The optimum decision rules in the same-different paradigm

Effect of stimulus type and pitch salience on pitch-sequence processing

Using a same-different discrimination task, it has been shown that discrimination performance for sequences of complex tones varying just detectably in pitch is less dependent on sequence length (1, 2, or 4 elements) when the tones contain resolved harmonics than when they do not [Cousineau, Demany, and Pessnitzer (2009). J. Acoust. Soc. Am. 126, 3179-3187]. This effect had been attributed to the activation of automatic frequency-shift detectors (FSDs) by the shifts in resolved harmonics. The present study provides evidence against this hypothesis by showing that the sequence-processing advantage found for complex tones with resolved harmonics is not found for pure tones or other sounds supposed to activate FSDs (narrow bands of noise and wide-band noises eliciting pitch sensations due to interaural phase shifts). The present results also indicate that for pitch sequences, processing performance is largely unrelated to pitch salience per se: for a fixed level of discriminability between sequence elements, sequences of elements with salient pitches are not necessarily better processed than sequences of elements with less salient pitches. An ideal-observer model for the same-different binary-sequence discrimination task is also developed in the present study. The model allows the computation of d' for this task using numerical methods.

A Computational Model of Perceptual and Mnemonic Deficits in Medial Temporal Lobe Amnesia

Damage to the medial temporal lobe (MTL) has long been known to impair declarative memory, and recent evidence suggests that it also impairs visual perception. A theory termed the representational-hierarchical account explains such impairments by assuming that MTL stores conjunctive representations of items and events, and that individuals with MTL damage must rely upon representations of simple visual features in posterior visual cortex, which are inadequate to support memory and perception under certain circumstances. One recent study of visual discrimination behavior revealed a surprising antiperceptual learning effect in MTL-damaged individuals: With exposure to a set of visual stimuli, discrimination performance worsened rather than improved [Barense, M. D., Groen, I. I. A., Lee, A. C. H., Yeung, L. K., Brady, S. M., Gregori, M., et al. Intact memory for irrelevant information impairs perception in amnesia. Neuron, 75, 157-167, 2012]. We extend the representational-hierarchical account to explain this paradox by assuming that difficult visual discriminations are performed by comparing the relative "representational tunedness"-or familiarity-of the to-be-discriminated items. Exposure to a set of highly similar stimuli entails repeated presentation of simple visual features, eventually rendering all feature representations maximally and, thus, equally familiar; hence, they are inutile for solving the task. Discrimination performance in patients with MTL lesions is therefore impaired by stimulus exposure. Because the unique conjunctions represented in MTL do not occur repeatedly, healthy individuals are shielded from this perceptual interference. We simulate this mechanism with a neural network previously used to explain recognition memory, thereby providing a model that accounts for both mnemonic and perceptual deficits caused by MTL damage with a unified architecture and mechanism.

A general formula for computing maximum proportion correct scores in various psychophysical paradigms with arbitrary probability distributions of stimulus observations

Proportion correct (Pc) is a fundamental measure of task performance in psychophysics. The maximum Pc score that can be achieved by an optimal (maximum-likelihood) observer in a given task is of both theoretical and practical importance, because it sets an upper limit on human performance. Within the framework of signal detection theory, analytical solutions for computing the maximum Pc score have been established for several common experimental paradigms under the assumption of Gaussian additive internal noise. However, as the scope of applications of psychophysical signal detection theory expands, the need is growing for psychophysicists to compute maximum Pc scores for situations involving non-Gaussian (internal or stimulus-induced) noise. In this article, we provide a general formula for computing the maximum Pc in various psychophysical experimental paradigms for arbitrary probability distributions of sensory activity. Moreover, easy-to-use MATLAB code implementing the formula is provided. Practical applications of the formula are illustrated, and its accuracy is evaluated, for two paradigms and two types of probability distributions (uniform and Gaussian). The results demonstrate that Pc scores computed using the formula remain accurate even for continuous probability distributions, as long as the conversion from continuous probability density functions to discrete probability mass functions is supported by a sufficiently high sampling resolution. We hope that the exposition in this article, and the freely available MATLAB code, facilitates calculations of maximum performance for a wider range of experimental situations, as well as explorations of the impact of different assumptions concerning internal-noise distributions on maximum performance in psychophysical experiments.

A transfer of technology from engineering: use of ROC curves from signal detection theory to investigate information processing in the brain during sensory difference testing

This article reviews a beneficial effect of technology transfer from Electrical Engineering to Food Sensory Science. Specifically, it reviews the recent adoption in Food Sensory Science of the receiver operating characteristic (ROC) curve, a tool that is incorporated in the theory of signal detection. Its use allows the information processing that takes place in the brain during sensory difference testing to be studied and understood. The review deals with how Signal Detection Theory, also called Thurstonian modeling, led to the adoption of a more sophisticated way of analyzing the data from sensory difference tests, by introducing the signal-to-noise ratio, d', as a fundamental measure of perceived small sensory differences. Generally, the method of computation of d' is a simple matter for some of the better known difference tests like the triangle, duo-trio and 2-AFC. However, there are occasions when these tests are not appropriate and other tests like the same-different and the A Not-A test are more suitable. Yet, for these, it is necessary to understand how the brain processes information during the test before d' can be computed. It is for this task that the ROC curve has a particular use.

Psychophysical reverse correlation with multiple response alternatives

Psychophysical reverse-correlation methods such as the "classification image" technique provide a unique tool to uncover the internal representations and decision strategies of individual participants in perceptual tasks. Over the past 30 years, these techniques have gained increasing popularity among both visual and auditory psychophysicists. However, thus far, principled applications of the psychophysical reverse-correlation approach have been almost exclusively limited to two-alternative decision (detection or discrimination) tasks. Whether and how reverse-correlation methods can be applied to uncover perceptual templates and decision strategies in situations involving more than just two response alternatives remain largely unclear. Here, the authors consider the problem of estimating perceptual templates and decision strategies in stimulus identification tasks with multiple response alternatives. They describe a modified correlational approach, which can be used to solve this problem. The approach is evaluated under a variety of simulated conditions, including different ratios of internal-to-external noise, different degrees of correlations between the sensory observations, and various statistical distributions of stimulus perturbations. The results indicate that the proposed approach is reasonably robust, suggesting that it could be used in future empirical studies.

On the choice of adequate randomization ranges for limiting the use of unwanted cues in same-different, dual-pair, and oddity tasks

A major concern when designing a psychophysical experiment is that participants may use a stimulus feature (cue) other than that intended by the experimenter. One way to avoid this problem is to apply random variations to the corresponding feature across stimulus presentations to make the unwanted cue unreliable. An important question facing experimenters who use this randomization (roving) technique is how large the randomization range should be to ensure that the participants cannot achieve a certain proportion correct by using the unwanted cue, while at the same time avoiding unnecessary interference of the randomization with task performance. Previous researchers have provided formulas for the selection of adequate randomization ranges in yes-no and multiple-alternative forced choice tasks. In this article, we provide figures and tables that can be used to select randomization ranges that are better suited to experiments involving a same-different, dual-pair, or oddity task.

Symmetry-based methodology for decision-rule identification in same--different experiments

The standard practice of reducing every same-different data set to two numbers (hits and false alarms) is wasteful, because the response pattern to all four stimulus pairs carries information about the decision rule adopted by the observer. We describe eight rules organized in three families: differencing, covert classification, and likelihood ratio. We prove that each family produces a characteristic pattern of (in)equalities among the response probabilities. We propose two simple qualitative tests. Is the performance on stimulus pairs AA and BB statistically indistinguishable? If not, differencing and likelihood-ratio strategies can be rejected. Is the performance on pairs AB and BA indistinguishable? If not, covert classification can be rejected. We present algorithms for fitting two covert-classification models and illustrate the new methodology in a perceptual learning experiment on visual motion-direction discrimination. The standard assumption of symmetric decision criteria was violated.

What makes a melody: The perceptual singularity of pitch sequences

This study investigated the ability of normal-hearing listeners to process random sequences of tones varying in either pitch or loudness. Same/different judgments were collected for pairs of sequences with a variable length (up to eight elements) and built from only two different elements, which were 200-ms harmonic complex tones. The two possible elements of all sequences had a fixed level of discriminability, corresponding to a d(') value of about 2, irrespective of the auditory dimension (pitch or loudness) along which they differed. This made it possible to assess sequence processing per se, independent of the accuracy of sound encoding. Pitch sequences were found to be processed more effectively than loudness sequences. However, that was the case only when the sequence elements included low-rank harmonics, which could be at least partially resolved in the auditory periphery. The effect of roving and transposition was also investigated. These manipulations reduced overall performance, especially transposition, but an advantage for pitch sequences was still observed. These results suggest that automatic frequency-shift detectors, available for pitch sequences but not loudness sequences, participate in the effective encoding of melodies.

Likelihood ratio, optimal decision rules, and relationship between proportion correct and d' in the dual-pair AB-versus-BA identification paradigm

The equal-variance Gaussian signal detection theory (SDT) decision model for the dual-pair (4IAX) change-detection paradigm has been described in earlier publications. In this research article, we consider the equal-variance Gaussian SDT model for the related 4IAX AB-versus-BA identification paradigm. The likelihood ratios, optimal decision rules, receiver-operating characteristics (ROCs), and relationships between d' and proportion correct (PC) are analyzed for two special cases: that of statistically independent observations, which typically applies in constant-stimuli experiments, and that of highly correlated observations, which typically applies in experiments where stimuli are roved widely across trials or pairs. A surprising outcome of this analysis is that, although these two situations lead to different optimal decision rules, the predicted ROCs and PC responses for these two cases are not substantially different and are either identical to or similar to those observed in the basic yes-no paradigm. Supplemental materials for this study can be downloaded from app.psychonomic-journals.org/content/supplemental.

Likelihood ratio, optimal decision rules, and correct response probabilities in a signal detection theoretic, equal-variance Gaussian model of the observer in the 4IAX paradigm

This article provides a synthetic account of the likelihood ratio, optimal decision rules, and correct response probabilities in a signal detection theoretic model of the observer in the dual-pair comparison, or four-interval AX (4IAX), paradigm. The model assumes a static sampling process, resulting in four, equal-variance normally distributed (i.e., Gaussian) observations on each trial First, a likelihood ratio equation allowing for an arbitrary degree of correlation between observations is provided. Specific solutions for the cases of independent and highly correlated observations are then derived. It is shown that these solutions, and the associated decision rules, correspond to those provided independently in earlier publications. A modified 4IAX paradigm involving, as a standard, an additional stimulus (C) located medially between the A and the B stimuli is also considered. It is shown that the optimal (static, equal-variance, Gaussian) decision model for this paradigm is unaffected by correlation between observations and is equivalent to the standard 4IAX with highly correlated observations. Finally, we consider how, under the considered (static, equal-variance, Gaussian) model, the proportion of correct responses in the different versions of the 4IAX paradigm is related to d', and a solution for the case of independent observations is provided.

Vibrotactile intensity and frequency information in the pacinian system: a psychophysical model

The objective of the study was to characterize the Pacinian representation of stimulus waveform. Subjects were presented with pairs of high-frequency vibrotactile stimuli that varied in intensity and/or frequency content and made same-different judgments under conditions of low-frequency adaptation designed to minimize the contribution of the RA system. We wished to infer the nature of the information conveyed by the Pacinian system about the stimuli from measured sensitivity (d') to stimulus differences. We first tested the hypothesis that the Pacinian system conveys only intensive information about vibratory stimuli and found that intensive cues could not account for much of the variance in the discrimination data. We then proposed a model characterizing the Pacinian-mediated representation of an arbitrary stimulus as a pattern of activation in a set of frequency-tuned minichannels. The model was shown to predict the discriminability of the stimulus pairs presented in the psychophysical experiments. Furthermore, the model parameters, optimized to fit the discrimination data, were compatible with analogous values obtained in other experimental contexts. One of the assumptions underlying the model is that information about individual spectral components is conveyed in parallel and quasi-independently. By simulating the response of a population of Pacinian afferents to a polyharmonic stimulus, we demonstrated that such a population can simultaneously convey information about multiple frequency components, despite having a homogeneous spectral profile.

Detection-theoretic analysis of same-different judgments for the amplitude discrimination of acoustic sinusoids

Three methods were applied to determine which decision model best accounted for same-different judgments about the amplitude of acoustic sinusoids. The methods were (1) analysis of the shape of the receiver operating characteristic; (2) analysis of an observer's decision space; and (3) a correlation method based on the conditional-on-single-stimulus procedure. In one experiment, observers rated their confidence that a pair of 1-kHz sinusoids was drawn from the same or from different sets. The two sets had identical Gaussian distributions of decibel amplitudes, but their mean amplitudes differed by 3 dB. The differencing model, in which observers base their decision on the absolute difference between the two observations, accounted for the data better than the independent-observation model, in which observers compute independent likelihood ratios for each observation. A second experiment added to every trial an interval that contained a 1-kHz sinusoid whose amplitude equaled the grand mean of both stimulus sets. Despite this additional information, which is needed to adopt the independent-observation model, the differencing model again better accounted for the data.

An assessment of response bias for the same-different task: implications for the single-interval task

Two auditory amplitude discrimination experiments were conducted using the same-different experimental design. Observer bias was manipulated, in the first experiment, by varying payoff matrices and, in the second, by varying prior probabilities of signal presentation. Five levels of bias manipulation and four levels of difficulty were employed in each experiment. Each observer received all combinations of bias manipulation and difficulty, but with only one of these combinations within each block of trials. Nine indices of bias were assessed by simultaneously fitting isosensitivity and isobias functions to the data and by fitting isobias functions only. Although none of the indices tested provided an exceptionally good fit to the data, two indices stood out from the rest. These were c*sd and ci, indices with isobias contours similar in shape to those for the c index derived from the yes-no task.

A Thurstonian model for the dual pair (4IAX) discrimination method

In the dual pair method, the subject is presented with two stimuli in two pairs: One pair is composed of two samples of the same stimulus; the other pair is composed of two samples of different stimuli, one being the same as that in the identical pair. The task of the judge is to select the most different pair. The psychometric function for the dual pair method is derived and expressed in terms of a singly noncentral beta distribution. A table is provided that connects a measure of the degree of difference, d, to the probability of a correct response. This table assumes an unbiased observer and differencing decision rule. A table is provided to give an estimate of the variance of d , the experimental estimate of d. The power of the dual pair method is also investigated, and a formula to determine the sample size required to meet Type I and Type II error specifications is given. The dual pair method appears to be slightly less powerful than the duo-trio and the triangular methods. Experimental investigation is needed to explore the dual pair in applied research work.

Complex tactile waveform discrimination

Complex vibrotactile waveforms consisting of two superimposed sinusoids at varying phases were presented to the fingertip, and observers made "same-different" judgments. It was found that the low-frequency (10Hz+30Hz) waveforms were discriminable from one another while discrimination of the high-frequency (100Hz+300Hz) vibrations was poor. High-frequency adaptation did not impair discrimination of the low-frequency waveforms, suggesting that the RA channel mediated discrimination. Low-frequency adaptation impaired discrimination of the high-frequency stimuli, suggesting that the RA channel likewise mediated the modest level of performance observed in the absence of an adapting stimulus. The results indicate that this channel encodes complex waveforms temporally. A simple model for low-frequency waveform discrimination is proposed. The results obtained with the high-frequency complex waveforms are compatible with the hypothesis that the PC channel integrates stimulus energy over time.

Latency and accuracy of discriminations of odor quality between binary mixtures and their components

Subjects made timed, same-different discriminations of odor quality, with the following principal findings: (i) latency reflected accuracy, with difficult discriminations, i.e. those between 50-50 mixtures and their components, requiring more time than less difficult discriminations, i.e. those between unmixed chemicals. This finding demonstrated the face validity of latency as a measure of qualitative similarity. (ii) Latency provided better resolution among pairs of odors than did errors of discrimination. This finding demonstrated the utility of collecting response times. (iii) Latency-based similarities among odors tested previously predicted similarities among pairs not yet tested. This finding demonstrated internal/predictive validity. (iv) A signal detection model assuming a differencing strategy best described the pattern of errors. Subjects appeared to make relative judgements regarding quality. (v) Finally, latency-based similarities between mixtures and their components demonstrated additivity. This finding suggested that binary mixtures fall on straight lines connecting their components in 'odor-space'.

An area theorem for the same-different experiment

If observers in a same-different experiment base their decisions on the absolute difference between observations on a trial, the area under the receiver operating characteristic equals the maximum proportion of correct decisions that an unbiased independent-observations observer could attain. Even though the differencing strategy is suboptimal, the area measure yields an index of optimal performance.

Likelihood-ratio decision strategy for independent observations in the same-different task: an approximation to the detection-theoretic model

An optimal decision strategy for deciding whether two things are the same or different is to adopt a likelihood-ratio criterion. The parametric equations for the receiver operating characteristic (ROC) based on the likelihood-ratio strategy when observations are independent are complicated; they require the numerical evaluation of a double integral. An approximation to the parametric equations for the likelihood-ratio strategy was developed. This approximation takes the form of a pair of equations that describe ROCs virtually indistinguishable from those of the full model.

The optimum decision rules for the oddity task

This paper presents the optimum decision rule for an m-interval oddity task in which m-1 intervals contain the same signal and one is different or odd. The optimum decision rule depends on the degree of correlation among observations. The present approach unifies the different strategies that occur with "roved" or "fixed" experiments (Macmillan & Creelman, 1991, p. 147). It is shown that the commonly used decision rule for an m-interval oddity task corresponds to the special case of highly correlated observations. However, as is also true for the same-different paradigm, there exists a different optimum decision rule when the observations are independent. The relation between the probability of a correct response and d' is derived for the three-interval oddity task. Tables are presented of this relation for the three-, four-, and five-interval oddity task. Finally, an experimental method is proposed that allows one to determine the decision rule used by the observer in an oddity experiment.