Departments of Psychology, Stanford University and Northwestern University
William S Verplanck and John W. Cotton
A. The Problem
This experiment was performed to determine how the alteration of two procedural variables involved in serially-ordered stimulation, such as that used in the classical Method of Limits in psychophysics, affects the probabilities that a subject will respond to dim flashes of light, and hence, the values of the threshold measures obtained. The study reflects an emphasis on variables more often encountered in research on learning, or more broadly, behavior, than in studies of thresholds as functions of physical dimensions of stimuli, of adaptation time, and so on. Although some of the effects which will be treated in this and succeeding reports have been alluded to in earlier reports in the literature (3; 10, pp. 41-42), they have been considered as the sources of one or another “error” or “so-called error,” or as “tendencies.” Systematic quantitative data are scanty.
In most forms of the method of limits a “threshold” is determined by computations of the values of the stimulus continuum at which there occur shifts from “yes” to “no” or the reverse (5, 110-114; 12, 397-400). Mean thresholds are computed for populations of such individual threshold measures. Theoretical treatments have stemmed in part from the special concepts of structural psychology and in part from complex statistical analyses. Only casual attention has been paid to a number of variables which have been acknowledged as important, but which do not enter into either theory or computations.
In this investigation, we are concerned not with the Method of Limits in this classical sense, but with a more generalized procedure. This also involves the presentation of stimuli to the subject in intensity-ordered series, to each member of which the subject must respond “yes” or “no.” We are also interested in the intensity values found under this procedure to yield various probabilities that a “yes” or “no” will be given. The procedure of stimulus presentation conforms with some variations of Method of Limits as it has been used, but the treatment of the data and the computation of the threshold do not.
In the intensity-ordered methods of measurement of absolute visual thresholds, S is presented with series of stimulations, each member of which differs from the preceding one by an increment (“ascending series”) or decrement (“descending series”) of fixed value (the “step”). Each such series may begin at a specified value and may end either when a fixed number of stimulations has been presented or when the subject has met some pre-established criterion of response, for example, when he has responded “yes” or “no” to three consecutive stimuli.
The number of series may be varied; the size of the step may be altered; the time intervening between successive stimulations may be varied; different beginning points and different criteria for termination may be employed; and series may be restricted in their direction. In this study, we are concerned with how the direction of series (i.e., whether ascending series, descending series, or both, are employed) and the length of series (as it depends upon the starting point of each) act to determine the probability that a positive response will be given to each stimulus intensity employed. Any effects these variables may have should appear in the parameters, the “mean,” or 50 per cent threshold, and “standard deviation,” or slope, of the psychometric functions generated by presentation to S of a number of series.
B. Apparatus and Procedure
The Hecht-Shlaer Model III Adaptometer was employed. It had been described fully elsewhere (7, 11). In the present experiment the pendulum shutter of the instrument was operated automatically by a programmer at 4-sec. intervals.
Six paid undergraduates, two men and four women, served as S. Each S met or exceeded Service visual norms for near and far acuity, phoria, and depth perception, as measured by the Bausch and Lomb Orthorater. No S had had previous experience in investigations of the visual threshold. All experimental Ss were run in the early afternoon, and insofar as possible, successive experimental days for each were not more than 72 hours apart.
3. Dark Adaptation and Darkroom Procedure
Before each experimental session, S was dark-adapted in X-DA3 plastic red goggles for 25 min., followed by 10 min. in darkness in the experimental room. S was then given standard instructions to fixate and to say “yes” each time he saw a flash of light when the shutter clicked and “no” if he did not. All responses were recorded except those after which the subject stated that he “wasn’t ready” when the shutter clicked. In such cases the stimulation was repeated and experimentation proceeded as before.
4. Experimental Design
A six-by-six Latin square design2was adopted to counterbalance three sets of variables: individual differences among the subjects, ordinal position of the experimental session, and the procedure of stimulation. We employed six such procedures, all of them involving intensity-ordered stimuli, on each of the six experimental days. In all six procedures, 20 series of stimulations were presented in each experimental session. A 4-sec. interval separated successive stimulations except for a 1-min. rest period between the tenth and eleventh series each day. A step of 0.10 log mmL in brightness3separated successive stimulations within each series; “yeses” to three successive stimuli ended ascending series and, similarly, three successive “noes” terminated descending ones.
The six procedures are termed in long descending (LD), the short descending (SD), the long ascending (LA), the short ascending (SA), the long ascending-and-descending (LAD), and the short ascending-and-descending (SAD) procedures. Each series of the LD procedure began at 3.40 log mmL and the SA series at 1.90 log mmL. Thus in all these procedures, each successive series began at a brightness fixed for the procedure. The point at which each ascending or descending series began was the same for each S, so that the average length of series varied for each S as a function of his threshold. In the two descending-and-ascending procedures, each ascending series followed a descending series and began 0.20 LU below the brightness of the last stimulus of the preceding series. A corresponding procedure was followed for each descending series except the first. The first series of the long descending-and-ascending procedure at 3.40 log mmL and the short descending-and-ascending procedure at 3.00 log mmL. Otherwise the two procedures were identical.
One day was spent in training before the six experimental days began, and this procedure was repeated on the day following the six experimental days. On these pre-experimental and post experimental days, all subjects responded through five series of each of the six experimental procedures plus 50 consecutive single brightness (2.5 log mmL stimulations (11).
The relative frequencies of “yeses” given to each stimulating brightness, in log mmL, were plotted on arithmetic probability paper, and a straight line was visually fitted to the experimental points following Beck’s method (1), yielding a conventional PR, or frequency of response function. The log brightness for which the relative frequency of “yeses” was .50 was taken as the 50 per cent threshold; the “standard deviation,” or slope constant, was measured by taking half the difference in log brightness between the points fitted for .84 and .16 relative frequency of “yes,” respectively. Such a threshold and slope measure were determined for each subject’s performance on each day.4Mean values for methods, subjects, and days are presented in Table 1.
An analysis of variance was performed on both the 50 per cent threshold values and “standard deviation” values of Table 1. The results are presented in Table 2. Although individuals differ significantly at better than the 1 per cent level in mean thresholds, they differ only at the 5 per cent level in the slope constant.5 Ordinal position of the experimental session yields no significance. Most important, the F test shows that the 50 per cent thresholds vary significantly with procedure at better than the 5 per cent level, but slope constants do not.
Mean Values of 50 Per Cent Threshold and “Standard Deviation”
Analyses of Variance of 50 Per Cent Thresholds and “Standard Deviations”
The finding that the value of a 50 per cent threshold depends upon the details of procedure demands that we investigate the exact dependence more closely. Table 1 shows that the ascending procedures, tend to give lower thresholds than the descending procedures, and that this difference is greater for the longer series. Employing Fisher’s estimate (4) of the standard error of each of the six means, t tests of the differences were made; these are presented in Table 3. The long descending procedure gives thresholds significantly higher, at better than the 5 per cent level of confidence, than either of the two ascending procedures, and the long ascending procedure yields a threshold significantly lower, at better than the 2 per cent level of confidence, than those of both descending procedures and the short ascending-descending procedure.
Mean thresholds and standard deviations for these data were also computed according to the classical procedure (5). The grand mean of thresholds computed by the classical method was 2.59 log mmL, as compared with the 2.60 log mmL mean shown in Table 1. The grand mean of standard deviations computed by the classical method was .20 as compared with the .18 in Table 1.
Results of t Tests Between Mean 50 Per Cent Thresholds for the Six Procedures
An examination of the mean classical thresholds shows that, without exception, those obtained with ascending procedures are lower than the corresponding 50 per cent thresholds derived from the visual fits, and those obtained for descending procedures were higher. The relationship is not an artifact of the procedure followed in obtaining the 50 per cent threshold.
These findings are not in accord with the widely held opinion that ascending and descending series yield “overlapping” thresholds. To determine what occurs when ascending and descending series are alternated, frequency of response functions were derived separately from the data obtained on the ascending and descending series of the alternating procedures. These curves yielded threshold values which did not differ significantly from one another; when ascending and descending series alternate, systematic differences associated with the direction of the series disappear.
The data suggest that when all series are in one direction, the longer the series (whether ascending or descending) the greater the effect. Unless this relationship holds, the ordinal position of the brightness at which the first altered response (from “no” to “yes” in an ascending, and from “yes” to “no” in a descending series) is given in a long series should be equal to the ordinal position of this brightness in a short plus 4 (the extra number of steps in the long series). We have determined the mean differences in position of ordinal brightness at the shift and find them to be 2.2 for ascending and 3.6 for descending series. A t test indicates that the first is significantly different from 4 at the 2 per cent level of confidence, t = 3.83, for 5 df), but that the second is not; (p > .6, t = 0.73 for 5 df). In the alternating procedures, where the initial step of a series is determined by the brightness that terminated the preceding series, no differences appear, and the mean ordinal position in a series at which the response change occurs is about the same in both ascending and descending series.
Three positive findings of this experiment may be stated: First, the 50 per cent thresholds taken from PR functions obtained when only ascending series of stimuli are employed systematically fall below those obtained when only descending series are used. This effect is greater for method of limits thresholds derived from the same data. Second, if the length of successive ascending series is increased by starting further below the threshold region, the 50 per cent threshold found is still lower. The opposite effect, for descending series, appears but not with statistical significance. Third, these effects do not occur when ascending and descending series are alternated, and thresholds are computed separately for each class. In this case both ascending and descending series yield the same 50 per cent threshold values.
The slope of the PR functions remained the same despite the changes in procedure we have investigated.
These results are in apparent contradiction with those of Thorne (9). In 1,299 pairs of threshold determinations on three subjects following the classical method of limits, he found that thresholds for ascending series were higher than for descending series.6 Thorne did not claim statistical significance for this conclusion. A simple explanation of the difference between Thorne’s results and our own is apparent: Thorne computed a “threshold” for each series, rather than for groups of series; this is the classical technique (5). In ascending series, his criterion for “arrival at the threshold” was a set of reports indicating that a specified stimulus brightness was seen two out of three times (two “yeses” to one “no”). In the descending series, however, he required that the subject fail two out of three (one “yes” to two “noes”). His ascending series thus yield a rough 67 per cent threshold, and his descending ones a 33 per cent threshold. “Overlap” in this case is to be expected without explanatory recourse to errors of habituation (12). The contradiction between Thorne’s findings and our own is only an apparent one.
Fernberger (3), in an intensive and complex investigation of procedural variations in measurements of the kinesthetic differential threshold, compared thresholds obtained by (a) the method of constant stimuli, (b) the method of limits, and (c) a series of computations based on the results of the method of constant stimuli series that were designed to predict the Method of Limits thresholds. He states, “The calculated thresholds in the direction of decrease by the method of just perceptible differences are constantly smaller.” This same finding occurs in his observed results, and is verified by the results reported here when method of limits thresholds taken for these data were compared with the 50 per cent thresholds obtained by the simple curve-fitting procedure. The greater differences found using classical techniques of determining the Method of Limits thresholds may hence be partially “artifactual.” On the other hand, Fernberger’s “thresholds in the direction of increase,” however obtained, are consistently higher than his “thresholds in the direction of decrease.” In any event the statistical model employed by Fernberger requires statistical independence.
The results reported here might be considered an artifact by the following reasoning: When a frequency of response curve is plotted for ascending series, the probability of a “yes” at higher brightness is determined on the assumption that responses to stimuli above the stopping point of the series would have been “yes.” Insofar as there is a finite probability of a “no” to stimuli brighter than those to which three “yeses” have just been made, this might tend to decrease the thresholds for ascending series. Similar reasoning might predict a higher threshold for descending series.
Analysis of the data shows that in 10. 8 per cent of the series in the ascending procedures, S said “no” after his first two “yeses”; and in 9.6 per cent of the series in the descending procedures, S said “yes” after his first two “noes.” So, with a criterion of three successive identical responses, the number of occasions on which a “no” followed the first three “yeses” in an ascending series, may approach a few per cent and introduce an underestimation of the occurrence of low frequency responses distributed over several points.
To determine whether this procedure of series termination introduced a bias into the thresholds obtained with the four procedures that did not involve alternating descending and ascending runs, 50 per cent thresholds for these procedures were computed on the assumption that all responses after the first two “noes” in a descending series would be “no,” and that all responses after the first two “yeses” in an ascending series would be “yes.” This resulted in average increases of 0.02 log mmL for thresholds based, respectively, upon the LA and SA procedures and an average increase of 0.02 log mmL for LD and SD. The argument that low ascending thresholds are artifacts of the procedure of threshold computation holds more strongly for the two-response criterion than for the three-response criterion used. Hence the failure to find differential effects of the change to a two-response criterion indicates that the method of threshold determination is not responsible for the findings.
Adaptation or summation processes as they are reported in the literature cannot account for our results. Light adaptation on descending series might, if sufficiently great, produce the effects found. But summation has not been observed over the time intervals which elapsed between successive stimuli (6), nor has anyone ever succeeded in demonstrating that measurable light adaptation can be produced by flashes as dim as those introduced in measurements of the threshold. The present data do not constitute evidence for such summation and adaptation effects, since no effect appears when ascending and descending series are alternated.
The “error of expectation” is a convenient generic term which has been employed in accounting for some data such as these. It is, perhaps, an apt label, but it is one that tells little of the conditions under which such findings can be made and less of the psychological laws that produce them. In the present experiment, the effect appears when all series are unidirectional and start with stimuli of a predetermined brightness; it is greater when these brightnesses are further from the range of varying response probability. The functional dependencies between values of PR and the values of these starting points have not been determined, nor are we yet in a position to relate the effect to other behavior.
Two possibilities may be considered. One emphasizes the events which terminate each series, and the second stresses the unidirectionality of series.
In conditioning, certain classes of events have been found to have the property of increasing the probability of the occurrence of events which have just preceded them. If the termination of a series is such an event, it would increase the probability of “yes” occurring at any point in an ascending series, and of a “no” in a descending one, and in this way produce the differences found between the ascending and descending series. More interesting, when alternating ascending and descending series are used, the effects should counterbalance one another, and leave the threshold unaltered.
The second possibility stems from the fact that in ascending series each stimulus is preceded by a less intense stimulus, and conversely in descending series. Serial dependencies of successive responses opposite to those found in other experimental investigations in the threshold range of intensities (1.5 to 3.4 log mmL) might account for these results (11). It remains to determine the conditions under which dependencies are negative rather than positive, and what relations, if any, these have to those of this experiment.
Six human subjects served in an experiment designed to evaluate the effects (a) of the direction of brightness change and (b) of the length of series, upon frequencies of response in methods where the brightnesses of successive stimuli were ordered according to intensity.
1. Mean 50 per cent thresholds varied significantly with shifts in the length and direction of series. Ascending series, taken alone, yielded lower thresholds and descending series higher ones. Starting a series further from the threshold, and hence lengthening it, increases the tendency for the ascending procedures to yield lower thresholds.
2. Individual differences in mean thresholds were found to be statistically significant.
3. The “standard deviation” of the PR functions did not vary with changes in the procedure.
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Department of Psychology Department of Psychology
Stanford University Northwestern University
Stanford, California Evanston, Illinois
* Received in the Editorial Office on August 7, 1953.
1The experiment reported in this paper was performed in the Department of Psychology, Indiana University, under ONR Contract N6onr-180, T. O. IV, Project 143-253, and with support from a grant by the Graduate School of Arts and Sciences.
2 At the time this experiment was designed and performed, McNemar’s note (8) had not appeared. In the present experiment, there is no reason not to assume that interactions are zero.
3 A setting on this instrument is accurate to approximately 0.02 log mmL. We are using the term “brightness” rather than “luminance” since the instrument is calibrated on the basis of the photopic visibility curve, and the low scale readings are derived by simple computation based on the transmittance of the filters.
4 This procedure does not assume that the PR function is analytically a cumulative probability function. Of course it is not. The fit is an empirical, i.e., purely descriptive, one, which makes it possible to characterize any given PR function by two numbers: a “mean” or the abscissa value for p = .50 and a “standard deviation” or slope constant (2). This procedure also introduces the assumption that reversal of response will rarely occur after the three consecutive “yeses” or “noes” which terminate a series. If such inversions occur, they will alter the frequencies of response at the upper or lower limbs of the function, and hence alter the fitted straight lines only slightly.
5 This finding is relevant to Crozier’s argument on the significance of this slope constant, s, in his notation.
6 An attempt to evaluate Thorne’s results is complicated by a typographical error in his Table 4. The mean difference between descending series and ascending series should be given as +.19, not -.19.