William J. McGill

The relation between information and variance analyses

Analysis of variance and uncertainty analysis are analogous techniques for partitioning variability. In both analyses negative interaction terms due to negative covariance terms that appear when non-orthogonal predictor variables are allowed may occur. Uncertainties can be estimated directly from variances if the form of distribution is assumed. The decision as to which of the techniques to use depends partly on the properties of the criterion variable. Only uncertainty analysis may be used with a non-metric criterion. Since uncertainties are dimensionless (using no metric), however, uncertainty analysis has a generality which may make it useful even when variances can be computed.

Neural counting mechanisms and energy detection in audition

Abstract A remarkable correspondence between data on detectability of auditory signals in noise and theoretical analysis, based on energy fluctuations of the stimulus, is considered in detail. In the present treatment the fluctuation statistics of acoustic noise are superimposed on a (Poisson) mean counting rate in the auditory system and the resulting counting distributions are analyzed. Detection laws derived from the latter are then compared with auditory data on masking and intensity discrimination, as well as with analogous predictions obtained from direct analysis of stimulus energy. The outcome can be characterized as establishing that internal counting states mimic the noise energy fluctuations. Hence, detection laws derived from either domain are found to be essentially equivalent. This result justifies the stimulus—based analysis of detectability now current in auditory psychophysics, but it also suggests that sensory detection models involving decision strategies with substantial information processing may prove to be unnecessarily complicated.

Multivariate information transmission

A multivariate analysis based on transmitted information is presented. It is shown that sample transmitted information provides a simple method for measuring and testing association in multidimensional contingency tables. Relations with analysis of variance are pointed out, and statistical tests are described.

The general-gamma distribution and reaction times ☆

Abstract The general-gamma distribution describes input-output times in a multistage process consisting of exponential components whose constants are all different. The distribution and its unique history are examined. A stochastic process that leads to it is presented. The conditional density (hazard) function is studied as a means for estimating parameters. Finally, the multistage process model is applied to simple reaction times in an effort to reveal underlying detection and response components.

Pure‐Tone Intensity Discrimination and Energy Detection

A study of 1000‐Hz intensity discrimination is reported in which single tones were identified as the louder or softer of two alternatives differing slightly in energy. Tones were phase locked and fixed in duration (150 msec). Feedback was given following each judgment. Psychometric functions are presented and a masking curve is computed. The masking curve is found to be linear above 25‐dB sensation level, and the slope is computed as 15/16. These results are interpreted as reflecting a form of energy discrimination similar to that proposed in noise masking.

Random fluctuations of response rate

A simple model for fluctuating interresponse times is developed and studied. It involves a mechanism that generates regularly spaced excitations, each of which can trigger off a response after a random delay. The excitations are not observable, but their periodicity is reflected in a regular patterning of responses. The probability distribution of the time between responses is derived and its properties are analyzed. Several limiting cases are also examined.

Multiplication noise in the human visual system at threshold: 1. Quantum fluctuations and minimum detectable energy

We have carried out a series of frequency-of-seeing experiments similar to those performed by Hecht, Shlaer, and Pirenne [ J. Gen. Physiol.25, 819– 840 ( 1942)], using an Ar+ laser operated at 514.5 nm as the source of light. In certain blocks of trials, our subjects were encouraged to report as seen those trials in which the stimulus might have been present. It was determined that sensitivity and reliability were traded against each other over a broad range: for our four subjects, the detection of 147 photons at the cornea with 60% frequency of seeing entailed, on the average, a 1% false-positive rate (FPR), whereas the detection of 34 photons at the cornea with 60% frequency of seeing was accompanied by a 33% FPR. A new neural-counting model has been developed in the framework of signal-detection theory. It combines Poisson stimulus fluctuations with additive and multiplicative neural noise, both of which are known to be present in the visual system at threshold. The resulting probability-of-detection curves, derived from the Neyman Type-A counting distribution, are in good accord with our experimental frequency-of-seeing data for sensible values of the model parameters. We deduce that, on the average, our four subjects are able to detect a single photon at the retina with 60% frequency of seeing, at the expense of a 55% FPR. In Part 2 of this set of papers [ PrucnalP. R.TeichM. C., Biol. Cybern.43, 87– 96 ( 1982)], we use the normalizing transform, together with probit analysis, to provide improved estimates of threshold parameters, whereas in Part 3 [ TeichM. C.PrucnalP. R.VannucciG.BretonM. E.McGillW. J., submitted to Biol. Cybern.], we consider the effects of non-Poisson quantum fluctuations.

A statistical description of verbal learning

Free-recall verbal learning is analyzed in terms of a probability model. The general theory assumes that the probability of recalling a word on any trial is completely determined by the number of times the word has been recalled on previous trials. Three particular cases of this general theory are examined. In these three cases, specific restrictions are placed upon the relation between probability of recall and number of previous recalls. The application of these special cases to typical experimental data is illustrated. An interpretation of the model in terms of set theory is suggested but is not essential to the argument.

The Slope of the Loudness Function: A Puzzle

This paper deals with the slope of the loudness function considered as a sensory parameter. Shapes of individual loudness functions are examined in a measurement situation in which loudness is estimated by marking a point on a line. It is shown that such personalized loudness functions can be converted to power functions by computing zero points differing widely from listener to listener. The slopes of the computed loudness functions also differ markedly.

Multiplication noise in the human visual system at threshold

Several kinds of light used in vision experiments produce photon statistics that are distinctly non-Poisson. Representative examples are light from a cathode-ray tube and an image-intensifier device. For the class of vision experiments in which the photon statistics play an important role, excess fluctuations produced by such light sources can alter the observed results and obscure the visual mechanisms being studied. They must therefore be accounted for in a proper way. We use the results of a Hecht-Shlaer-Pirenne type experiment, carried out with modulated Poisson light, to illustrate the point. Sensitivity and modulation depth, as well as sensitivity and reliability, are shown to be traded against each other. Finally, we demonstrate that number-state light, which is comprised of photons of an ideal kind, provides the ultimate tool for extracting information about the intrinsic noise distribution in the visual system at threshold. The state of the art in producing such light is discussed.