A comparison of six methods to estimate thresholds from psychometric functions

There are many ways in which to estimate thresholds from psychometric functions. However, almost nothing is known about the relationships between these estimates. In the present experiment, Monte Carlo techniques were used to compare psychometric thresholds obtained using six methods. Three psychometric functions were simulated using Naka-Rushton and Weibull functions and a probit/logit function combination. Thresholds were estimated using probit, logit, and normit analyses and least-squares regressions of untransformed orz-score and logit-transformed probabilities versus stimulus strength. Histograms were derived from 100 thresholds using each of the six methods for various sampling strategies of each psychometric function. Thresholds from probit, logit, and normit analyses were remarkably similar. Thresholds fromz-score- and logit-transformed regressions were more variable, and linear regression produced biased threshold estimates under some circumstances. Considering the similarity of thresholds, the speed of computation, and the ease of implementation, logit and normit analyses provide effective alternatives to the current “gold standard”—probit analysis—for the estimation of psychometric thresholds.     

A statistical model for discriminating between subliminal and near-liminal performance

The concept of a psychophysical threshold is foundational in perceptual psychology. In practice, thresholds are operationalized as stimulus values that lead to a fairly high level of performance such as .75 or .707 in two-choice tasks. These operationalizations are not useful for assessing subliminality—the state in which a stimulus is so weak that performance is at chance. We present a hierarchical Bayesian model of performance that incorporates a threshold that divides subliminal from near-liminal performance. The model provides a convenient means to measure at-chance thresholds and therefore is useful for testing theories of subliminal priming. The hierarchical nature of the model is critical for efficient analysis as strength is pooled across people and stimulus values. A comparison to Rasch psychometric models is provided.     

Threshold estimation in two-alternative forced-choice (2AFC) tasks: the Spearman-Kärber method

The Spearman-K?rber method can be used to estimate the threshold value or difference limen in two-alternative forced-choice tasks. This method yields a simple estimator for the difference limen and its standard error, so that both can be calculated with a pocket calculator. In contrast to previous estimators, the present approach does not require any assumptions about the shape of the true underlying psychometric function. The performance of this new nonparametric estimator is compared with the standard technique of probit analysis. The Spearman-K?rber method appears to be a valuable addition to the toolbox of psychophysical methods, because it is most accurate for estimating the mean (i.e., absolute and difference thresholds) and dispersion of the psychometric function, although it is not optimal for estimating percentile-based parameters of this function.     

Personality disorder diagnostic thresholds correspond to different levels of latent pathology

To meet a diagnosis for a particular personality disorder (PD), among other requirements, a person must exhibit a specified minimum number of the features outlined for that PD, otherwise referred to as the diagnostic threshold. Despite many years of research on the DSM PDs, there is little empirical basis for the chosen thresholds. The present study used mechanisms of item response theory (IRT) to link the diagnostic thresholds of six PDs to their corresponding level of latent PD pathology. Consistent with our hypothesis, analyses of the data from 41,227 participants revealed that PD diagnostic thresholds corresponded to a wide range of latent pathology. For example, the diagnostic threshold for schizoid PD corresponded to 1.54 SDs of pathology whereas the threshold for dependent PD corresponded to 2.72 SDs of pathology. The current analyses have demonstrated for the first time that the latent pathology associated with each PD threshold varies widely, and thus has quantified what others accurately have speculated. Implications and considerations for the future directions of PD classification are discussed.     

Auditory threshold,signal-to-noise ratio and signal detection theory

The use of Signal Detection Theory has demonstrated that there is not an auditory threshold at the level supposed by classical psychophysics, but has not disproved the existence of an auditory threshold at a lower level. Following the recommended signal detection theory procedure fairly closely, an experiment obtained evidence either of the existence of a signal-to-noise ratio audi tory threshold, or a nonlinear relationship between stimulus and response which could not be statistically distiguished from such a threshold , Using the same apparatus and subjects, audi tory thresholds were also obtained using the classical method of limits. The acoustic pressure for the classical thresholds was about five times as great as for the signal detection theory method.     

Converting between measures of slope of the psychometric function

The psychometric function’s slope provides information about the reliability of psychophysical threshold estimates. Furthermore, knowing the slope allows one to compare, across studies, thresholds that were obtained at different performance criterion levels. Unfortunately, the empirical validation of psychometric function slope estimates is hindered by the bewildering variety of slope measures that are in use. The present article provides conversion formulas for the most popular cases, including the logistic, Weibull, Quick, cumulative normal, and hyperbolic tangent functions as analytic representations, in both linear and log coordinates and to different log bases, the practical decilog unit, the empirically based interquartile range measure of slope, and slope in a? representation of performance.     

Threshold adjustment in response to asymmetric loss functions: The case of auditors’ “substantial doubt” thresholds

Thresholds play a critical role in linking judgments and choices. One way they do so is by dividing a continuous judgment variable, such as probability, into two regions that imply different choices and actions, such as operate/do not operate, admit/do not admit, or invest/do not invest. We examine this link by focusing on the tendency of professional decision makers to adjust their thresholds when the risks associated with negative decision outcomes are elevated. We report two studies involving experienced auditors that investigate thresholds in a “going-concern” setting. In this setting, the auditor assesses the probability that a business firm will be unable to continue in existence for the coming year; if that probability exceeds a “substantial doubt” threshold (also assessed by the auditor), the auditor is required to disclose the relevant information in the firm’s annual report to the public. Our studies use two experimental cases based on actual business firms that differ in their objective likelihoods of not continuing as going concerns. We derive the auditors’ substantial doubt thresholds from the relationship between their probability assessments and their disclosure choices, unlike all prior research which has simply asked auditors to state the probability they believe represents substantial doubt. We find that auditors’ derived thresholds are adjusted downward for the more problematic firm, a result that we attribute to the asymmetric loss functions inherent in going-concern settings. The second study has some of the auditors directly provide their substantial doubt thresholds (SDTs). The downward adjustment is again found for derived thresholds but not for elicited thresholds, suggesting that the method of capturing thresholds may be an important issue in understanding judgment and choice.     

Computation of psychophysical thresholds using the probit technique

Probit analysis is a statistical technique that can be used for the computation of psychophysical thresholds, although it was originally developed for the analysis of pharmacological and toxicological data. Probit analysis may be applied to psychophysical data if the method of constant stimuli has been used for data collection. A computer program written in APPLESOFT BASIC that performs probit analyses on such psychophysical data is presented and explained.     

Interaction effects in successive visual displays: An extension of the Eriksen-Collins paradigm

The paradigm of Eriksen and Collins, in which one-haft of the elements of a display is assigned randomly to each of two successively presented fields, was applied to the visual discrimination of one- and two-dimensional Markov constraints. In contrast with the two-flash masking paradigm, performance in the Eriksen-Collins paradigmimproves as the interval between the successively presented fields is shortened. Constraint thresholds may increase with interfield intervals as short as 20 msec. In contrast with the results of most visual information-processing tasks, increasing the duration of each field may result in sharplyelevated thresholds. The elevation of thresholds can be reduced through repeated presentation. Sharp asymmetries are also obtained with differential brightness of the two fields.     

The dynamics of choice among multiple alternatives

We consider neurally based models for decision-making in the presence of noisy incoming data. The two-alternative forced-choice task has been extensively studied, and in that case it is known that mutually inhibited leaky integrators in which leakage and inhibition balance can closely approximate a drift-diffusion process that is the continuum limit of the optimal sequential probability ratio test (SPRT). Here we study the performance of neural integrators in n?2 alternative choice tasks and relate them to a multihypothesis sequential probability ratio test (MSPRT) that is asymptotically optimal in the limit of vanishing error rates. While a simple race model can implement this ‘max-vs-next’ MSPRT, it requires an additional computational layer, while absolute threshold crossing tests do not require such a layer. Race models with absolute thresholds perform relatively poorly, but we show that a balanced leaky accumulator model with an absolute crossing criterion can approximate a ‘max-vs-ave’ test that is intermediate in performance between the absolute and max-vs-next tests. We consider free and fixed time response protocols, and show that the resulting mean reaction times under the former and decision times for fixed accuracy under the latter obey versions of Hick's law in the low error rate range, and we interpret this in terms of information gained. Specifically, we derive relationships of the forms log(n-1), log(n), or log(n+1) depending on error rates, signal-to-noise ratio, and the test itself. We focus on linearized models, but also consider nonlinear effects of neural activities (firing rates) that are bounded below and show how they modify Hick's law.     

Studies in olfactory acuity: III. relative detectability of n-aliphatic acetates by the rat

The ability of black and albino rats to detect the first seven n-aliphatic acetates in the vapour phase has been investigated by use of an odour choice box. Both the median threshold and the slope of the probit regression line (which describes the relation between concentration and response) tend to decrease logarithmically in value as the series is ascended. The position of methyl acetate in these relations appears anomalous. When expressed as thermodynamic activities median thresholds for acetates of intermediate chain length are about equally stimulating, whilst short chain alcohols excite at decreasing activities with increasing chain length. The close similarity of these findings to those reported in the previous study on n-aliphatic alcohols supports the view that such relations may be of more general occurrence in olfaction in the rat.

The stimulus-response curve for each acetate studied shows one or more reversals in slope. It is suggested that certain of these are due to the presence of two or more olfactory receptor types one or more of which attain maximum response at lower concentrations of a given acetate than do the remainder. The asymptotes of the curves are reached in about 2-3 log units of concentration.     

Operator behavior in a dynamic visual signal detection task

The effects of changes in signal-to-noise ratio (SIN) and feedback state on the operator’s detection behavior, particularly derived operator thresholds, were investigated using a discrete visual signal. It was found that: (1) the operator behaves as if he has established different thresholds as a function of S/N; (2) his decision thresholds are relatively independent of the presence or absence of feedback; and (3) the operator will change his threshold accordingly for a change from one S/N to another. A model of the operator’s behavior, which introduced the concept of “operator” noise. yielded predictions of detection and false alarm probabilities which were extremely close to the observed values.     

Effect of stimulus duration on stereo and vernier displacement thresholds

Stereo and vernier thresholds were determined as a function of stimulus duration in both misalignment and displacement paradigms. All four thresholds increase as stimulus duration decreases. The increase is greatest for the stereo displacement threshold, less for the vernier displacement threshold, and least for the misalignment thresholds. The results are consistent with the hypothesis that depth signals are averaged over time.     

The effect of warning interval on the electric phosphene and auditory thresholds

Using different warning signals and threshold stimuli, the thresholds, as determined by a method of limits, were found to rise monotonically as the interval between warning signal and threshold stimulus increased from I to 9 sec. It was found that the variability of the threshold did not increase as the threshold increased. Similar results were obtained for phosphene and auditory thresholds and with visual and auditory warnings; therefore the effect was considered to be central. Motokawa's finding of a minimum in the phosphene threshold 2 sec. after a flash of white light was not repeated. The rise in threshold was not obtained when the warning intervals were randomized and so seemed to depend on the use of fixed warning intervals. A model was developed relating threshold level to accuracy of anticipation of the end of the warning interval.     

A rating formulation for ordered response categories

A rating response mechanism for ordered categories, which is related to the traditional threshold formulation but distinctively different from it, is formulated. In addition to the subject and item parameters two other sets of parameters, which can be interpreted in terms of thresholds on a latent continuum and discriminations at the thresholds, are obtained. These parameters are identified with the category coefficients and the scoring function of the Rasch model for polychotomous responses in which the latent trait is assumed uni-dimensional. In the case where the threshold discriminations are equal, the scoring of successive categories by the familiar assignment of successive integers is justified. In the case where distances between thresholds are also equal, a simple pattern of category coefficients is shown to follow.This work was conducted in part in the first half of 1977 while the author was on study leave at the Danish Institute for Educational Research. The Institute provided required research facilities while The University of Western Australia provided financial support.     

A threshold theory account of psychometric functions with response confidence under the balance condition

The study of thresholds for discriminability has been of long‐standing interest in psychophysics. While threshold theories embrace the concept of discrete‐state thresholds, signal detection theory discounts such a concept. In this paper we concern ourselves with the concept of thresholds from the discrete‐state modelling viewpoint. In doing so, we find it necessary to clarify some fundamental issues germane to the psychometric function (PF), which is customarily constructed using psychophysical methods with a binary‐response format. We challenge this response format and argue that response confidence also plays an important role in the construction of PFs, and thus should have some impact on threshold estimation. We motivate the discussion by adopting a three‐state threshold theory for response confidence proposed by Krantz (1969, Psychol. Rev., 76, 308–324), which is a modification of Luce's (1963, Psychol. Rev., 70, 61–79) low‐threshold theory. In particular, we discuss the case in which the practice of averaging over order (or position) is enforced in data collection. Finally, we illustrate the fit of the Luce–Krantz model to data from a line‐discrimination task with response confidence.     

Flexible Bayesian modelling in dichotomous item response theory using mixtures of skewed item curves

Most item response theory (IRT) models for dichotomous responses are based on probit or logit link functions which assume a symmetric relationship between the probability of a correct response and the latent traits of individuals taking a test. This assumption restricts the use of those models to the case in which all items behave symmetrically. On the other hand, asymmetric models proposed in the literature impose that all the items in a test behave asymmetrically. This assumption is inappropriate for great majority of tests which are, in general, composed of both symmetric and asymmetric items. Furthermore, a straightforward extension of the existing models in the literature would require a prior selection of the items' symmetry/asymmetry status. This paper proposes a Bayesian IRT model that accounts for symmetric and asymmetric items in a flexible but parsimonious way. That is achieved by assigning a finite mixture prior to the skewness parameter, with one of the mixture components being a point mass at zero. This allows for analyses under both model selection and model averaging approaches. Asymmetric item curves are designed through the centred skew normal distribution, which has a particularly appealing parametrization in terms of parameter interpretation and computational efficiency. An efficient Markov chain Monte Carlo algorithm is proposed to perform Bayesian inference and its performance is investigated in some simulated examples. Finally, the proposed methodology is applied to a data set from a large-scale educational exam in Brazil.     

Academic and Behavior Combined Support: Evaluation of an integrated supplemental intervention for early elementary students

In current multi-tier models, students who are not responsive to universal instruction receive supplementary support. Despite most students having co-occurring academic and behavior challenges, their needs are often addressed through separate academic or behavior interventions. This approach may not only be costly for schools, but it also fails to acknowledge the well-documented link between behavior and academic performance. The purpose of this study was to evaluate Academic and Behavior Combined Support (ABC Support), a newly developed supplemental intervention that merges a combined focus on reading fluency and engagement. Six teachers implemented the intervention for 6–8 weeks with Grade 2 students. Students' oral fluency on standard and training reading passages and occurrence of engagement and disruptive behaviors during classroom reading instruction were measured repeatedly across baseline and intervention phases in a multiple-baseline design. Visual and statistical analyses revealed significant improvement from baseline to intervention for both reading and behavior outcomes. Post-intervention surveys and interviews revealed a high level of acceptability among teachers and students. Theoretical and empirical contributions as well as practice implications are addressed.     

Frequency selectivity for hearing-impaired and broadband-noise-masked normal listeners

When evaluating frequency selectivity, it is difficult to determine if the deviation from normal performance observed for hearing-impaired listeners reflects abnormal cochlear function or normal, level-dependent changes in frequency selectivity. This experiment was designed to investigate the dependence of frequency-selectivity measures on threshold and signal level. Auditory-filter characteristics, critical ratios, forward-masked psychophysical tuning curves, and narrowband-noise masking patterns were obtained from normal-hearing listeners in quiet and in the presence of broadband noise at five levels. These findings were then compared to previously obtained frequency-selectivity measurements for subjects with mild-to-moderate sensorineural hearing loss whose absolute thresholds are comparable to the normal-hearing listeners' masked thresholds. The results suggest that frequency selectivity deterioriates as threshold increases for all subjects. However, the poor frequency selectivity exhibited by hearing-impaired listeners may not be explained entirely by the effects of threshold and signal level. Nevertheless, because frequency selectivity is poorer in the normal auditory system at higher stimulus levels, the deviation from normal performance observed for hearing-impaired listeners may not be as large as previously suggested.     

Tracking of nociceptive thresholds using adaptive psychophysical methods

Psychophysical thresholds reflect the state of the underlying nociceptive mechanisms. For example, noxious events can activate endogenous analgesic mechanisms that increase the nociceptive threshold. Therefore, tracking thresholds over time facilitates the investigation of the dynamics of these underlying mechanisms. Threshold tracking techniques should use efficient methods for stimulus selection and threshold estimation. This study compares, in simulation and in human psychophysical experiments, the performance of different combinations of adaptive stimulus selection procedures and threshold estimation methods. Monte Carlo simulations were first performed to compare the bias and precision of threshold estimates produced by three different stimulus selection procedures (simple staircase, random staircase, and minimum entropy procedure) and two estimation methods (logistic regression and Bayesian estimation). Logistic regression and Bayesian estimations resulted in similar precision only when the prior probability distributions (PDs) were chosen appropriately. The minimum entropy and simple staircase procedures achieved the highest precision, while the random staircase procedure was the least sensitive to different procedure-specific settings. Next, the simple staircase and random staircase procedures, in combination with logistic regression, were compared in a human subject study (n = 30). Electrocutaneous stimulation was used to track the nociceptive perception threshold before, during, and after a cold pressor task, which served as the conditioning stimulus. With both procedures, habituation was detected, as well as changes induced by the conditioning stimulus. However, the random staircase procedure achieved a higher precision. We recommend using the random staircase over the simple staircase procedure, in combination with logistic regression, for nonstationary threshold tracking experiments.