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How might the human mind be studied, scientiﬁcally?

It is a commonplace notion that the brightness of a light, the loudness of a sound, or the heaviness of a weight is one thing, and the luminance, the sound pressure level, or the physical weight in gm quite another. If a way could be found to measure brightness, loudness, and heaviness—not the physical attributes of luminance, sound pressure level, and weight, but the subjective experiences they engender—then such measurements might provide the foundation for a scientiﬁc study of the mind, analogous to physics.

The name of this enterprise is psychophysics. It has led to two quite distinct traditions of experimental research that I call sensory discrimination and sensory judgment.

## 1. Sensory Discrimination

Fechner ([1860] 1966) conceived the idea of using the ‘just noticeable diﬀerence’ (JND) between two stimulus magnitudes as the unit of subjective measurement. Earlier, Weber ([1834, 1846] 1978) had enunciated the law that now bears his name. If X_{1} and X_{2} are two sensory magnitudes such that X is just noticeably greater than X_{2}, then

Writing the constant, which is known as the Weber Fraction, as Θ,

or

another constant, which Fechner identiﬁed as the subjective value of a JND. It follows that the subjective experience of stimulus magnitude increases as ln X, a relation known as Fechner’s Law.

### 1.1 Psychophysical Methods

The project of measuring subjective sensation by means of Eqn. (3) depends on accurate measurement of JNDs. Fechner directed systematic research towards the development of three methods.

1.1.1 The Method Of Limits. Envisage that X_{2} , greater than X_{1} , is reduced progressively by small amounts until it is judged ‘equal’ and then by further small amounts until it is ‘less than’, or the process might be reversed with X_{2} initially less than X_{1} . This procedure identiﬁes ‘thresholds’ above and below X_{1} at which the judgment changes.

1.1.2 The Method Of Average Error. Envisage that X_{2} is adjusted until it is judged ‘equal’ to X_{1} , which is ﬁxed. The probable error of adjustment can then be used as an alternative to the upper and lower thresholds identiﬁed by the method of limits.

1.1.3 The Method Of Constant Stimuli. Envisage that various X_{2}s are presented together with a ﬁxed X_{1} and the subject asked to say which is greater. It turns out that the probability of a correct judgment increases smoothly with the diﬀerence between X_{1} and X_{2} . That relationship is known as the psychometric function.

### 1.2 Signal-Detection Theory

The hundred years following Fechner’s seminal work were spent arguing, ﬁrst, how properly to analyze the judgments from these three psychophysical methods and, second, what the relationship between them was. That argument was resolved only with the advent of signal detection theory, which showed that these were the wrong questions to be asking. To put the matter succinctly, the presentation of a stimulus, be its magnitude X_{1} or X_{2} , provides the subject with information whether the stimulus magnitude is, in fact, X_{1} or X_{2} . This information is a random variable with a diﬀerent distribution depending on which stimulus is presented. The optimum strategy for the subject is to report ‘X_{2} ’ when the information in favour of X_{2} exceeds a certain criterion and ‘X_{1} ’ otherwise.

Signal-detection theory brought the idea of hypothesis testing by bisection of the likelihood-ratio (Neyman and Pearson 1933) into psychological theory. It identiﬁed two fundamental factors in sensory discrimination; there is (a) the information aﬀorded to the subject (a property of the stimuli to be discriminated), and (b) the decision criterion (which is at the subject’s disposal). These two factors are confounded in the traditional psychophysical methods.

It is common to model signal detection in terms of two normal distributions (unit variance, diﬀerence of mean d ), but it should be borne in mind that this is a model of the information and its function is simply to facilitate calculations. Nevertheless, in that spirit, let the mean of the distribution representing magnitude X be ln X and let the variance be [ln(1 +Θ)]^{2}. If a JND be deﬁned as that stimulus diﬀerence which gives d` = 1, Eqn. (3) is obtained.

We cannot say, however, that signal-detection theory validates Fechner’s Law (Eqn. (3)) because the signal-detection model is no more than a model and any other model delivering the same numerical predictions, more or less, will do as well. Such another model is composed of χ^{2 }distributions multiplied by scaling factors proportional to the magnitudes of the stimuli. It arises in this manner.

### 1.3 Diﬀerential Coupling

Weber’s Law (Eqn. (1)) holds rather accurately for discrimination between two separate stimuli (two ﬂashes of light in darkness, two bursts of noise in silence), and for many stimulus attributes, down to, but not below, absolute threshold. The amplitude of a pure tone (but not Gaussian noise) and the contrast of a sinusoidal grating are known exceptions. But if the task is to detect a ﬂash of light added to a pre-existing background, the relationship is more complicated. For the detection of a sinusoidal grating van Nes and Bouman (1967) demonstrated a square-root relationship up to a transition illuminance, above which detection conformed to Weber’s Law, and the transition illuminance increased as the square of the wavenumber of the grating. In addition, the psychometric function for detection is steeper than that for discrimination between two separate stimuli; and the Weber fraction for discrimination increases when one of the stimulus magnitudes lies below the absolute threshold. In short, Weber’s Law, on which Fechner’s Law is based, is just one component of a large body of related phenomena, a body suﬃciently large and complex as to admit substantially only one explanation.

(a) The sensory pathway is diﬀerentially coupled to the physical stimulus so that only changes in sensory input are available as a basis for discrimination. When the two stimuli to be distinguished are presented separately, that means intrinsic random variability only. The energy of that in trinsic variability (a sample of Gaussian noise) has a χ^{2 }distribution multiplied by the noise power density, which, in this case, is proportional to the stimulus magnitude X. The logarithm of a χ^{2 }variable is, to a very good approximation, normal with variance independent of X. Weber’s Law falls directly out of diﬀerential coupling.

(b) The steeper gradient of the psychometric function for detection is equivalent to the interposition of a square law transform, which, in turn, is directly analogous to the phenomenon of low-signal suppression in radio reception (Davenport and Root 1958). The stimulus to be detected adds a small perturbation to the Gaussian noise giving a noncentral χ^{2 }distribution of energy. The resultant central/noncentral χ^{2} model accommodates additional phenomena within the general theory and is to be preferred to the logarithmic model underlying Fechner’s Law. The distributions in this model are everywhere scaled in proportion to physical stimulus magnitude, so that there is no empirical justiﬁcation (arising from the study of sensory discrimination) for any measure of sensation distinct from the physical (Laming 1997, Chaps. 3, 4).

(c) The diﬀerential coupling is realized in the balanced excitatory/inhibitory receptive ﬁelds of sensory neurons. The low-signal suppression is a consequence of transmission by all-or-none action potentials of one polarity only. The theory of sensory discrimination describes the ﬂow of information through the early stages of sensory pathways and annexes that preconscious component of perception to the physical domain. There is no possibility of basing a scientiﬁc study of the mind on resolving power.

## 2. Sensory Judgment

A very wide range of sound energy levels are encountered in nature and sound pressure level is measured on a logarithmic scale; 10 decibels equates to a 10-fold increase in power. If Fechner’s Law were valid, people should say that 100 dB (a subway train entering a station) should sound twice as loud as 50 dB (quiet conversation in a library). Most people would say that the subway train is much more than twice as loud. Around 1930 this posed a problem for acoustic engineers explaining sound levels to their clients and led to the development of the sone scale (Stevens 1936). Loudness in sones increases twofold for each 10 dB increase in sound power.

### 2.1 Stevens’ Power Law

Stevens (1957) returned to the problem of subjective measurement in the 1950s with three new experimental methods.

(a) Magnitude estimation, in which subjects assigned numbers to stimulus magnitudes;

(b) Magnitude production, in which subjects adjusted a variable stimulus magnitude to match a given number; and

(c) Cross-modality matching, in which subjects adjusted a stimulus magnitude on one continuum to match a given magnitude on another.

Taking the numbers assigned to the stimuli to be direct measures of subjective sensation, Stevens found that

where N is the mean assignment, a a constant, and β an exponent typical of the stimulus attribute. Equation (4) is known as Stevens’ Power Law and its validity has been demonstrated for more than 30 diﬀerent attributes.

### 2.2 Explanation Of The Power Law

It is tempting to suppose, as Stevens did, that the sheer volume of experimental demonstration cracks the problem of subjective measurement. But those experiments invariably employed a geometric ladder of stimulus values. If subjects could be induced to assign numbers in like manner (to ‘judge ratios’), then a power law relationship is inevitable, simply by regression of log number onto log stimulus magnitude. There are several reasons for supposing that this is all Eqn. (4) tells us.

(a) The power law relation can be perturbed by spacing the stimuli other than geometrically.

(b) The subjects in Stevens’ laboratory consistently used numbers extending over a range of about 1 to 30. The estimate of the exponent β can therefore be calculated from the log range of stimulus values employed.

(c) Instructions to subjects are invariably illustrated with numerical examples. A wider spread of numbers in those examples, but examples which conform to the same nominal scale, produces a wider spread of estimates in the experiment and therefore an increased value of β. Subjects accept guidance about how wide a range of numbers they should be using.

(d) The variance of the numerical assignments in experiments from Stevens’s laboratory is about 100fold greater than the equivalent variability implicit in sensory discrimination.

(e) But that variability is much reduced when two similar stimulus values ( 5 dB) are presented in succession. The correlation between successive log numerical assignments (then about 0.8) implies that each stimulus is used as a point of reference for the succeeding judgment.

So magnitude estimates are very imprecisely determined and are easily inﬂuenced by extraneous factors. Good power law data depends on balancing these extraneous inﬂuences by appropriate randomization of the schedule of stimulus presentations.

### 2.3 Relative Judgment

The correlation (e) above supports a theory of relative judgment in which each stimulus is compared to its predecessor. More to the point, that comparison is little better than ordinal; we can say that this sound is louder/softer/about the same as the previous one, but that is all. This idea (Laming 1984) accounts for several otherwise quite unrelated phenomena, including the quantitative variation of the autocorrelation (e) with stimulus diﬀerences. By focusing on the precision with which such judgments are made, this idea also relates magnitude estimation to category judgment and absolute identiﬁcation—the stimuli and the subject are the same, only the response categories are diﬀerent—and especially to the often repeated ﬁnding that the amount of information in a category judgment is limited to 2.3 bits, which is equivalent to the use of no more than ﬁve categories without error (Garner 1962, Chap. 3). But if human judgment is ultimately no better than ordinal, it aﬀords no possibility of establishing a metric internal to the mind that is distinct from the physical.

## 3. How Might The Human Mind Be Studied?

There are three ways in which these traditions of research might be continued.

### 3.1 Information Theory

(Note that I refer here to the mathematical theory of statistical information, not to the information theory (properly communication theory) of Shannon (1949), nor to the colloquial use of ‘information’ which has subsequently become fashionable in psychology.)

Discriminations between two separate stimuli conform accurately to the normal, equal variance, signal detection model (Tanner and Swets 1954). In that model the parameter d^{2 }measures the mean information aﬀorded by a single presentation of either stimulus. A discrimination between two indistinguishable (identical) stimuli provides a natural zero; two looks at the same stimulus (cf. Swets 1964), or two simultaneous, correlated, signals, provide a natural concatenation of separate means; and a comparison between the informativeness of two diﬀerent discriminations can easily be fashioned from statistical analysis. These are the criteria needed to establish ratio- scale measurement, and it is ratio-scale measurement of fundamental quantities such as mass, length, and time that has enabled the physical sciences to develop their present-day precision.

With that lesson in mind, psychologists have long striven to establish their own scales of measurement, but hitherto have always lacked empirical foundations suﬃcient to e stablish a ratio scale (Michell 1999). The parameter d^{2 }bids fair to provide the ﬁrst ratio-scale measure of speciﬁcally psychological origin; other ratio scales in psychology have been borrowed from extraneous disciplines. By this means experimental performance may be related to physical parameters of the stimuli (see Laming 1986, 1988).

### 3.2 The Precision Of Sensory Judgment

Magnitude estimation and category judgment may be meaningfully analyzed using a multistimulus elaboration of the normal signal detection model, with a separate normal distribution for each stimulus value (the Law of Categorical Judgment, Case IC; Torgerson 1958). The inverse variance in units of log stimulus magnitude measures the precision of the judgments. That precision varies inversely with the square of the log stimulus range and also depends on the randomness of the stimulus presentation schedule. An ‘analysis of variance’ of sensory judgment provides a second vehicle of enquiry.

Inverse variance is Fisher’s (192 2) measure of information and is closely related to d . So this second suggestion employs the same mathematical theory as the ﬁrst, but now directed toward the relationships between diﬀerent experimental paradigms, including the classical psychophysical methods listed above. The lesson to be absorbed is that a magnitude estimate provides some limited information about the subject’s assessment of the stimulus, but is never to be taken at its face value.

### 3.3 Sequential Interactions

It has been known since 1920 that successive psychophysical judgments are statistically related, but there is, as yet, no theoretical understanding what those interactions signify. Aggregate performance in an experiment results from a pattern of sequential interactions driven by a (usually) random sequence of stimulus values. Trial-by-trial analysis provides a third way to study the mechanics of the human mind.

### 3.4 Internal Sensation

But none of these approaches measures subjective sensation. Internal experience is private to the subject; it cannot be shared with the experimenter except to the extent that it can be expressed in terms of some feature of the external world perceptible to both. Experimental paradigms are simply schemes for expressing that otherwise private experience. The strictly limited precision of any and every subject’s judgments suggests that there is no more precise internal metric to be expressed.

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