Signal detection theory with finite mixture distributions: theoretical developments with applications to recognition memory

An extension of signal detection theory (SDT) that incorporates mixtures of the underlying distributions is presented. The mixtures can be motivated by the idea that a presentation of a signal shifts the location of an underlying distribution only if the observer is attending to the signal; otherwise, the distribution is not shifted or is only partially shifted. Thus, trials with a signal presentation consist of a mixture of 2 (or more) latent classes of trials. Mixture SDT provides a general theoretical framework that offers a new perspective on a number of findings. For example, mixture SDT offers an alternative to the unequal variance signal detection model; it can also account for nonlinear normal receiver operating characteristic curves, as found in recent research.     

Signal detection theory with item effects

Applications of signal detection theory (SDT) often involve presentations of different items on each trial, such as slides in a medical imaging study or words in a memory study. If factors particular to the items themselves, apart from being a signal or noise, affect observers’ responses, then ‘item effects’ are present. One way to model these effects is to use a latent continuous variable as an item ‘factor’, such as item ‘difficulty’. Details of SDT models with item effects are clarified via derivations of their implied conditional means, variances, and covariances. Intra-item correlations are defined and suggested as measures of the magnitude of item effects. The SDT-item models are simple random coefficient models and can be fit with standard software. More general models, such as item models with mixing and/or with random observer effects, are also considered.     

Poisson signal-detection theory: Link between threshold models and the Gaussian assumption

The Gaussian model of signal detection cannot fit asymmetric data as long as the variances of the distributions are kept equal. It is therefore common practice to assume unequal variances in order to fit these data. But this assumption leads to the well-known crossover problem. The present paper provides new arguments for the abandonment of the Gaussian model with unequal variances. In its stead, this paper reevaluates multiple-parallel-threshold models. In particular, the Poisson model turns out to be very useful: it can handle data with any degree of asymmetry, giving a reasonable interpretation of the two parameters of the receiver-operating characteristic. The three-state-threshold model (Krantz, 1969) is given a new interpretation in light of the Poisson model. The slope of Poisson double-probability plots turns out to be much closer to unity than is predicted by the Gaussian approximation.     

Poisson signal-detection theory: link between threshold models and the Gaussian assumption.

The Gaussian model of signal detection cannot fit asymmetric data as long as the variances of the distributions are kept equal. It is therefore common practice to assume unequal variances in order to fit these data. But this assumption leads to the well-known crossover problem. The present paper provides new arguments for the abandonment of the Gaussian model with unequal variances. In its stead, this paper reevaluates multiple-parallel-threshold models. In particular, the Poisson model turns out to be very useful: it can handle data with any degree of asymmetry, giving a reasonable interpretation of the two parameters of the receiver-operating characteristic. The three-state-threshold model (Krantz, 1969) is given a new interpretation in light of the Poisson model. The slope of Poisson double-probability plots turns out to be much closer to unity than is predicted by the Gaussian approximation.     

Thresholds for detection and awareness of masked facial stimuli

It has been suggested that perception without awareness can be demonstrated by a dissociation between performance in objective (forced-choice) and subjective (yes–no) tasks, and such dissociations have been reported both for simple stimuli and more complex ones including faces. However, signal detection theory (SDT) indicates that the subjective measures used to assess awareness in such studies can be affected by response bias, which could account for the observed dissociation, and this was confirmed by Balsdon and Azzopardi (2015) using simple visual targets. However, this finding may not apply to all types of stimulus, as the detectability of complex targets such as faces is known to be affected by their configuration as well as by their stimulus energy. We tested this with a comparison of forced-choice and yes–no detection of facial stimuli depicting happy or angry or fearful expressions using a backward masking paradigm, and using SDT methods including correcting for unequal variances in the underlying signal distributions, to measure sensitivity independently of response criterion in 12 normal observers. In 47 out 48 comparisons there was no significant difference between sensitivity (d_a) in the two tasks: hence, across the range of expressions tested it appears that the configuration of complex stimuli does not enhance detectability independently of awareness. The results imply that, on the basis of psychophysical experiments in normal observers, there is no reason to postulate that performance and awareness are mediated by separate processes.     

The multiple dual-pair method

A model for the multiple dual-pair method, a generalization of the traditional dual-pair (4IAX) paradigm, is given. This model is expressed in terms of normal and beta distributions. This generalization allows for the simultaneous estimation of the perceptual distances among three or more stimuli. This model has applications in cases in which multiple two-sample comparisons would be too time consuming and labor intensive. The theory discussed shows how unequal variances can be estimated on the basis of results from that method. Two numerical examples that illustrate the ability of the beta distribution-based model to retrieve the appropriate parameters are given. It is also shown how the traditional dual-pair model is a special case of the multiple dual-pair model.     

The multiple dual-pair method

A model for the multiple dual-pair method, a generalization of the traditional dual-pair (4IAX) paradigm, is given. This model is expressed in terms of normal and beta distributions. This generalization allows for the simultaneous estimation of the perceptual distances among three or more stimuli. This model has applications in cases in which multiple two-sample comparisons would be too time consuming and labor intensive. The theory discussed shows how unequal variances can be estimated on the basis of results from that method. Two numerical examples that illustrate the ability of the beta distribution-based model to retrieve the appropriate parameters are given. It is also shown how the traditional dual-pair model is a special case of the multiple dual-pair model.     

A relation between a between-item multidimensional IRT model and the mixture rasch model

Two generalizations of the Rasch model are compared: the between-item multidimensional model (Adams, Wilson, and Wang, 1997), and the mixture Rasch model (Mislevy & Verhelst, 1990; Rost, 1990). It is shown that the between-item multidimensional model is formally equivalent with a continuous mixture of Rasch models for which, within each class of the mixture, the item parameters are equal to the item parameters of the multidimensional model up to a shift parameter that is specific for the dimension an item belongs to in the multidimensional model. In a simulation study, the relation between both types of models also holds when the number of classes of the mixture is as small as two. The relation is illustrated with a study on verbal aggression. Frank Rijmen was supported by the Fund for Scientific Research Flanders (FWO). This research is also funded by the GOA/2000/02 granted from the KU Leuven. We would like to thank Kristof Vansteelandt for providing the data of the study on verbal aggression.     

A note on preliminary tests of equality of variances

Preliminary tests of equality of variances used before a test of location are no longer widely recommended by statisticians, although they persist in some textbooks and software packages. The present study extends the findings of previous studies and provides further reasons for discontinuing the use of preliminary tests. The study found Type I error rates of a two‐stage procedure, consisting of a preliminary Levene test on samples of different sizes with unequal variances, followed by either a Student pooled‐variances t test or a Welch separate‐variances t test. Simulations disclosed that the twostage procedure fails to protect the significance level and usually makes the situation worse. Earlier studies have shown that preliminary tests often adversely affect the size of the test, and also that the Welch test is superior to the t test when variances are unequal. The present simulations reveal that changes in Type I error rates are greater when sample sizes are smaller, when the difference in variances is slight rather than extreme, and when the significance level is more stringent. Furthermore, the validity of the Welch test deteriorates if it is used only on those occasions where a preliminary test indicates it is needed. Optimum protection is assured by using a separate‐variances test unconditionally whenever sample sizes are unequal.     

The Fisher-Pitman permutation test when testing for differences in mean and variance

The Fisher-Pitman permutation test can detect any type of difference between two samples: hence, a significant Fisher-Pitman permutation test does not necessarily provide evidence for a difference in means. It is possible, however, to test separately for differences in means and variances. Here, we present a recently proposed two-stage procedure to decide whether there are differences in means or variances that can be applied when samples may come from nonnormal distributions with possibly unequal variances.     

Source monitoring and multivariate signal detection theory, with a model for selection

Participants in source monitoring studies, in addition to determining whether an item is old or new, also discriminate the source of the item, such as whether the item was presented in a male or female voice. This article shows how to apply multivariate signal detection theory (SDT) to source monitoring. An interesting aspect of one version of the source monitoring procedure, from the perspective of multivariate SDT, is that it involves a type of selection, in that a discrimination response is observed only if the detection decision is that an item is old. If the selection is ignored, then the estimate of the discrimination parameter can be biased; the nature and magnitude of the bias are illustrated. A bivariate signal detection model that recognizes selection is presented and its application is illustrated. The approach to source monitoring via multivariate SDT provides new results that are informative about underlying psychological processes.     

Motivational predictors of learning strategies,participation, exertion,and performance in physical education: A randomized controlled trial

Teachers’ support of student autonomy in physical education (PE) is believed to be important for students’ motivation and outcomes in PE. We tested the hypothesis that an intervention designed to help teachers to be more autonomy supportive in teaching their students to use learning strategies (relative to standard teaching) would increase students’ perceived autonomy support from the teachers, perceived competence, autonomous motivation, use of learning strategies and their exertion, participation, and grades in PE over a school year. We also tested a self-determination theory (SDT) process model. Experimental effects of the intervention yielded significant positive effects on changes in perceived autonomy support, learning strategies defined as absorption and effort regulation, as well as for performance (i.e., grades). In testing the SDT process model with SEM, most of the predicted paths were significantly supported.     

Heterogeneous heterogeneity by default: Testing categorical moderators in mixed-effects meta-analysis

Categorical moderators are often included in mixed-effects meta-analysis to explain heterogeneity in effect sizes. An assumption in tests of categorical moderator effects is that of a constant between-study variance across all levels of the moderator. Although it rarely receives serious thought, there can be statistical ramifications to upholding this assumption. We propose that researchers should instead default to assuming unequal between-study variances when analysing categorical moderators. To achieve this, we suggest using a mixed-effects location-scale model (MELSM) to allow group-specific estimates for the between-study variance. In two extensive simulation studies, we show that in terms of Type I error and statistical power, little is lost by using the MELSM for moderator tests, but there can be serious costs when an equal variance mixed-effects model (MEM) is used. Most notably, in scenarios with balanced sample sizes or equal between-study variance, the Type I error and power rates are nearly identical between the MEM and the MELSM. On the other hand, with imbalanced sample sizes and unequal variances, the Type I error rate under the MEM can be grossly inflated or overly conservative, whereas the MELSM does comparatively well in controlling the Type I error across the majority of cases. A notable exception where the MELSM did not clearly outperform the MEM was in the case of few studies (e.g., 5). With respect to power, the MELSM had similar or higher power than the MEM in conditions where the latter produced non-inflated Type 1 error rates. Together, our results support the idea that assuming unequal between-study variances is preferred as a default strategy when testing categorical moderators.     

Robust and powerful nonorthogonal analyses

Numerous types of analyses for factorial designs having unequal cell frequencies have been discussed in the literature. These analyses test either weighted or unweighted marginal means which, in turn, correspond to different model comparisons. Previous research has indicated, however, that these analyses result in biased (liberal or conservative) tests when cell variances are heterogeneous. We show how to obtain a generally robust and powerful analysis with any of the recommended nonorthogonal solutions by adapting a modification of the Welch-James procedure for comparing means when population variances are heterogeneous.This research was supported by a Social Sciences and Humanities Research Council (SSHRC) grant (# 410-92-0430) to the first author, a Manitoba Health Research Council Scholar Award and a grant from Natural Sciences and Engineering Research Council to the second author, and a SSHRC Doctoral Fellowship (# 752-92-1628) to the third author. The authors would like to express their gratitude to Joanne Keselman and three anonymous reviewers for their many helpful substantive comments on earlier drafts of this paper.     

The mirror effect and mixture signal detection theory

The mirror effect for word frequency refers to the finding that low-frequency words have higher hit rates and lower false alarm rates than high-frequency words. This result is typically interpreted in terms of conventional signal detection theory (SDT), in which case it indicates that the order of the underlying old item distributions mirrors the order of the new item distributions. However, when viewed in terms of a mixture version of SDT, the order of hits and false alarms does not necessarily imply the same order in the underlying distributions because of possible effects of mixing. A reversal in underlying distributions did not appear for fits of mixture SDT models to data from 4 experiments.     

Meaning making,self‐determination theory,and the question of wisdom in personality

Self‐determination theory (SDT) has advanced the most comprehensive model of motives for human flourishing in the field of personality psychology and beyond. In this article, we evaluate SDT relative to the process of meaning making, particularly from a narrative perspective, showing what SDT can and cannot explain about the construction of self‐identity and its relation to human flourishing. On the one hand, SDT explains how subjective assessments of need fulfillment drive the process of self‐determined living. The internal motives that follow such fulfillment serve as important themes in people's life stories that predict several markers of hedonic and eudaimonic well‐being. On the other hand, SDT's focus on subjective fulfillment limits what SDT can explain about how wisdom, which is a canonical good of both eudaimonia and meaning making, helps people make sense of life's more difficult or unfulfilling events. SDT may facilitate a facet of wisdom that is more subjective and experiential but not the critical facet of wisdom defined by objectively more complex structures of interpretation.     

Interval estimates of weighted effect sizes in the one‐way heteroscedastic ANOVA

A framework for comparing normal population means in the presence of heteroscedasticity and outliers is provided. A single number called the weighted effect size summarizes the differences in population means after weighting each according to the difficulty of estimating their respective means, whether the difficulty is due to unknown population variances, unequal sample sizes or the presence of outliers. For an ANOVA weighted for unequal variances, we find interval estimates for the weighted effect size. In addition, the weighted effect size is shown to be a monotone function of a suitably defined weighted coefficient of determination, which means that interval estimates of the former are readily transformed into interval estimates of the latter. Extensive simulations demonstrate the accuracy of the nominal 95% coverage of these intervals for a wide range of parameters.     

Properties of hypothesis testing techniques and (Bayesian) model selection for exploration‐based and theory‐based (order‐restricted) hypotheses

In this paper, the performance of six types of techniques for comparisons of means is examined. These six emerge from the distinction between the method employed (hypothesis testing, model selection using information criteria, or Bayesian model selection) and the set of hypotheses that is investigated (a classical, exploration‐based set of hypotheses containing equality constraints on the means, or a theory‐based limited set of hypotheses with equality and/or order restrictions). A simulation study is conducted to examine the performance of these techniques. We demonstrate that, if one has specific, a priori specified hypotheses, confirmation (i.e., investigating theory‐based hypotheses) has advantages over exploration (i.e., examining all possible equality‐constrained hypotheses). Furthermore, examining reasonable order‐restricted hypotheses has more power to detect the true effect/non‐null hypothesis than evaluating only equality restrictions. Additionally, when investigating more than one theory‐based hypothesis, model selection is preferred over hypothesis testing. Because of the first two results, we further examine the techniques that are able to evaluate order restrictions in a confirmatory fashion by examining their performance when the homogeneity of variance assumption is violated. Results show that the techniques are robust to heterogeneity when the sample sizes are equal. When the sample sizes are unequal, the performance is affected by heterogeneity. The size and direction of the deviations from the baseline, where there is no heterogeneity, depend on the effect size (of the means) and on the trend in the group variances with respect to the ordering of the group sizes. Importantly, the deviations are less pronounced when the group variances and sizes exhibit the same trend (e.g., are both increasing with group number).     

Psychological needs,purpose in life,and problem video game playing among Chinese young adults

The negative impacts of excessive and problematic video game playing on both children and adults are attracting increasing concern. Based on self‐determination theory (SDT; Ryan & Deci, 2000), this study hypothesized that the three basic psychological needs of autonomy, competence, and relatedness are positively associated with purpose in life, which in turn acts as a protective factor against problem video game playing among Chinese young adult players. Through a questionnaire survey with a sample of 165 Chinese adults aged between 18 and 30 years (mean age = 22.7 years), we found that perceived autonomy, competence, relatedness, and purpose in life were all negatively correlated with problem game playing. The demographic and psychological factors explained 38% of the variances of problem game playing. Specifically, gender, perceived relatedness, and purpose in life emerged as the three most salient predictors of problem game playing among the Chinese young adults. The mediating role of purpose in life was evidenced and it was found that purpose in life mediated the influences of the psychological needs proposed by SDT on problem game playing. Moreover, young men were significantly more susceptible to problem game playing than their female counterparts. To conclude, psychological needs and purpose in life influenced Chinese young adults’ vulnerability to problem game playing directly or indirectly. Intervention programs that encourage social involvement and voluntary work, as well as counseling service that helps clients to search for life purpose, are suggested for intervening in problem game playing among Chinese young adults.     

A self-determination process model of physical activity adoption in the context of a randomized controlled trial

Objective: The PA Counseling (PAC) trial tested a primary care intervention using Self-Determination Theory (SDT) as the guiding framework. This paper presents specific results related to SDT constructs in a physical activity (PA) context. Specifically, whether patients who received both brief autonomy supportive PA counseling from their health-care provider and intensive (3 month) autonomy supportive counseling from a PA counselor would report greater levels of autonomy support, autonomous motivation, perceived competence and PA adoption, than patients receiving only brief counseling. In addition, we tested Williams’ SDT process model in the context of PA adoption.Method: Measures of autonomous motivation and perceived competence for PA were measured at baseline and 6 weeks. PA was assessed at baseline and 13 weeks. An autonomy support index was calculated by multiplying minutes of counseling by perceptions of autonomy support. Group differences in autonomy support, autonomous motivation, perceived competence and PA were examined. The SDT process model of PA adoption was tested via path analysis.Results: The results showed higher autonomy support and autonomous motivation at 6 weeks and higher PA levels at 13 weeks for the experimental group. The SDT process model for PA adoption showed that autonomous motivation and perceived competence at 6 weeks significantly predicted 13-week PA for the experimental group.Conclusions: This study provides a rigorous field test of SDT theory in a PA context. It demonstrates the versatility and applicability of the SDT model for health behavior change. SDT-trained PA counselors appear to provide valuable contribution to facilitating patient behavior change, by increasing patient autonomous motivation for PA.