The slope of the psychometric function and non-stationarity of thresholds in spatiotemporal contrast vision

The slope of the two-interval, forced-choice psychometric function (e.g. the Weibull parameter, β) provides valuable information about the relationship between contrast sensitivity and signal strength. However, little is known about how or whether β varies with stimulus parameters such as spatiotemporal frequency and stimulus size and shape. A second unresolved issue concerns the best way to estimate the slope of the psychometric function. For example, if an observer is non-stationary (e.g. their threshold drifts between experimental sessions), β will be underestimated if curve fitting is performed after collapsing the data across experimental sessions. We measured psychometric functions for 2 experienced observers for 14 different spatiotemporal configurations of pulsed or flickering grating patches and bars on each of 8 days. We found β ≈ 3 to be fairly constant across almost all conditions, consistent with a fixed nonlinear contrast transducer and/or a constant level of intrinsic stimulus uncertainty (e.g. a square law transducer and a low level of intrinsic uncertainty). Our analysis showed that estimating a single β from results averaged over several experimental sessions was slightly more accurate than averaging multiple estimates from several experimental sessions. However, the small levels of non-stationarity (SD ≈ 0.8 dB) meant that the difference between the estimates was, in practice, negligible.     

Variations in the slope of the psychometric acuity function with acuity threshold and scale

Psychometric measurement of visual acuity not only determines an acuity threshold, but also, by the slope of the psychometric function, reveals how the percentage of correct responses changes with increasing target size. In this study we determined how the slope of the psychometric function changed for different threshold acuities. Psychometric visual acuity functions were obtained using a clinical contour-interaction chart for 35 naive, normal subjects with best refractive correction and with 1.5 and 3.0 diopters of spherical blur. The slope of the psychometric acuity function remained essentially constant for different threshold acuities when acuity was scaled logarithmically (log MAR), but varied with the threshold when inverse Snellen (MAR) and Snell-Sterling scales were used. Our results lend support to the use of the log MAR scale for acuity but, because normal subjects' slopes range tenfold, the use of this parameter in making clinical distinctions may be problematic.     

Bayesian adaptive estimation of threshold versus contrast external noise functions: the quick TvC method

External noise paradigms, measuring contrast threshold as a function of external noise contrast (the "TvC" function), provide a valuable tool for studying perceptual mechanisms. However, measuring TvC functions at the multiple performance criteria needed to constrain observer models has previously involved demanding data collection (often>2000 trials). To ease this task, we developed a novel Bayesian adaptive procedure, the "quick TvC" or "qTvC" method, to rapidly estimate multiple TvC functions, by adapting a strategy originally developed to estimate psychometric threshold and slope [Cobo-Lewis, A. B. (1996). An adaptive method for estimating multiple parameters of a psychometric function. Journal of Mathematical Psychology, 40, 353-354; Kontsevich, L. L., and Tyler, C. W. (1999). Bayesian adaptive estimation of psychometric slope and threshold. Vision Research, 39(16), 2729-2737]. Exploiting the regularities observed in empirical TvC functions, the qTvC method estimates three parameters: the optimal threshold c(0), the critical noise level N(c), and the common slope, eta, of log-parallel psychometric functions across external noise conditions. Before each trial, the qTvC uses a one-step-ahead search to select the stimulus (jointly defined by signal and noise contrast) that minimizes the expected entropy of the three-dimensional posterior probability distribution, p(N(c),c(0),eta). The method's accuracy and precision, for estimating TvC functions at three performance criteria (65%, 79%, and 92% correct), were evaluated using Monte-Carlo simulations and a psychophysical task. Simulations showed that less than 300 trials were needed to estimate TvC functions at three widely separated criteria with good accuracy (bias<5%) and precision (mean root mean square error <1.5 dB). Using an orientation identification task, we found excellent agreement (weighted r(2)>.95) between TvC estimates obtained with the qTvC and the method of constant stimuli, although the qTvC used only 12% of the data collection (240 vs 1920 trials). The qTvC may hold considerable practical value for applying the external noise method to study mechanisms of observer state changes and special populations. We suggest that the same adaptive strategy can be applied to directly estimate other classical functions, such as the contrast sensitivity function, elliptical equi-discrimination contours, and sensory memory decay functions.     

The slope of the psychometric function at different wavelengths

Many current models of visual detection predict that the slope of psychometric functions for detection will be independent of the spectral power distribution of the test light once the spatial and temporal characteristics of the light are fixed. This article examines reports that the slope of the psychometric function depends on the wavelength of a spectrally-narrowband test light of fixed size and duration. Three sources of difficulty that any experimental measurement of slope must encounter are addressed: (1) the different spatial distributions of the photoreceptor classes across the retina; (2) possible variations in threshold over the course of the experiment; and (3) the large variability of estimates of slope and the need for a method of assessing this variability. Measurements of slope with 2 and 4 deg test flashes against a bright 510 nm field show no significant trend with the wavelength of the test. A novel statistical test bounds the magnitude of possible variations in slope across the visible spectrum.     

Bayesian adaptive estimation of psychometric slope and threshold.

We introduce a new Bayesian adaptive method for acquisition of both threshold and slope of the psychometric function. The method updates posterior probabilities in the two-dimensional parameter space of psychometric functions and makes predictions based on the expected mean threshold and slope values. On each trial it sets the stimulus intensity that maximizes the expected information to be gained by completion of that trial. The method was evaluated in computer simulations and in a psychophysical experiment using the two-alternative forced-choice (2AFC) paradigm. Threshold estimation within 2 dB (23%) precision requires less than 30 trials for a typical 2AFC detection task. To get the slope estimate with the same precision takes about 300 trials.     

Binocular contrast detection with unequal monocular illuminance

Binocular summation was measured in eight normal subjects by means of psychometric functions for contrast detection. An average 47% increase in binocular over monocular performance was obtained. Our data agreed with the simple summation model of Signal Detection Theory (Legge, 1984). Binocular psychometric functions were also measured when the sensitivity of one eye was decreased by means of a 1.0 neutral density (ND) filter. We found that binocular detectability in this case was reduced to below that of the better eye. This binocular inhibition was seen in all subjects. The slope of a contrast detection function gives a measure of the rate of change in detectability with contrast. If the slopes of two functions are equal, then the difference in detectability between these functions remains constant for all the contrast values used. When the slopes of the measured functions were analysed, no significant differences were found under any of the testing conditions. This indicates that the magnitude of summation (with equal monocular sensitivities), and of inhibition (with unequal monocular sensitivities), remains constant across the range of stimulus contrasts. The clinical implications of binocular inhibition are discussed.     

Asymmetries in visual acuity around the visual field

Human vision is heterogeneous around the visual field. At a fixed eccentricity, performance is better along the horizontal than the vertical meridian and along the lower than the upper vertical meridian. These asymmetric patterns, termed performance fields, have been found in numerous visual tasks, including those mediated by contrast sensitivity and spatial resolution. However, it is unknown whether spatial resolution asymmetries are confined to the cardinal meridians or whether and how far they extend into the upper and lower hemifields. Here, we measured visual acuity at isoeccentric peripheral locations (10 deg eccentricity), every 15° of polar angle. On each trial, observers judged the orientation (± 45°) of one of four equidistant, suprathreshold grating stimuli varying in spatial frequency (SF). On each block, we measured performance as a function of stimulus SF at 4 of 24 isoeccentric locations. We estimated the 75%-correct SF threshold, SF cutoff point (i.e., chance-level), and slope of the psychometric function for each location. We found higher SF estimates (i.e., better acuity) for the horizontal than the vertical meridian and for the lower than the upper vertical meridian. These asymmetries were most pronounced at the cardinal meridians and decreased gradually as the angular distance from the vertical meridian increased. This gradual change in acuity with polar angle reflected a shift of the psychometric function without changes in slope. The same pattern was found under binocular and monocular viewing conditions. These findings advance our understanding of visual processing around the visual field and help constrain models of visual perception.     

Observer-based rather than population-based confidence limits for determining probability of change in visual fields

Determining confidence intervals on psychophysical thresholds is straight forward if the psychometric function is known. In clinical settings, however, there is only partial information about the psychometric function, hence confidence limits are usually derived from test-retest data collected from many subjects. In this paper, we introduce a computational technique for deriving confidence limits for an individual's endpoint threshold using data typically obtained in a clinical setting, rather than a database of test-retest performance. The technique uses probabilistic analysis of all possible response sequences in a test procedure. We then extend this procedure to allow for levels of typical uncertainty in data measurement.     

The Freiburg Visual Acuity Test-Variability unchanged by post-hoc re-analysis

Background The Freiburg Visual Acuity and Contrast Test (FrACT) has been further developed; it is now available for Macintosh and Windows free of charge at . The present study sought to reduce the test-retest variability of visual acuity on short runs (18 trials) by post-hoc re-analysis. Methods The FrACT employs advanced computer graphics to present Landolt Cs over the full range of visual acuity. The sequence of optotypes presented follows an adaptive staircase procedure, the Best-PEST algorithm. The Best-PEST threshold obtained after 18 trials was compared to the result of a post-hoc re-analysis of the acquired data, where both threshold and slope of the psychometric function were estimated via a maximum-likelihood fit. Results Testing time was 1.7 min per run on average. Test-retest reproducibility was ±2 lines (or ±0.2 logMAR) for a 95% confidence band (using 18 optotype presentations per test run). Post-hoc psychometric fitting reproduced the Best-PEST result within 1%, although the individual slopes varied widely; test-retest reproducibility was not improved. Conclusions The FrACT offers advantages over traditional chart testing with respect to objectivity and reliability. The similarity between the results of the Best-PEST vs. post-hoc analysis, fitting both slope and threshold, suggest that there is no disadvantage to the constant slope assumed by Best PEST. Furthermore, since variability was not reduced by post-hoc analysis, for high reliability more trials should be employed than the 18 trials per run used here.     

Binocular contrast summation—I. Detection and discrimination

Binocular summation was evaluated for contrast detection and discrimination. Monocular and binocular forced-choice psychometric functions were measured for the detection of 0.5-c/deg sine-wave gratings presented alone (simple detection), or superimposed on identical background gratings (discrimination). The dependence of detectability d′ on signal contrast C could be described by: d′ = (C/C′)ⁿ · C′ is threshold contrast, and n is an index of the steepness of the psychometric function, n was near 2 for simple detection, near 1 for discrimination, and was approximately the same for monocular and binocular viewing. Monocular thresholds were about 1.5 times binocular thresholds for detection, but the ratio dropped for suprathreshold discrimination. These results reveal a dependence of binocular summation on background contrast. For simple detection, binocular detectabilities were at least twice monocular detectabilities. For contrast discrimination, the amount of binocular summation decreased. For a 25%-contrast background, there was little or no binocular summation. It is concluded that binocular contrast summation decreases as background contrast rises.     

The Freiburg Visual Acuity Test-variability unchanged by post-hoc re-analysis.

BACKGROUND: The Freiburg Visual Acuity and Contrast Test (FrACT) has been further developed; it is now available for Macintosh and Windows free of charge at http://www.michaelbach.de/fract.html . The present study sought to reduce the test-retest variability of visual acuity on short runs (18 trials) by post-hoc re-analysis. METHODS: The FrACT employs advanced computer graphics to present Landolt Cs over the full range of visual acuity. The sequence of optotypes presented follows an adaptive staircase procedure, the Best-PEST algorithm. The Best-PEST threshold obtained after 18 trials was compared to the result of a post-hoc re-analysis of the acquired data, where both threshold and slope of the psychometric function were estimated via a maximum-likelihood fit. RESULTS: Testing time was 1.7 min per run on average. Test-retest reproducibility was +/-2 lines (or +/-0.2 logMAR) for a 95% confidence band (using 18 optotype presentations per test run). Post-hoc psychometric fitting reproduced the Best-PEST result within 1%, although the individual slopes varied widely; test-retest reproducibility was not improved. CONCLUSIONS: The FrACT offers advantages over traditional chart testing with respect to objectivity and reliability. The similarity between the results of the Best-PEST vs. post-hoc analysis, fitting both slope and threshold, suggest that there is no disadvantage to the constant slope assumed by Best PEST. Furthermore, since variability was not reduced by post-hoc analysis, for high reliability more trials should be employed than the 18 trials per run used here.     

Covert attention affects the psychometric function of contrast sensitivity

We examined the effect of transient covert attention on the psychometric function for contrast sensitivity in an orientation discrimination task when the target was presented alone in the absence of distracters and visual masks. Transient covert attention decreased both the threshold (consistent with a contrast gain mechanism) and, less consistently, the slope of the psychometric function. We assessed performance at 8 equidistant locations (4.5 degrees eccentricity) and found that threshold and slope depended on target location-both were higher on the vertical than the horizontal meridian, particularly directly above fixation. All effects were robust across a range of spatial frequencies, and the visual field asymmetries increased with spatial frequency. Notwithstanding the dependence of the psychometric function on target location, attention improved performance to a similar extent across the visual field.Given that, in this study, we excluded all sources of external noise, and that we showed experimentally that spatial uncertainty cannot explain the present results, we conclude that the observed attentional benefit is consistent with signal enhancement.     

Temporal determinants of spatial sine-wave masking

A temporal forced-choice procedure was used to measure the contrast threshold for a sinusoidal test grating (spatial frequency = f) superimposed upon a sinusoidal background or masking grating (spatial frequency = 3f). The spatial contrast of the background grating was varied, and threshold measurements were made at each of a number of background contrasts to describe a threshold versus masking contrast (tvc) function.Tvc functions were obtained when the background and test grating contrasts were, independently of each other, held steady (0 Hz) or modulated at 5 Hz. The changes of temporal modulation frequency affected the slopes of thetvc functions. In some cases thetvc functions for steady and flickering test gratings crossed one another. The changes of slope suggest, and the crossovers imply, that some steady and flickering patterns are detected by separate visual mechanisms.     

Spatial summation in visual noise

Contrast thresholds were measured for sinusoidal gratings, with Gaussian spatial and temporal envelopes, as a function of spatial extent in the presence and absence of dynamic white noise. Spatial frequencies ranged from 0.5 to 32 c/deg. Efficiency is defined as the ratio of the ideal's contrast-energy threshold to that of the observer under study, each at the same performance level. For spatial frequencies from 0.5 to 8 c/deg, efficiencies for gratings in noise ranged from 8 to 30% for widths less than one cycle, but dropped rapidly as the width was increased beyond one cycle. Spatial summation of gratings in noise resembles the performance of a cross-correlator whose template is matched to a signal about 1 cycle wide (between 1/e points). The psychometric function slope is consistent with this idea.     

Sensory and decisional factors in motion-induced blindness

The processes underlying motion-induced blindness (MIB) are widely debated. Ultimately, however, they must reduce to a sensitivity drop and/or to an upward shift of the decision criterion. The first possibility was tested by assessing the detection threshold for a contrast (or luminance) increment applied to the MIB target under its visible and suppressed phases. This was performed over a whole range of reference target contrasts (yielding the standard Threshold vs. Contrast [TvC] function) with a 2AFC staircase procedure. The second possibility was tested with a Yes/No procedure, allowing the assessment of both the sensitivities (d') and decision criteria (c) yielded by four contrast increments applied to a fixed reference target contrast (the psychometric function). The 2AFC procedure yielded a global upward shift of the TvC function of about 5.3 dB (a factor of 1.84) under the suppressed phase. The Yes/No procedure yielded a commensurate d' drop (of about 0.8 sigma) under the suppressed phase with no change in the slope of the psychometric function and an upward shift of c of about 0.7 sigma. The presently observed vertical shift (in log-log coordinates) of the TvC function in the suppressed phase is indicative of a divisive inhibition occurring after the transducing stage. The invariance of the psychometric function slope under the visible and suppressed MIB phases supports this conclusion. The present experiments cannot settle the issue of whether the upward shift of the decision criteria is yet another cause of the MIB or a consequence of its underlying inhibitory process. Instead, they make clear that MIB (and perhaps other unstable perceptual phenomena) is associated with both sensory and decisional processes.     

Comparing integration rules in visual search

Search performance for a target tilted in a known direction among vertical distractors is well explained by signal detection theory models. Typically these models use a maximum-of-outputs rule (Max rule) to predict search performance. The Max rule bases its decision on the largest response from a set of independent noisy detectors. When the target is tilted in either direction from the reference orientation and the task is to identify the sign of tilt, the loss of performance with set size is much greater than predicted by the Max rule. Here we varied the target tilt and measured psychometric functions for identifying the direction of tilt from vertical. Measurements were made at different set sizes in the presence of various levels of orientation jitter. The orientation jitter was set at multiples of the estimated internal noise, which was invariant across set sizes and measurement techniques. We then compared the data to the predictions of two models: a Summation model that integrates both signal and noise from local detectors and a Signed-Max model that first picks the maxima on both sides of vertical and then chooses the value with the highest absolute deviation from the reference. Although the function relating thresholds to set size had a slope consistent with both the Signed-Max and the Summation models, the shape of individual psychometric functions was in the most crucial conditions better predicted by the Signed-Max model, which chooses the largest tilt while keeping track of the direction of tilt.     

Best's macular dystrophy in Australia: phenotypic profile and identification of novel BEST1 mutations

Purpose

(1) To evaluate the spectrum of BEST1mutations within Australian Best Disease or vitelliform macular dystrophy (VMD) pedigrees, including any novel mutations; (2) to analyse the range of clinical presentations of this cohort; (3) to determine any possible genotype–phenotype correlations and (4) to compare clinical data of patients with phenotypic VMD, both with and without a BEST1mutation.

Patients and methods

Patients with suspected VMD were referred to clinical centres for ophthalmological assessment and genetic screening. When a mutation was identified in a proband, further family members were invited for clinical and genetic screening.

Results

We identified 42 patients with one of 13 BEST1mutations. Seven mutations were novel. There were a further 14 probands in whom a BEST1mutation was not identified. Median visual acuity in both VMD (mutation positive) and clinical VMD (no BEST1mutation identified) groups reached driving standards (6/12 or better).

Conclusion

We did not identify any firm genotype–phenotype correlations in our Australian VMD pedigrees, in which there was a spectrum of BEST1mutations and marked variation in clinical presentation. Genetic screening remains the gold standard for VMD diagnosis. Patients should be counselled that visual acuity might remain at or above driving standards in at least one eye even in the presence of a BEST1mutation.     

A failure to observe negative preference in infant acuity testing

Held et al. (1979) Vision Res.19, 1377–1379, reported that infants' psychometric functions in preferential looking experiments exhibit a region of below chance performance or “negative preference”. They argued from this that previous preferential looking experiments may have systematically underestimated acuity because they ignored this negative preference. We present data from an experiment designed to reveal negative preference in the version of the preferential looking paradigm used in our laboratory. The results do not exhibit negative preference apart from random variations about 50%. Gwiazda et al. (1980) Am. J. Optom. physiol. Opt.57, 420–427; 428–432, developed a psychophysical procedure for infant testing that is designed to yield accurate threshold estimates in a short period of time. We present the results of computer simulations of more conventional psychophysical procedures. The simulations demonstrate that an up-down staircase procedure is more efficient than the Gwiazda et al. procedure.     

Visual acuity for vertical and diagonal gratings in human infants

The present paper describes the development of a two-alternative forced-choice psychophysical method for use with human infants. The essence of the technique (see Fantz, 1965, 1967) is that an adult observer must, on each trial, state in which of two possible positions an acuity target is presented, by observing only the infant's eye and head movements. The per cent correct responses of the observer, for each of several acuity targets, are plotted to yield psychometric functions.The technique was used to test the performance of infants between 1 and 6 months of age, toward vertical and diagonal acuity gratings. Twenty psychometric functions, accumulated from a total of 8 individual infants at two illumination levels, and a longitudinal series on one infant, are presented. The data are orderly and reveal trends toward better performance with increasing age.No differences were found between psychometric functions generated in the presence of gratings in the two different orientations. The results are compared to earlier data in which orientational differences in acuity have been shown to exist in human adults. The implications of the results are discussed in relation to the development of orientational acuity differences.