A test of the optimal classifier's independence assumption in perceptual categorization

Observers completed perceptual categorization tasks that included separate base-rate/payoff manipulations, corresponding simultaneous base-rate/payoff manipulations, and conflicting simultaneous base-rate/payoff manipulations. Performance (1) was closer to optimal for 2:1 than for 3:1 base-rate/payoff ratios and when base rates as opposed to payoffs were manipulated, and (2) was more in line with the predictions from the flat-maxima hypothesis than from the independence assumption of the optimal classifier in corresponding and conflicting simultaneous base-rate/payoff conditions. A hybrid model that instantiated simultaneously the flat-maxima and the competition between reward and accuracy maximization (COBRA) hypotheses was applied to the data. The hybrid model was superior to a model that incorporated the independence assumption, suggesting that violations of the independence assumption are to be expected and are well captured by the flat-maxima hypothesis without requiring any additional assumptions. The parameters indicated that observers' reward-maximizing decision criterion rapidly approaches the optimal value and that more weight is placed on accuracy maximization in separate and corresponding simultaneous base-rate/payoff conditions than in conflicting simultaneous base-rate/payoff conditions.     

A test of the optimal classifier’s independence assumption in perceptual categorization

Observers completed perceptual categorization tasks that included separate base-rate/payoff manipulations, corresponding simultaneous base-rate/payoff manipulations, and conflicting simultaneous base-rate/payoff manipulations. Performance (1) was closer to optimal for 2:1 than for 3:1 baserate/ payoff ratios and when base rates as opposed to payoffs were manipulated, and (2) was more in line with the predictions from the flat-maxima hypothesis than from the independence assumption of the optimal classifier in corresponding and conflicting simultaneous base-rate/payoff conditions. A hybrid model that instantiated simultaneously the flat-maxima and the competition between reward and accuracy maximization (COBRA) hypotheses was applied to the data. The hybrid model was superior to a model that incorporated the independence assumption, suggesting that violations of the independence assumption are to be expected and are well captured by the flat-maxima hypothesis without requiring any additional assumptions. The parameters indicated that observers’ reward-maximizing decision criterion rapidly approaches the optimal value and that more weight is placed on accuracy maximization in separate and corresponding simultaneous base-rate/payoff conditions than in conflicting simultaneous base-rate/payoff conditions.     

Probability matching, accuracy maximization, and a test of the optimal classifier’s independence assumption in perceptual categorization

Observers completed perceptual categorization tasks that included 25 base-rate/payoff conditions constructed from the factorial combination of five base-rate ratios (1:3, 1:2, 1:1, 2:1, and 3:1) with five payoff ratios (1:3, 1:2, 1:1, 2:1, and 3:1). This large database allowed an initial comparison of the competition between reward and accuracy maximization (COBRA) hypothesis with a competition between reward maximization and probability matching (COBRM) hypothesis, and an extensive and critical comparison of the flat-maxima hypothesis with the independence assumption of the optimal classifier. Model-based instantiations of the COBRA and COBRM hypotheses provided good accounts of the data, but there was a consistent advantage for the COBRM instantiation early in learning and for the COBRA instantiation later in learning. This pattern held in the present study and in a reanalysis of Bohil and Maddox (2003). Strong support was obtained for the flat-maxima hypothesis over the independence assumption, especially as the observers gained experience with the task. Model parameters indicated that observers’ reward-maximizing decision criterion rapidly approaches the optimal value and that more weight is placed on accuracy maximization in separate base-rate/payoff conditions than in simultaneous base-rate/payoff conditions. The superiority of the flat-maxima hypothesis suggests that violations of the independence assumption are to be expected, and are well captured by the flat-maxima hypothesis, with no need for any additional assumptions.     

Probability matching, accuracy maximization, and a test of the optimal classifier's independence assumption in perceptual categorization

Observers completed perceptual categorization tasks that included 25 base-rate/payoff conditions constructed from the factorial combination of five base-rate ratios (1:3, 1:2, 1:1, 2:1, and 3:1) with five payoff ratios (1:3, 1:2, 1:1, 2:1, and 3:1). This large database allowed an initial comparison of the competition between reward and accuracy maximization (COBRA) hypothesis with a competition between reward maximization and probability matching (COBRM) hypothesis, and an extensive and critical comparison of the flat-maxima hypothesis with the independence assumption of the optimal classifier. Model-based instantiations of the COBRA and COBRM hypotheses provided good accounts of the data, but there was a consistent advantage for the COBRM instantiation early in learning and for the COBRA instantiation later in learning. This pattern held in the present study and in a reanalysis of Bohil and Maddox (2003). Strong support was obtained for the flat-maxima hypothesis over the independence assumption, especially as the observers gained experience with the task. Model parameters indicated that observers' reward-maximizing decision criterion rapidly approaches the optimal value and that more weight is placed on accuracy maximization in separate base-rate/payoff conditions than in simultaneous base-rate/payoff conditions. The superiority of the flat-maxima hypothesis suggests that violations of the independence assumption are to be expected, and are well captured by the flat-maxima hypothesis, with no need for any additional assumptions.     

Linear transformations of the payoff matrix and decision criterion learning in perceptual categorization

The effects of payoff-matrix multiplication, payoff-matrix addition, the presence of long-run gains versus long-run losses, category discriminability, and base rate on decision criterion learning were examined in 2 perceptual categorization experiments. Observers were found to be sensitive to the effects of payoff-matrix multiplication (and category discriminability) on the steepness of the objective reward function in line with predictions from the flat-maxima hypothesis and contrary to the predictions from the payoff-variance hypothesis. Decision criterion learning was best in base-rate conditions, was worst when losses were associated with incorrect responding, and was intermediate when no losses were associated with incorrect responding. This performance profile was well captured by the competition between reward and accuracy (COBRA) hypothesis. A hybrid model framework that instantiates both the flat-maxima and COBRA hypotheses was necessary to account for the data from both experiments.     

Category discriminability, base-rate, and payoff effects in perceptual categorization

The optimality of perceptual categorization performance under manipulations of category discriminability (i.e., d' level), base rates, and payoffs was examined. Base-rate and payoff manipulations across two category discriminabilities allowed a test of the hypothesis that the steepness of the objective reward function affects performance (i.e., the flat-maxima hypothesis), as well as the hypothesis that observers combine base-rate and payoff information independently. Performance was (1) closer to optimal for the steeper objective reward function, in line with the flat-maxima hypothesis, (2) closer to optimal in base-rate conditions than in payoff conditions, and (3) in partial support of the hypothesis that base-rate and payoff knowledge is combined independently. Implications for current theories of base-rate and payoff learning are discussed.     

Toward a unified theory of decision criterion learning in perceptual categorization

Optimal decision criterion placement maximizes expected reward and requires sensitivity to the category base rates (prior probabilities) and payoffs (costs and benefits of incorrect and correct responding). When base rates are unequal, human decision criterion is nearly optimal, but when payoffs are unequal, suboptimal decision criterion placement is observed, even when the optimal decision criterion is identical in both cases. A series of studies are reviewed that examine the generality of this finding, and a unified theory of decision criterion learning is described (Maddox & Dodd, 2001). The theory assumes that two critical mechanisms operate in decision criterion learning. One mechanism involves competition between reward and accuracy maximization: The observer attempts to maximize reward, as instructed, but also places some importance on accuracy maximization. The second mechanism involves a flat-maxima hypothesis that assumes that the observer's estimate of the reward-maximizing decision criterion is determined from the steepness of the objective reward function that relates expected reward to decision criterion placement. Experiments used to develop and test the theory require each observer to complete a large number of trials and to participate in all conditions of the experiment. This provides maximal control over the reinforcement history of the observer and allows a focus on individual behavioral profiles. The theory is applied to decision criterion learning problems that examine category discriminability, payoff matrix multiplication and addition effects, the optimal classifier's independence assumption, and different types of trial-by-trial feedback. In every case the theory provides a good account of the data, and, most important, provides useful insights into the psychological processes involved in decision criterion learning.     

On the logic of testing the independence assumption in the process-dissociation procedure

Theprocess-dissociation procedure is designed to provide quantitative estimates of the influence of explicit and implicit memory in a variety of tasks. The procedure relies on the assumption that these two forms of memory produce independent influences on performance. Prior investigators have attempted to test this assumption by determining whether the parameter representing the influence of implicit memory (denotedA) is constant across experimental conditions. I argue that the constancy of A cannot provide an appropriate test of the independence assumption, because (1) the prediction of constancy can be generated without the assumption of independence, obviating the need to posit independence; and (2) the constancy ofA does not necessarily imply independence, even if one assumes that a dependency hypothesis, supplemented by ancillary assumptions (Curran & Hintzman, 1995), predicts differences inA. I close by emphasizing that we can test the independence assumption by using standard procedures that compare the fit of a model that assumes independence with the fit of a model that assumes dependence.     

Encoding variability theory and the spacing effect in associate learning

Encoding variability theory accounts for the spacing effect by assuming that, as the lag between repetitions increases, the memorial representations approach independence. A method for testing this assumption in determining repetition effects in associative learning is suggested. In the independence hypothesis, we simply assume that each presentation of a word pair is represented independently, so that the expected proportion recalled is simple, P=P1+P2 - PIP2. Two studies are reported using a continuous paired-associate task representing a factorial combination of lag and retention interval as well as single-presentation control conditions. The results indicate that for moderately long retention intervals, as the lag between repetitions increases, the observed proportion recalled is first less, then eventually exceeds, and then returns to the level expected by an independence hypothesis.     

Decomposable Models: A New Look at Interdependence and Dependence Structures in Psychological Research

Decomposable models represent interdependence structures for observable variables. Each model is fully characterized by a set of conditional independence restrictions, and can be visualized with an undirected as well as a special type of a directed graph. As a consequence each decomposable model can be interpreted either in terms of interdependencies only or as a particular kind of dependence structure, as a recursive system or path analysis model. Under the assumption of normally distributed variables, decomposable models determine the structure of correlation matrices, and maximum-likelihood estimates of these can be calculated with the help of ordinary least squares estimation. Using several examples from psychological research, we discuss the interpretation of decomposable models. Furthermore, it is demonstrated how recursive dependence structures can be specified with the help of decomposable models in a hypothesis generating (exploratory) as well as in a hypothesis testing (confirmatory) manner.     

A multiplicative model of the utility of survival duration and health quality

Survival duration and health quality are fundamentally important aspects of health. A utility model for survival duration and health quality is a model of the subjective value of these attributes. We investigate the hypothesis that the utility (subjective value) of survival duration and health quality is determined by a multiplicative model. According to this model, there are separate subjective scales for the utility of survival duration and health quality. If F(Y) equals the utility of surviving Y years, and G(Q) equals the utility of living in health state Q, then the multiplicative model proposes that F(Y)G(Q) equals the utility of surviving Y years in health state Q. This model provides a simple explanation for several intuitively compelling relationships. First, the distinction between better-than-death and worse-than-death health states corresponds to the assignment of positive or negative utilities to different health states. Second, a zero duration of survival removes any reason to prefer one health state over any other, just as multiplying the utility of health quality by zero eliminates differences between the utilities of different health states. Third, the subjective difference between Y years in pain and Y years free from pain increases as Y increases as if the difference in utility between pain and no pain were being multiplied by the utility of surviving Y years. A critical prediction of the multiplicative model is the hypothesis that preferences between gambles for health outcomes satisfy a property called utility independence. Individual analyses revealed that most subjects satisfy utility independence, thereby supporting the multiplicative utility model. Some subjects appear to violate a fundamental assumption of utility theory: They appear to violate the assumption that a single utility scale represents both the ordinal preference relations between certain outcomes and the subjective averaging that underlies the utility of gambles. The violation is inferred from an inconsistency between preferences for multiattribute outcomes when they are viewed as certain outcomes and when they are viewed as the outcomes of gambles.     

Feature inference with uncertain categorization: Re-assessing Anderson’s rational model

A key function of categories is to help predictions about unobserved features of objects. At the same time, humans are often in situations where the categories of the objects they perceive are uncertain. In an influential paper, Anderson (Psychological Review, 98(3), 409–429, 1991) proposed a rational model for feature inferences with uncertain categorization. A crucial feature of this model is the conditional independence assumption—it assumes that the within category feature correlation is zero. In prior research, this model has been found to provide a poor fit to participants’ inferences. This evidence is restricted to task environments inconsistent with the conditional independence assumption. Currently available evidence thus provides little information about how this model would fit participants’ inferences in a setting with conditional independence. In four experiments based on a novel paradigm and one experiment based on an existing paradigm, we assess the performance of Anderson’s model under conditional independence. We find that this model predicts participants’ inferences better than competing models. One model assumes that inferences are based on just the most likely category. The second model is insensitive to categories but sensitive to overall feature correlation. The performance of Anderson’s model is evidence that inferences were influenced not only by the more likely category but also by the other candidate category. Our findings suggest that a version of Anderson’s model which relaxes the conditional independence assumption will likely perform well in environments characterized by within-category feature correlation.     

Independent Sampling vs Interitem Dependencies in Whole Report Processing: Contributions of Processing Architecture and Variable Attention

All current models of visual whole report processing assume perceptual independence among the displayed items in which the perceptual processing of individual items is not affected by other items in the display. However, models proposed by Townsend (1981, Acta Psychologica 47, 149-173), Shibuya and Bundesen (1988, Journal of Experimental Psychology: Human Perception and Performance 14, 591-600), and Bundesen (1990, Psychological Review 97, 523-547) contain postperceptual buffers that must predict negative dependencies. The perceptual-independence assumption forms what we term the modal model class. A recent example of a model that assumes perceptual independence is the Independent Sampling Model of Loftus, Busey, and Senders (1993, Perception and Psychophysics 54, 535-554). The fundamental independence assumption has only been directly tested once before, where tests revealed no dependencies except those produced by guessing. The present study tests the independence assumption using several different statistics and, contrary to most extant models of whole report, finds significant positive dependence. Poisson models do predict a positive dependence and we develop a succinctly parameterized version, the Weighted Path Poisson Model, which allows the finishing order to be a weighted probabilistic mechanism. However, it does not predict the data quite as well as a new model, the Variable Attention Model, which allows independence within trials (unlike the Poisson models). This model assumes that attention (or, potentially, other aspects such as signal quality) varies widely across trials, thus predicting an overall positive dependence. Intuitions for and against the competing models are discussed. In addition, we show, through mimicking formulae, that models which contain the proper qualitative type of dependence structure can be cast in either serial or parallel form. Copyright 2001 Academic Press.     

Naïve and Robust: Class‐Conditional Independence in Human Classification Learning

Humans excel in categorization. Yet from a computational standpoint, learning a novel probabilistic classification task involves severe computational challenges. The present paper investigates one way to address these challenges: assuming class‐conditional independence of features. This feature independence assumption simplifies the inference problem, allows for informed inferences about novel feature combinations, and performs robustly across different statistical environments. We designed a new Bayesian classification learning model (the dependence‐independence structure and category learning model, DISC‐LM) that incorporates varying degrees of prior belief in class‐conditional independence, learns whether or not independence holds, and adapts its behavior accordingly. Theoretical results from two simulation studies demonstrate that classification behavior can appear to start simple, yet adapt effectively to unexpected task structures. Two experiments—designed using optimal experimental design principles—were conducted with human learners. Classification decisions of the majority of participants were best accounted for by a version of the model with very high initial prior belief in class‐conditional independence, before adapting to the true environmental structure. Class‐conditional independence may be a strong and useful default assumption in category learning tasks.     

Defense mechanisms and coping strategies in conjugal relationships: An integration

Coping strategies and defence mechanisms are used to describe people's responses to stressful situations. The concept of coping strategies comes from the social psychological tradition, whereas the concept of defence mechanisms comes from the psychoanalytic tradition. According to the traditional view, the two concepts are very different. However, recently, a growing number of researchers suggested that coping strategies and defence mechanisms are more interrelated than what was assumed previously. This study was aimed at documenting the relationship between both concepts in the context of predicting adjustment in a specific situation, marital relationships, one of the most well‐suited situations to investigate people's responses to stressful situations. The sample consisted of 157 couples. Each partner completed the Ways of Coping Questionnaire, the Defense Style Questionnaire, and the Dyadic Adjustment Scale. Two theoretical models, the independence model and the effectiveness model, accounting for the relationship between coping strategies and defence mechanisms were first tested, using confirmatory factor analyses. According to the independence hypothesis, one latent dimension should measure defence mechanisms and the other latent dimension should measure coping strategies. According to the effectiveness hypothesis, coping strategies and defence mechanisms could vary along two dimensions: adaptive and maladaptive ways of dealing with marital difficulties. Results revealed that the effectiveness hypothesis was a better representation of the relationship between the two concepts than the independence hypothesis. The unique contribution of defence mechanisms and coping strategies to the prediction of marital adjustment was next examined using multiple regression analyses. Again, results were generally congruent with the effectiveness hypothesis. Positive and negative relationships with marital adjustment were observed for both concepts. In addition, coping strategies did not result in more positive outcomes on marital adjustment than defence mechanisms. It seems that coping strategies and defence mechanisms have both their specificity and their shared characteristics.     

Statistical Tests of Conditional Independence Between Responses and/or Response Times on Test Items

Three plausible assumptions of conditional independence in a hierarchical model for responses and response times on test items are identified. For each of the assumptions, a Lagrange multiplier test of the null hypothesis of conditional independence against a parametric alternative is derived. The tests have closed-form statistics that are easy to calculate from the standard estimates of the person parameters in the model. In addition, simple closed-form estimators of the parameters under the alternatives of conditional dependence are presented, which can be used to explore model modification. The tests were applied to a data set from a large-scale computerized exam and showed excellent power to detect even minor violations of conditional independence.     

A dual-task analysis of detection accuracy for the case of high target-distractor similarity: Further evidence for independent processing

Previous studies of visual detection performance indicate that perceptual span has a negligible effect on detection accuracy; hence, previous results suggest that individual characters within a multielement array are perceived independently. In this study, additional experimental factors were examined in order to determine whether conditions exist in which perceptual span influences detection performance: targets and distractors were chosen to be maximally similar, and detection accuracy was studied in relation to display size, target redundancy, and homogeneity of distractor characters. To test for perceptual span effects, a three-state model was proposed which incorporates the assumption that individual characters are perceived independently. The model may be applied to (a) both yes-no and forced-choice tasks, and (b) the case when effects of display characteristics differ for response alternatives. Predictions based on the three-state model agree satisfactorily with observed proportions. Moreover, interactions between display characteristics and response alternatives accord well with model predictions, and accuracy differences between subjects are consistent with individual differences in model parameters. The results provide further evidence for the independence assumption and illustrate additional conditions for which perceptual span has a negligible effect on detection accuracy.     

Independence of reinforcement delay and magnitude in concurrent chains

A three-component concurrent-chains procedure was used to investigate preference between terminal-link schedules that differed in delay and magnitude of reinforcement. Response and time allocation data were well described by a generalized matching model. Sensitivity to delay appeared to be lower when reinforcement magnitudes were unequal than when they were equal, but when obtained rather than programmed time spent responding in the initial links was used in the model, the difference vanished. The results support independence of delay and magnitude as separate dimensions of reinforcement value, as required by the matching law, and the assumption of the contextual choice model (Grace, 1994) that sensitivities to delay and magnitude are affected similarly by temporal context. Although there was statistical evidence for interaction between successive components, the effects were small and transient. The multiple-component concurrent-chains procedure should prove useful in future research on multidimensional preference, although it may be necessary to control obtained initial-link time more precisely.