The psychometric function relates an observer’s performance to an independent variable, usually some physical quantity of a stimulus in a psychophysical task. This paper, together with its companion paper (Wichmann & Hill, 2001), describes an integrated approach to (1) fitting psychometric functions, (2) assessing the goodness of fit, and (3) providing confidence intervals for the function’s parameters and other estimates derived from them, for the purposes of hypothesis testing. The present paper deals with the first two topics, describing a constrained maximum-likelihood method of parameter estimation and developing several goodness-of-fit tests. Using Monte Carlo simulations, we deal with two specific difficulties that arise when fitting functions to psychophysical data. First, we note that human observers are prone to stimulus-independent errors (orlapses). We show that failure to account for this can lead to serious biases in estimates of the psychometric function’s parameters and illustrate how the problem may be overcome. Second, we note that psychophysical data sets are usually rather small by the standards required by most of the commonly applied statistical tests. We demonstrate the potential errors of applying traditionalX2 methods to psychophysical data and advocate use of Monte Carlo resampling techniques that do not rely on asymptotic theory. We have made available the software to implement our methods.

https://link.springer.com/content/pdf/10.3758%2FBF03194544.pdf

The psychometric function: I. Fitting, sampling, and goodness of fit

WHO Regional Office for Europe.

The psychometric function relates an observer's performance to an independent variable, usually a physical quantity of an experimental stimulus Even if a model is successfully fit to the data and its goodness of fit is acceptable, experimenters require an estimate of the variability of the parameters to assess whether differences across conditions are significant Accurate estimates of variability are difficult to obtain, however, given the typically small size of psychophysical data sets: Traditional statistical techniques are only asymptotically correct and can be shown to be unreliable in some common situations Here and in our companion paper (Wichmann & Hill, 2001), we suggest alternative statistical techniques based on Monte Carlo resampling methods The present paper's principal topic is the estimation of the variability of fitted parameters and derived quantities, such as thresholds and slopes First, we outline the basic bootstrap procedure and argue in favor of the parametric, as opposed to the nonparametric, bootstrap Second, we describe how the bootstrap bridging assumption, on which the validity of the procedure depends, can be tested Third, we show how one's choice of sampling scheme (the placement of sample points on the stimulus axis) strongly affects the reliability of bootstrap confidence intervals, and we make recommendations on how to sample the psychometric function efficiently Fourth, we show that, under certain circumstances, the (arbitrary) choice of the distribution function can exert an unwanted influence on the size of the bootstrap confidence intervals obtained, and we make recommendations on how to avoid this influence Finally, we introduce improved confidence intervals (bias corrected and accelerated) that improve on the parametric and percentile-based bootstrap confidence intervals previously used Software implementing our methods is available

/pdf/the-psychometric-function-ii-bootstrap-based-confidence-2s4vkhd2o4.pdf

The psychometric function: II. Bootstrap-based confidence intervals and sampling.

Steady-state evoked potentials can be recorded from the human scalp in response to auditory stimuli presented at rates between 1 and 200 Hz or by periodic modulations of the amplitude and/or frequency of a continuous tone. Responses can be objectively detected using frequency-based analyses. In waking subjects, the responses are particularly prominent at rates near 40 Hz. Responses evoked by more rapidly presented stimuli are less affected by changes in arousal and can be evoked by multiple simultaneous stimuli without significant loss of amplitude. Response amplitude increases as the depth of modulation or the intensity increases. The phase delay of the response increases as the intensity or the carrier frequency decreases. Auditory steady-state responses are generated throughout the auditory nervous system, with cortical regions contributing more than brainstem generators to responses at lower modulation frequencies. These responses are useful for objectively evaluating auditory thresholds, assessing suprathreshold hearing, and monitoring the state of arousal during anesthesia.

Human auditory steady-state responses.

Objective:This study was designed to determine the prevalence of minimal sensorineural hearing loss (MSHL) in school-age children and to assess the relationship of MSHL to educational performance and functional status.Design:To determine prevalence, a single-staged sampling frame of all schools in t

Children with minimal sensorineural hearing loss : Prevalence, educational performance, and functional status

Objective:Pure-tone thresholds for conventional and extended high frequencies were analyzed for 188 older adult human subjects (91 females, 97 males). The objectives were to study longitudinal changes in thresholds as well as the effects of initial threshold levels, age, gender, and noise history on

Longitudinal study of pure-tone thresholds in older persons.

Age-related changes in auditory potentials of Mongolian gerbil.

A maximum-likelihood method was applied in measurements of frequency and intensity discrimination for aged and young normal-hearing subjects with closely matched audiograms. This method was preferred over other psychophysical procedures because it is efficient and controls experimental variance, features that are highly desirable for testing aged subjects. In order to implement the method, psychometric functions for each task were also measured from young subjects using a constant-stimuli procedure. For the young subjects, the differential thresholds obtained from these two procedures were generally comparable. Further, both sets of data were consistent with previous literature, indicating that the maximum-likelihood method was successfully applied for frequency and intensity discrimination. A frequency-dependent difference between young and aged subjects in both frequency and intensity discrimination was observed. Even with closely matched audiograms, aged subjects demonstrated poorer discrimination abilities than young subjects. The age-related difference was always largest at 500 Hz and decreased as frequency increased.

Frequency and intensity discrimination measured in a maximum-likelihood procedure from young and aged normal-hearing subjects

Psychometric functions for detection of temporal gaps in wideband noise were measured in a “yes/no” paradigm from normal-hearing young and aged subjects with closely matched audiograms. The effects of noise-burst duration, gap location, and uncertainty of gap location were tested. A typical psychometric function obtained in this study featured a steep slope, which was independent of most experimental conditions as well as age. However, gap thresholds were generally improved with increasing duration of the noise burst for both young and aged subjects. Gap location and uncertainty had no significant effects on the thresholds for the young subjects. For the aged subjects, whenever the gap was sufficiently away from the onset or offset of the noise burst, detectability was robust despite uncertainty about the gap location. Significant differences between young and aged subjects could be observed only when the gap was very close to the signal onset and offset.

Psychometric functions for gap detection in noise measured from young and aged subjects.

Detection of amplitude modulation (AM) in 500 and 4000 Hz tonal carriers was measured as a function of modulation frequency from younger and older adults with normal hearing through 4000 Hz. The modulation frequency above which sensitivity to AM increased (“transition frequency”) was similar for both groups. Temporal modulation transfer function shapes showed significant age-related differences. For younger subjects, AM detection thresholds were generally constant for low modulation frequencies. For a higher carrier frequency, AM detection thresholds then increased as modulation frequency further increased until the transition frequency. In contrast, AM detection for older subjects continuously increased with increasing modulation frequency, indicating an age-related decline in temporal resolution for faster envelope fluctuations. Significant age-related differences were observed whenever AM detection was dependent on temporal cues. For modulation frequencies above the transition frequency, age-related differences were larger for the lower frequency carrier (where both temporal and spectral cues were available) than for the higher frequency carrier (where AM detection was primarily dependent on spectral cues). These results are consistent with a general age-related decline in the synchronization of neural responses to both the carrier waveform and envelope fluctuation.

John H. Mills

Papers

Longitudinal study of pure-tone thresholds in older persons.

Age-related changes in auditory potentials of Mongolian gerbil.

Frequency and intensity discrimination measured in a maximum-likelihood procedure from young and aged normal-hearing subjects

Psychometric functions for gap detection in noise measured from young and aged subjects.

Age-related differences in the temporal modulation transfer function with pure-tone carriers.