The effect of constant outcome value on judgments and decision making given linguistic probabilities

Two experiments were performed to determine whether judgments of the relative chances of two independent events occurring are biased by constant outcome values contingent on the events when the uncertainties are specified by linguistic expressions (e.g. doubtful). In Experiment 1, subjects directly judged the relative chances of the two events, of which one was represented by a spinner and the other by a linguistic probability expression. In Experiment 2, only linguistic probability expressions were used to describe the two events and a betting procedure was used. A bias was evident in both studies, such that the relative judgments tended to favour the event with the positive rather than the negative contingent outcome. The bias was smaller for the low- than for the high-probability phrases. Individual differences were great, with the bias appearing strongly in only about one-third of the population. Theoretical implications of the present and related results are discussed.     

Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis

One of the intriguing questions of factor analysis is the extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries. We show in this paper that the set of matrices, which can be reduced to rankr, has positive (Lebesgue) measure if and only ifr is greater or equal to the Ledermann bound. In other words the Ledermann bound is shown to bealmost surely the greatest lower bound to a reduced rank of the sample covariance matrix. Afterwards an asymptotic sampling theory of so-called minimum trace factor analysis (MTFA) is proposed. The theory is based on continuous and differential properties of functions involved in the MTFA. Convex analysis techniques are utilized to obtain conditions for differentiability of these functions.     

On the estimation of parameters in latent structure analysis

In this note, we describe the iterative procedure introduced earlier by Goodman to calculate the maximum likelihood estimates of the parameters in latent structure analysis, and we provide here a simple and direct proof of the fact that the parameter estimates obtained with the iterative procedure cannot lie outside the allowed interval. Formann recently stated that Goodman's algorithm can yield parameter estimates that lie outside the allowed interval, and we prove in the present note that Formann's contention is incorrect.This research was supported in part by Research Contract No. NSF SOC 76-80389 from the Division of the Social Sciences of the National Science Foundation. The author is indebted to C. C. Clogg for helpful comments and for the numerical results reported here (see, e.g., Table 1).     

On Bayesian estimation in unrestricted factor analysis

It is shown that the common and unique variance estimates produced by Martin & McDonald's Bayesian estimation procedure for the unrestricted common factor model have a predictable sum which is always greater than the maximum likelihood estimate of the total variance. This fact is used to justify a suggested simple alternative method of specifying the Bayesian parameters required by the procedure.     

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.     

Sample Size Planning for Detecting Mediation Effects: A Power Analysis Procedure Considering Uncertainty in Effect Size Estimates

Abstract

When planning mediation studies, researchers are often interested in the sample size needed to achieve adequate power for testing mediation. Power depends on population effect sizes, which are unknown in practice. In conventional power analysis, effect size estimates, however, are often used as population values, which could result in underpowered studies. Uncertainty in effect size estimates has been considered in other sample size planning contexts (e.g., t-test, ANOVA), but has not been handled properly for planning mediation studies. In the current study, we proposed an easy-to-use sample size planning method for testing mediation with uncertainty in effect size estimates considered. We conducted simulation studies to demonstrate the impact of uncertainty in effect size estimates on power of testing mediation, and to provide sample size suggestions under different levels of uncertainty. Empirical examples were provided to illustrate the application of our method. R functions and a web application were developed to facilitate implementation.     

Making probability judgments of future product failures: The role of mental unpacking

When consumers mentally unpack (i.e., imagine) the reasons for product failure, their probability judgments of future product failures are higher than when no mental unpacking is undertaken. However, increasing the level of mental unpacking does not lead to monotonically increasing effects on probability judgments but results in inverted U-shaped relationships. Using a two-factor structure, we propose that when consumers undertake mental unpacking, there will be two conflicting processes; while imagining causes for an event will lead to greater perceived probability, the greater difficulty in generating reasons for an event will lead to lower perceived probability.     

Non-bayesian inference: causal structure trumps correlation

The study tests the hypothesis that conditional probability judgments can be influenced by causal links between the target event and the evidence even when the statistical relations among variables are held constant. Three experiments varied the causal structure relating three variables and found that (a) the target event was perceived as more probable when it was linked to evidence by a causal chain than when both variables shared a common cause; (b) predictive chains in which evidence is a cause of the hypothesis gave rise to higher judgments than diagnostic chains in which evidence is an effect of the hypothesis; and (c) direct chains gave rise to higher judgments than indirect chains. A Bayesian learning model was applied to our data but failed to explain them. An explanation-based hypothesis stating that statistical information will affect judgments only to the extent that it changes beliefs about causal structure is consistent with the results.     

The where and when of likely and unlikely events

Do people think likelihood is distributed evenly or do they have an intuition about the types of contexts in which likely and unlikely events tend to occur? Drawing on a probability-as-distance framework, the current research suggests that people relate probability to other distance dimensions, expecting unlikely events to more often happen in distant contexts and likely events in near contexts. Evidence for this association emerges using within-subject designs where participants directly assign low and high likelihood events to near and distant contexts (Studies 1 and 2), as well as between-subjects designs that focus on a variety of related judgments, including willingness to bet on favorites and long-shots (Study 3), decisions about insurance purchases (Study 4), and expectations regarding games of chance (Study 5). Results appear consistent across outcomes of differing valence (Study 5). Implications and future research directions are discussed.