The validity of the successive intervals method of psychometric scaling

The degree to which scale values computed by the method of successive intervals diverge from theoretically true values is seen to be due to three types of error: error due to inequalities in variances of the distributions from which the scale values are computed, error due to non-normality of the distributions, and sampling error. The contribution of each type of error to the total error is evaluated; the latter is seen to be surprisingly small under appropriate conditions. Certain aspects of the formal methodology underlying scaling procedures are also briefly considered.This paper reports research undertaken in cooperation with the Quartermaster Food and Container Institute for the Armed Forces, and has been assigned number 475 in the series of papers approved for publication. The views or conclusions contained in this report are those of the authors. They are not to be construed as necessarily reflecting the views or indorsement of the Department of Defense.     

The estimation of the discriminal dispersion in the method of successive intervals

A new algebraic formula is derived for estimation of the discriminal dispersion in the method of successive intervals. The legitimate use of the formula requires that as many normal deviates as possible be present in the matrix. For this reason, it is recommended that deviates corresponding to the interval (0.01, 0.99) of the cumulative proportions be used, instead of those corresponding to (0.05, 0.95), the interval used by Edwards and Thurstone. Computations on data published by Edwards and Thurstone showed that when adjustment was made for variability in dispersions calculated by the formula of this paper, a reduction of fifty per cent in mean absolute discrepancy was produced. Since the formula is easy to use and avoids the disadvantages of its predecessors, it should have fairly wide applicability in psychological research.This research was supported in part by the United States Air Force under Contract No. AF 33(038)-25726 monitored by Air Force Personnel and Training Research Center. Permission is granted for reproduction, translation, publication, use and disposal in whole and in part by or for the United States Government. The writer is grateful to Dr. A. L. Edwards for a critical reading of an earlier version of this paper, and to Dr. L. H. Lanier and Dr. L. M. Stolurow for editorial advice on the present version, which was written at the University of Illinois. The editors ofPsychometrika have informed the writer that H. J. A. Rimoldi and M. Hormaeche (7) have independently derived the same formula for the discriminal dispersion from a different set of postulates using the law of comparative judgment.     

Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations

Thurstone's Case V of the method of paired comparisons assumes equal standard deviations of sensations corresponding to stimuli and zero correlations between pairs of stimuli sensations. It is shown that the assumption of zero correlations can be relaxed to an assumption of equal correlations between pairs with no change in method. Further the usual approach to the method of paired comparisons Case V is shown to lead to a least squares estimate of the stimulus positions on the sensation scale.This research was performed in the Laboratory of Social Relations under a grant made available to Harvard University by the RAND Corporation under the Department of the Air Force, Project RAND.     

A modification of the method of successive intervals

A modification of the method of successive intervals is presented which yields scale values correlating .995 with those from Thurstone's method described by Saffir. Values yielded by the present method can be obtained in 25 per cent of the time required by the older method and are shown to be, on apriori grounds, more reliable as well.     

The timing of interresponse intervals

Analogs of models of duration discrimination are here related to the timing of discrete motor responses. The measure of interest is the variability in duration of intervals collected in short interval reproduction tasks. For data from a Morse key-tapping task, it is shown that, taken separately, neither of the models described can completely account for the relation between the mean and the variance of the interresponse intervals.     

Remarks on the method of paired comparisons: II. The effect of an aberrant standard deviation when equal standard deviations and equal correlations are assumed

If customary methods of solution are used on the method of paired comparisons for Thurstone's Case V (assuming equal standard deviations of sensations for each stimulus), when in fact one or more of the standard deviations is aberrant, all stimuli will be properly spaced except the one with the aberrant standard deviation. A formula is given to show the amount of error due to the aberrant stimulus.This research was performed in the Laboratory of Social Relations under a grant made available to Harvard University by the RAND Corporation under the Department of the Air Force, Project RAND.     

The threat of appearing prejudiced and race-based attentional biases

The current work tested whether external motivation to respond without prejudice toward Blacks is associated with biased patterns of selective attention that reflect a threat response to Black individuals. In a dot-probe attentional bias paradigm, White participants with low and high external motivation to respond without prejudice toward Blacks (i.e., low-EM and high-EM individuals, respectively) were presented with pairs of White and Black male faces that bore either neutral or happy facial expressions; on each trial, the faces were displayed for either 30 ms or 450 ms. The findings were consistent with those of previous research on threat and attention: High-EM participants revealed an attentional bias toward neutral Black faces presented for 30 ms, but an attentional bias away from neutral Black faces presented for 450 ms. These attentional biases were eliminated, however, when the faces displayed happy expressions. These findings suggest that high levels of external motivation to avoid prejudice result in anxious arousal in response to Black individuals, and that this response affects even basic attentional processes.     

The discrimination of brief temporal intervals

The discrimination of brief durations (0.25 to 3.75 sec) defined by the presentation of a visual stimulus was examined. The results indicate that the Weber fraction is invariant over this range of temporal intervals, and that the value of the fraction is about 0.05. The relation of these results to the literature on temporal discriminability was discussed.     

The emotional meaning of harmonic intervals

The present study reexamines the hypothesis that there exist emotional attributions specific to simple musical elements. In Experiment 1, groups of participants, with varying musical expertise, rated the emotional meaning of four natural intervals heard as two harmonic sine waves. In Experiment 2, the higher tone was kept constant at an octave above the low tone used in Experiment 1, while the lower tone was constant. Attributions for each interval were positively correlated from one experimental session to another; despite the intervals differed in terms of their component pitches. Musicians gave the most reliable choices of meaning. In a third experiment, participants rated the emotional meaning of various unfamiliar ethnic melodies with expressions describing the intervals’ meaning based on the results of Experiment 1 and 2. There were distinct profiles of emotional meanings for each melody and these coincided with the meaning of intervals that constituted the surface and deep structure of each melody. The intervallic structures (i.e., the main intervals of the tunes) and respective chords for each melody were also presented aurally and participants’ ratings showed similar emotional profiles for these when compared to those of the melodies themselves.     

The perception of brief temporal intervals: power functions for auditory and visual stimulus intervals

Two experiments were performed to examine the role of method of estimation and the employment of a standard stimulus on the judged duration of auditory and visual stimuli presented for brief temporal intervals (0.25 to 5.0 s). The results indicate that the relationship between judged and physical duration is nearly direct and linear. Psychophysical methodology and stimulus modality exerted little influence on the obtained power functions.     

The fallacy of placing confidence in confidence intervals

Interval estimates – estimates of parameters that include an allowance for sampling uncertainty – have long been touted as a key component of statistical analyses. There are several kinds of interval estimates, but the most popular are confidence intervals (CIs): intervals that contain the true parameter value in some known proportion of repeated samples, on average. The width of confidence intervals is thought to index the precision of an estimate; CIs are thought to be a guide to which parameter values are plausible or reasonable; and the confidence coefficient of the interval (e.g., 95 %) is thought to index the plausibility that the true parameter is included in the interval. We show in a number of examples that CIs do not necessarily have any of these properties, and can lead to unjustified or arbitrary inferences. For this reason, we caution against relying upon confidence interval theory to justify interval estimates, and suggest that other theories of interval estimation should be used instead.     

Remarks on the method of paired comparisons: III. A test of significance for paired comparisons when equal standard deviations and equal correlations are assumed

A test of goodness of fit is developed for Thurstone's method of paired comparisons, Case V. The test involves the computation of , wheren is the number of observations per pair, and and are the angles obtained by applying the inverse sine transformation to the fitted and the observed proportions respectively. The number of degrees of freedom is (k–1) (k–2)/2.This research was performed in the Laboratory of Social Relations under a grant made available to Harvard University by the RAND Corporation under the Department of the Air Force, Project RAND.     

Contours of equal perceived amplitude and equal perceived frequency for electrocutaneous stimuli

Previous measurements of equal-sensation contours for electrocutaneous stimuli consisting of repeated bursts of biphasic pulses have shown that stimulus frequency has little effect on perceived amplitude, and that stimulus amplitude has no effect on perceived frequency. These earlier contours, however, were measured over a very restricted range of amplitude and frequency or for a single perceived amplitude or perceived frequency. Contours of equal perceived amplitude and equal perceived frequency were measured in the present study for stimuli covering most of the useable range of amplitudes and frequencies: 3–12 dB SL and 4–64 Hz. Eight naive subjects generated contours of equal perceived amplitude at four reference amplitudes via Békésy tracking, and 8 additional subjects generated contours of equal perceived frequency at three reference frequencies. The contours of equal perceived amplitude declined slightly but significantly with increases in stimulus frequency, consistent with previous results. The shape of the contours was also slightly dependent on the amplitude of the reference stimulus. Contours of equal perceived frequency were unaffected by stimulus amplitude on the average, but the contour shape did vary modestly, though erratically, with reference frequency.     

An internal consistency check for scale values determined by the method of successive intervals

The method of successive intervals is a psychological scaling procedure in which stimuli are classified into successive intervals according to the degree of some defined attribute which they are judged to possess. A psychological continuum is defined and the scale values are then taken as the medians of the distributions of judgments on the psychological continuum. It is assumed that the distributions of judgments for each stimulus are normal on the psychological continuum as defined.An internal consistency check indicates that the cumulative distributions of empirical judgments for the various stimuli can be reproduced by means of a limited number of parameters with an average error that compares favorably with that usually reported for paired comparison data. Furthermore, the scale values obtained by successive interval scaling, for the data reported, are shown to be linearly related to those obtained by the method of paired comparisons.This paper was written while the first author was a post-doctoral Research Training Fellow of the Social Science Research Council studying at the University of Chicago. It reports research undertaken in cooperation with the Quartermaster Food and Container Institute for the Armed Forces. The views or conclusions contained in this report are those of the authors. They are not to be construed as necessarily reflecting the views or endorsement of the Department of Defense.