment 1 showed that the incidence of the Gambler's Fallacy (GF) was reduced when the number of sets was increased. In Experiment 2, a reduction of the GF also occurred but, more importantly, the percentage of correct responses of the participants increased when three sets of possibilities instead of two were used. Therefore, both Experiments 1 and 2 demonstrated that an increase in the extensional complexity of a problem can, under certain circumstances, lead to facilitation. These results support the importance of the extensional features in solving chance problems and are consistent with the model theory of reasoning.     

On muthén’s maximum likelihood for two-level covariance structure models

Data in social and behavioral sciences are often hierarchically organized. Special statistical procedures that take into account the dependence of such observations have been developed. Among procedures for 2-level covariance structure analysis, Muthén’s maximum likelihood (MUML) has the advantage of easier computation and faster convergence. When data are balanced, MUML is equivalent to the maximum likelihood procedure. Simulation results in the literature endorse the MUML procedure also for unbalanced data. This paper studies the analytical properties of the MUML procedure in general. The results indicate that the MUML procedure leads to correct model inference asymptotically when level-2 sample size goes to infinity and the coefficient of variation of the level-1 sample sizes goes to zero. The study clearly identifies the impact of level-1 and level-2 sample sizes on the standard errors and test statistic of the MUML procedure. Analytical results explain previous simulation results and will guide the design or data collection for the future applications of MUML.This research was supported by NSF Grant DMS04-37167.We thank Dr.Bengt Muthén for providing key references. We are also grateful to three expert reviewers for their constructive comments that have led the paper to an improvement over the previous version.This revised article was published online in August 2005 with the PDF paginated correctly.     

Factor analysis for non-normal variables

Factor analysis for nonnormally distributed variables is discussed in this paper. The main difference between our approach and more traditional approaches is that not only second order cross-products (like covariances) are utilized, but also higher order cross-products. It turns out that under some conditions the parameters (factor loadings) can be uniquely determined. Two estimation procedures will be discussed. One method gives Best Generalized Least Squares (BGLS) estimates, but is computationally very heavy, in particular for large data sets. The other method is a least squares method which is computationally less heavy. In one example the two methods will be compared by using the bootstrap method. In another example real life data are analyzed.This paper has partly been written while the author was a visiting scholar at the Department of Psychology, University of California, Los Angeles. He wants to thank Peter Bentler who made this stay at UCLA possible and for his valuable contributions to this paper. This research was supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O) under number R56-150 and by USPHS Grant DA01070.     

Measuring coherence

This paper aims to contribute to our understanding of the notion of coherence by explicating in probabilistic terms, step by step, what seem to be our most basic intuitions about that notion, to wit, that coherence is a matter of hanging or fitting together, and that coherence is a matter of degree. A qualitative theory of coherence will serve as a stepping stone to formulate a set of quantitative measures of coherence, each of which seems to capture well the aforementioned intuitions. Subsequently it will be argued that one of those measures does better than the others in light of some more specific intuitions about coherence. This measure will be defended against two seemingly obvious objections.     

Explaining the gambler's fallacy: Testing a gestalt explanation versus the “law of small numbers”

The present study tests a gestalt (closure) explanation for the gambler's fallacy which posits that runs in random events will be expected to reverse only when the run is open or ongoing. This is contrasted with the law of small numbers explanation suggesting that people expect random outcomes to balance out generally. Sixty-one university students placed hypothetical guesses and bets on a series of coin tosses. Either heads or tails were dominant (8 versus 4). In a closed run condition the run ended prior to the critical trial (e.g., HHHT), and in an open run condition the run remained open (e.g., THHH). As hypothesised, participants showed the gambler's fallacy in the open run condition, but not in the closed run condition. This difference is not due to differential memory for the outcomes. Men, and people with more previous experience gambling, were also found to be more prone to the gambler's fallacy. It is argued that the gestalt explanation best explains the results.