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Non-independence of successive responses in measurements of the visual threshold

1)1. The 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. Work is continuing at Harvard University under Contract N5ori-07639.William S. Verplanck, 3)2. Now at Harvard University.George H. Collier, 4)3. Now at Duke University. and John W. Cotton 5)4. Now at Northwestern University.

Historically, psychophysics has been concerned with two kinds of functions. The first kind, that with which we shall be concerned here, includes functions that relate the probability of a response (e.g., “yes,” “greater,” “equal”) to the spatial, temporal, and energy characteristics of stimuli; these are variously termed “psychometric,” “threshold,” or, as we shall call them here, “PR” (probability of response) functions. The second type includes those that relate certain parameters (e.g., the abscissa of inflexion or 50% threshold and the SD) of functions of the first type to some physical dimension, again, of the stimulus. An example, in vision, of the first type is the function relating log intensity of a visual stimulus to the probability of response (3, 18). Examples of the second type are the visibility (4, 31), DI/I (14, 16), and dark-adaptation curves (3, 13).

Operationally, then, the great majority of investigations, both basic and applied, in the sensory and perceptual area have been concerned with first obtaining functions of the first type, that is, with determining the frequency of occurrence of some response as a function of the physical properties of the stimulating situation. The quantitative values of the various parameters of those functions have been treated as isomorphic indicators of other processes, such as mental events (sensations), events in the nervous system (nerve fibre discharges or additions of “elements”), or physical events at the receptors (absorption of certain numbers of quanta of light by the retina). The sensory, neural, photochemical, or physical processes that have been postulated to generate the observed functions have been inferred from the functions themselves, and have seldom, if ever, been available to direct or collateral observation. Limits have been placed on such inference by information obtained in other disciplines. Physiological, biochemical, and histological evidence, often scanty, has served to keep in check the variety of inferences drawn. Thus, the histological structure of the retina has placed limitations on retinal models proposed in theories of visual acuity. In vision, such men as Hecht (15), Wald (30), Crozier (3), and Graham (9), have proposed theories concerning the physical, photochemical, or neural processes responsible for observed functions.

In repeated determinations of PR functions, it is found that the relative frequencies of a categorized response, such as “yes” or “no,” to a series of presentations of stimuli varying along a single dimension under fixed experimental conditions (7) converge stochastically to stable values. It has, moreover, usually been assumed that, for an intertrial interval greater than the “relaxation time” of the eye, approximately .1 sec.,2)5. A best estimate from the literature on temporal summation. successive trials yield a population of responses that are independent events (e.g., 6, 11, 18, 25, 26, 28). 6)6. The assumption of independence is seldom, if ever, made explicitly, but it is implicit in the statistical methods employed.Various distribution functions (phi-gamma, Poisson, log-normal) have been proposed as relating, empirically or theoretically, the probability of response to variations in the stimulus dimension. As with the second class of function, various psychological, physiological, and physical processes have been suggested as generating the distribution functions fitted to the data, e. g., Guilford’s fluctuating attention (10), Crozier’s fluctuating thresholds of the excitable elements of the central nervous system (3), Hecht’s quantal variability of the stimulus (18), and perhaps even Hull’s SOR (20).

One interesting example of a theory of the PR function is that of Hecht (18). He found that under his conditions of stimulation, the amount of energy necessary to produce a “yes (I saw it)” on 60% of all the occasions on which it was presented fell in the range 2.0 to 6.0 ´ 10 -10 ergs at the cornea, or, after correction for reflection, transmission, and absorption, 5 to 14 quanta at the retina. If it is assumed with Hecht that it is necessary for one quantum of light to be absorbed by each of 5 to 14 rods if a “yes” is to occur, then the relationship of response probability to stimulus intensity may be generated solely in terms of the quantal variability of the stimulus. On this hypothesis, Hecht derived a Poisson distribution function (predicting the absorption of n quanta by the rods) for the PR functions which he obtained experimentally. The theoretical curves appeared to fit his data very well. One obvious corollary of this theory is that, since by the basic physical theory, the probability of n quanta arriving at the retina on any trial is independent of what may have occurred on preceding trials, then, if the time since the preceding stimulation is greater than the “relaxation time,” or critical duration, the probability of a response is dependent solely on the delivery, by the stimulating flash, of the appropriate number of quanta, and is, hence, independent of preceding responses.

As we have seen, the generation of a threshold function involves the repeated presentation of stimuli, and the measurement of probabilities of response to them. Trials are defined and intertrial intervals as well. The dependent variable, relative frequency, or probability of response is one which has been employed by many behavior theorists. Operationally, a “threshold measurement” is not unlike the kind of experiment performed in studies of learning, performance, and endogenous behavior (1, 2, 5, 8, 21, 22, 23, 27). In view of this similarity, it seemed likely to the writers that many previous formulations must necessarily have neglected the operation of a number of variables governing behavior, and thus must be dependent for their verification on the use of a relatively limited set of values of these variables. Hence, too, such theories may be incomplete.

With this point of view, we have been concerned with the investigation of behavioral variables, such as the number of trials, intertrial intervals, and “reinforcing stimuli” on the threshold function. This paper reports one of a series of such investigations.

The first indication of the operation of behavioral variables in threshold measurements appeared in some exploratory investigations that yielded data suggesting that statistical sequential dependence of successive responses may be found with intertrial intervals greater than the critical duration, and beyond those over which temporal summation occurs. The assumption of independence of successive responses is involved in all attempts to fit theoretically the psychophysical function by the normal probability integral, by its phi-gamma modification, or by the Poisson function, and therefore, to all theories of receptor process based on such fits. Hartline (12), in as yet unpublished research, found that his assumption holds for the elaboration of single nerve impulses in single fibre preparations of the Limulus eye, when these are stimulated some 300 successive times by a stimulus of fixed and low intensity at invariant stimulus intervals of 5 sec. or greater. At intervals of from 2 to 5 sec., the occurrence of a response may reduce the probability that a second flash would yield a response.

The present experiment is designed to investigate systematically the statistical independence of responses to successive visual stimulations of the human, following a procedure modeled after that of Hartline.

Apparatus and Procedure

Apparatus.—The apparatus used in this experiment was a Model III Hecht-Shlaer Adaptometer (17). This instrument presents binocularly a circular stimulus area subtending 3° at the retina, that is presented 7 ° below a red fixation point. The viewing distance is held at 25 cm. by a headrest.

A pendulum shutter provides for the presentation of the illuminated stimulus area to S for a period of .2 sec. The wavelength distribution of the flash is determined by a glass filter with maximum transmittance at 480 mm. Both color temperature and intensity of the light source are held constant by the use of a fixed filament current. The luminance of the stimulus patch is continuously variable from 1. 5 log mmL to 5.4 log mmL by means of a neutral optical wedge and an auxiliary neutral filter. Calibration is maintained photometrically. As the instrument was used in this experiment, flashes could be presented either at fixed mechanically determined intertrial intervals, or whenever S pressed a key. The apparatus was housed in a sound-deadened and light-tight room.

Subjects.—Sixteen paid Indiana University undergraduates, 12 men and 4 women, served as Ss. All met or exceeded Naval visual standards as tested by the Orthorater (20/20 visual acuity O. D., O. S.; no phorias; “color normal”). All were previously inexperienced in visual experimentation, so that complete control of their experimental histories was possible.7)7. Despite the frequent use of highly trained Ss in visual-threshold research, we chose to employ inexperienced persons. In view of Hecht and Shlaer’s report (17) and of our own experience that a naive S will begin to make highly reliable judgments following a short period of training on threshold measurements, no loss of consistency of report need be expected. Certain advantages are attached to the use of such Ss. When a response is suspected to be a function of pervious behavior in a situation, it is desirable to have complete control over, and records of, such behavior, which is possible only with Ss untrained prior to the beginning of the experiment. Further, Ss are less likely to divine the purpose of an experiment if their previous experience is limited, and spurious results, attributable to “suggestion,” may be precluded.

Our Ss remained entirely uniformed of the purpose, procedures, and design of the experiment. They were instructed to maintain fixation, and to say “yes” if they saw a flash of light when the shutter operated, and “no” if they did not. When the experimental periods began, each S had spend 30 min. in dark-adaptation goggles and 10 min. in complete darkness. No special treatments distinguished between training, control, and experimental days, or among procedures of stimulation, except as specified later.

Procedures.—Two procedures of stimulation were employed. The first follows the method of limits, in which each series, ascending or descending, consisted of flashes differing successively in steps of .10 log units (LU), presented at 5-sec. intervals. Each series was terminated when three successive responses (“yes” in ascending, “no” in descending series) were made, and the next series, in the opposite direction, began with luminance .20 LU higher or lower, respectively, than the terminal luminance of the preceding series. The second procedure is the single luminance method. Since the problem to be investigated in this experiment is the stochastic independence of successive responses, it is necessary to eliminate the effect of changing intensities of stimulation on the probability of response. The method, therefore, consists of the presentation to S of a series of 300 flashes of constant luminance. The luminance is selected so that the probability of a “yes” in response to it by each S will be approximately .50. These flashes were presented in two ways. On some days, they automatically followed one another at 5-sec. intervals (A). On others, S was instructed to present a flash to himself whenever he was “ready’ for it, by pressing a key (S).8)8 This procedure aimed at duplicating the method used by Hecht in his experiments on the quantum theory of the threshold (18). In the latter procedure, the interstimulus interval was determined by S, who presented stimuli to himself at “optimal” times. The Ss were allowed to take a 30-sec. rest when they wished. Insofar as possible, all sessions were run at the same time of day, on alternate days.

Experimental design.—Five preliminary sessions were given. Each such session consisted of ten ascending and ten descending series of the method of limits. From the data of each of the last four sessions, the luminance yielding a relative frequency of “yes” of .50 was determined for each S. The mean of these values was taken as the best estimate of S‘s 50% threshold. These preliminary sessions provided each S with a standardized practice, and ensured that Ss yielded results indicating that they followed the instructions, and that were comparable to the results usually obtained in threshold measurements. Four experimental sessions followed. Each began and ended with six ascending and six descending series on the method of limits. The middle period of these sessions was given over to the single-luminance method, in which each S was presented with 300 flashes; the luminance of each of these was set at the 50% threshold measured in the last four preliminary sessions. On two days, the intertrial interval was 5 sec. (A); on the other two days, the intertrial interval was controlled by S (S). The 16 Ss were divided into four groups and each group was assigned one of four combinations of the two intertrial intervals over the four days of experimentation. The first group followed the order AASS, the second SSAA, the third ASAS, and the fourth SASA.

One control session was run for six randomly selected Ss. This day differed from the experimental days only in that approximately 60 “catch” trials were introduced into the series of 300 single-luminance trials. Since it might well be argued that any results might be due to “false” responses (“yeses” not contingent on stimulation), and since the statistical treatment forbade that these be introduced on the experimental days, this final day was considered necessary. All other conditions were the same as on preceding days and Ss were not informed of the change in procedure.

Results

Method of limits: Thresholds.—The relative frequency of “yes” responses was plotted against log luminance (log mmL) on normal-probability paper for each of the last four preliminary sessions on the method of limits. A straight line was visually fitted to these plots and the threshold (luminance yielding 50% “yes” responses) to the nearest tenth of a log unit was determined by interpolation.9)9. We do not assume that the relative frequency of “yes” versus log luminance curves employed are normal-probability ogives; this type of plot was used because it gives a convenient and reproducible graphical method for interpolating the 50% threshold values. It works. For each S a mean threshold for the four days was determined. This luminance value is presented in the second column of Table 1. The average of the mean 50% thresholds for all Ss was 2. 68 log mmL, with an SD of .30mmL.

Method of limits and single-luminance frequencies of response.—Table 1 presents the relative frequency of “yes” responses given by each S in each of the four experimental sessions. The mean of the distribution of differences between the .50 (predicted from the mean threshold values) and these relative frequencies was .065 and the sigma .182. A t test of the hypothesis that the population mean of this difference distribution is zero does not permit its rejection (p > 5% with t = 1.57 for 15 df). Within the limits of these experimental procedures, and assuming a normal distribution of thresholds, the 50% threshold, determined in the method of limits sessions, yielded, when employed in the single luminance method, on the average, 50% “yes” responses.10)10 This test is valid if the population sampled is normally distributed. A chi-square test of this assumption did not lead to its rejection here (p > 5% with X2= 3.92 for 2 df). Because of the small number of df, the probability of a beta-type error is large; so that we cannot accept the hypothesis of normality with any assurance. That the values sampled are means lends weight to the hypothesis by the Central Limit Theorem.

False responses; the control session.—The last column of Table 1 presents the ratio of “yeses” to the total number of blank trials given. In two cases, this approaches 5%; in the other four, it is less. This proportion of “false” responses is small and cannot seriously affect our conclusions, although, to be sure, one would prefer that statistical necessity did not make it impossible to evaluate this basal PR in each experimental session. This resting rate of response is somewhat analogous to an operant level.

Frequencies of response as a function of method of stimulus presentation.—The relative frequencies of “yes” responses for each S were averaged separately for the days when the Ss stimulated themselves (S), and when they were stimulated automatically (A). Two corresponding distributions of differences from .50 were derived. The mean and SD of the A distribution were, respectively, .048 and .179, and those of the S distribution were 0.078 and .208. When the variances of the two distributions were compared, the null hypothesis could not be rejected (p > 5% with F = 1.34 for 15, 15 df). To test the homogeneity of means, a distribution of differences between the mean relative frequency of “yes” for the A and S conditions was obtained. The mean difference was equal to .033 and the SD .169. A test of the hypothesis that this mean differed significantly from zero did not lead to its rejection (p > 5% with t = 0.79 for 15 df). Once again, our conclusions with respect to homogeneity of variance and homogeneity of mean relative frequencies are subject to the restriction that the tests of the normality of the distributions are not very good with so few df available.

Summarizing, within the limits of the present experimental procedures, and the assumption of normal distributions, the method of limits trials yielded 50% thresholds which on the average gave 50% “yes” responses on the single-luminance trials. The two intertrial procedures, self-stimulation and automatic stimulation at 5-sec. intervals, did not alter significantly the percentages of “yeses” given to the fixed luminance nor did one condition result in a more variable percentage of “yeses.”

Sequential analysis: The serial correlation from trial to trial (lag 1).—In the determination of probability of response functions, a sequence of trials is given; in our case, Ss respond with “yes (I saw it)” or “no (I did not see it)” to 300 successive presentations of a stimulus patch of invariant brightness. This procedure yields a time-ordered sequence of responses such as Yes, Yes, No, No, No, Yes, No, Yes, Yes, Yes, . . . From the relative frequency of “yeses” to all 300 stimulations, an estimate of the probability of a “yes” to any given stimulation, under these conditions, can be obtained.

It has been typically assumed, implicitly or explicitly, that the successive responses, YYNNY . . ., are independent of one another, that is, that the probability of obtaining a “yes” (or a “no”) on any trial does not depend on the response given to the preceding stimulus in the series. From this, it follows that the frequency of runs of responses (a run is defined as a sequence of identical responses preceded and followed by a different response) of various lengths will occur according to certain distribution laws. If significantly more runs occur than are predicted by these laws (i.e., if S alternates YNYNYN . . .), or fewer than predicted (YYYNNN . . .), then the assumption of independence must be rejected, and the data exhibit serial dependencies.

In order to test the hypothesis that successive responses are independent of one another, we have employed a nonparametric serial-correlation test described by Hoel (19).11)

11 This test is an approximation of one developed by Wald and Wolfowitz (29). It is based on the cross product term

R = Sxixi – 1.

R = 300 – 4(r),
E ( R ) = ( S12 – 300)/299,
sR = 300 + (S14 – 1196 S12 + 89400)/ 89102 – E2 (R),
where: r — number of runs, S1 — number of “yeses” minus the number of “noes.”

The serial coefficient of correlation, R, in this test, as we have used it, is a function of the sample size, the number of “yeses” in the total number of responses, and the number of “yes” runs. R can vary between zero and plus infinity. Since the number of “yeses” varies from session to session, and since n is here relatively small for the computation of serial correlations, R‘s obtained from different Ss on different days are not directly comparable. We have, therefore, used the critical ratio (CRR) for their evaluation and comparison. This statistic has an n (0, 1) distribution. A significant positive CRRindicates too few runs and hence a tendency to repeat a response just made; a significant negative CRR indicates too many runs and hence a tendency to alternate. The results of this analysis are presented in Table 1, which presents both the number of runs and the CRR for each experimental session.

Of the sixty-four CRR‘s (16 Ss over four days) calculated, only seven failed to reach the 5% level of significance. The distribution of CRR‘s, averaged over all days for each S, gave a mean of 5.87 and a sigma of 2.25. The probability of this distribution of CRR ‘s12)12 Again, this test assumes a normal distribution of CRR‘s. Again, there are insufficient degrees of freedom to make a reliable test of this hypothesis for the present data. however, evidence from other experiments indicates that the distribution of CRR‘s is normal, with a mean different from zero and a sigma different from one in cases where significant serial correlation is present. A nonparametric sign test of the same hypothesis on the present data yields a probability between 5 and 10%. on the assumption of M = 0 and s = 1, is very much less than 1% (p < 1% with X2 = 631.55 for 15 df) and (p < 1% with X2 = ns2/s2 = 33.75 for 15 df). Serial correlation is significantly present.

A comparison between the automatic- and self-stimulation conditions was made by subtracting the average CRR obtained with automatic stimulation at 5-sec. intervals from the average CRR obtained with self-stimulation. The mean of this distribution was 1.22, and the sigma 2.33. A test of the hypothesis that the mean difference was zero could not be rejected (p > 5% with t = 2.03 for 15 df).

A test of the homogeneity of variance between the distribution of the mean CRR‘s (over two days for each S) for self- and automatic-stimulation conditions again does not permit us to reject the null hypothesis (p > 5% for F = 1.88 with 15, 15 df). For further analyses the two sets of data were combined.

We may conclude that there is a significant serial correlation under both experimental conditions, and that there is no significant difference between the two conditions in the degree of serial correlation, as measured by the critical-ratio test. All but one of the CRR‘s are positive, that is, runs are longer than expected on the hypothesis of a random sequence of responses. Subjects tend to repeat responses that they have just made. Serial dependencies are exhibited.

Sequential analysis: Serial correlations at lags greater than one.—The demonstration of dependent probabilities between successive responses may suggest cyclical or long-range effects of some sort. These should reveal themselves in the form of systematic changes in the index of serial correlation as the temporal interval between responses correlated increases. In order to investigate this possibility, serial-correlation coefficients at lags greater than one may be computed. That is, the correlation of the first response with the fourth, the fourth with the seventh, the seventh with the tenth, and so on (Lag 3 ) may be determined. These computations have been made on the present data at Lags 3, 5, 7, 9, and 20. If cyclic or regular changes in response probability occur, then the CR of the serial correlation coefficients should show corresponding changes. In Figure 1 are presented the mean critical ratios (over days and Ss) plotted against lag. Up to Lag 20, the critical ratio appears to be a monotonic decreasing function of the lag. Long-range effects appear, but this analysis shows no evidence of orderly cycles.

Examination of individual records, made by plotting subsets of a single S‘s Data on a given day on binomial-probability paper (24)13)13 This statistical tool exhibits serial dependencies in a single session’s data rapidly. We used it extensively in preliminary analyses. also reveals no effects at lags greater than 20. We do not find statistically significant lack of independence beyond about Lag 11, which, under our condition “automatic stimulation,” covers about 1 min.

Discussion

Successive responses were found to be dependent upon one another under all conditions of this experiment. Runs of “yeses” and “noes” of length greater than expected on the hypothesis of independence occur. If S reports “yes” on one flash, he is more likely to report “yes” on the next.

Furthermore, no statistically significant differences can be exhibited between either the percentage of the stimuli seen, or the dependent probabilities of successive responses when the stimulations are presented automatically, at a fixed 5-sec. interval, or when they are presented to S by himself when his is “ready.” Further work is necessary before intertrial interval can be considered to be an irrelevant variable, since our Ss may have stimulated themselves at a rate close to the automatic one.

Statistical nonindependence may be produced in either of two ways. First, the probability of a particular response may depend upon the identity of the previous response. Second, some as yet unidentified “state” variables may alter the probabilities of responses on trials close to one another in time—that is, there may be trends cyclical or otherwise, in a time-ordered series, although its members are not dependent upon one another. Four categories of variables suggest themselves as possibly responsible for such effects: (a) stimulus variables; (b) nonstimulus variables independent of preceding responses, of response history, and of preceding stimulation; (c) nonstimulus variables independent of preceding stimulation; (d) nonstimulus variables dependent on preceding response history.

Category (a) includes variables such as fluctuations in the intensity of the stimulating light, continuous versus discontinuous presentation of the fixation point, concurrent stimulation (shutter noises, etc.), and so on. Fluctuations of the intensity of the stimulating light were controlled by our calibration and regulatory apparatus and cannot be considered a contributing variable. Extraneous stimulation was controlled by the use of a sound-deadened room, and shutter-masking noises.

Category (b) includes such variables as physiological state (blood pressure, etc.), changes in pupillary diameter, changes in the retinal site of stimulation owing to fluctuations in fixation, gross motivational fluctuations that are not a function of the immediate stimulating conditions, changes in the criterion of the S, and so on.

Category (c) includes such variables as adaptation, summation, changes in pupillary diameter produced by preceding stimulation, fluctuations in the place of stimulation dependent on preceding stimulation, etc. Adaptation is probably not a contributing variable, since its effect would be to reduce sensitivity, thereby decreasing the probability of response following a supraliminal flash, and so yielding negative serial correlations. Retinal summation over a five-sec. intertrial interval has never been reported to our knowledge.

Category (d) includes such variables as conditioning (i.e., an increment to the probability of response produced by the occurrence of some reinforcing event after a preceding response), peripheral or central “orientation” behavior dependent upon preceding response history, such as might result from preliminary training on the method of limits, motivational factors that are a function of the immediately preceding response history, and so on. At the present time, reports of experiments on the effects of many of these variables are in preparation. Other variables are still under investigation.

In any event, these results have a number of implications, not only for theories about the visual (and other) thresholds, but also for all procedures which investigate the sensitivity of the intact human. Most particularly, theories of the threshold which either assume or predict independence may need revision, and conversely, some account of the variables which produce nonindependence must be made. The results raise other questions as well: What, for example, might be done to keep S at peak responsiveness?

Summary and Conclusions

This experiment was performed in order to determine whether the successive responses of S given in measurements of his visual threshold are statistically independent of one another. This is implicit in several theories of sensory thresholds. Sixteen Ss were presented, on four successive days, with 300 consecutive stimuli of a luminance which had been found previously to be reported 50% of the time. On two days these stimuli were presented automatically at 5-sec. intervals. On the other two, S presented them to himself. Critical ratios of serial-correlation coefficients were computed on the time-ordered sequences of responses at Lags 1, 3, 5, 7, 9, 11, and 20.

1. Under the conditions of this experiment the mean 50% absolute visual threshold for each subject elicited, on the average, 50% frequencies of responses on immediately subsequent days.

2. When S is presented with a discrete, dim flash of light, his response to it will depend not only upon the luminance of the flash, but also, among other things, on how he has responded to preceding flashes of light of the same luminance. Each response is dependent upon previous response, or perhaps both are dependent upon a third variable which varies in time.

3. This nonindependence is exhibited with statistical significance when as many as 10 responses intervene between the correlated responses. The value ofCRR seems to be a monotonically decreasing function of the lag.

4. No statistically significant differences appeared between the data obtained under the two conditions of stimulation.

(Received for priority publication, May 19, 1952)

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Footnotes   [ + ]

1. 1. The 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. Work is continuing at Harvard University under Contract N5ori-07639.
2. 5. A best estimate from the literature on temporal summation.
3. 2. Now at Harvard University.
4. 3. Now at Duke University.
5. 4. Now at Northwestern University.
6. 6. The assumption of independence is seldom, if ever, made explicitly, but it is implicit in the statistical methods employed.
7. 7. Despite the frequent use of highly trained Ss in visual-threshold research, we chose to employ inexperienced persons. In view of Hecht and Shlaer’s report (17) and of our own experience that a naive S will begin to make highly reliable judgments following a short period of training on threshold measurements, no loss of consistency of report need be expected. Certain advantages are attached to the use of such Ss. When a response is suspected to be a function of pervious behavior in a situation, it is desirable to have complete control over, and records of, such behavior, which is possible only with Ss untrained prior to the beginning of the experiment. Further, Ss are less likely to divine the purpose of an experiment if their previous experience is limited, and spurious results, attributable to “suggestion,” may be precluded.
8. 8 This procedure aimed at duplicating the method used by Hecht in his experiments on the quantum theory of the threshold (18).
9. 9. We do not assume that the relative frequency of “yes” versus log luminance curves employed are normal-probability ogives; this type of plot was used because it gives a convenient and reproducible graphical method for interpolating the 50% threshold values. It works.
10. 10 This test is valid if the population sampled is normally distributed. A chi-square test of this assumption did not lead to its rejection here (p > 5% with X2= 3.92 for 2 df). Because of the small number of df, the probability of a beta-type error is large; so that we cannot accept the hypothesis of normality with any assurance. That the values sampled are means lends weight to the hypothesis by the Central Limit Theorem.
11.

11 This test is an approximation of one developed by Wald and Wolfowitz (29). It is based on the cross product term

R = Sxixi – 1.

R = 300 – 4(r),
E ( R ) = ( S12 – 300)/299,
sR = 300 + (S14 – 1196 S12 + 89400)/ 89102 – E2 (R),
where: r — number of runs, S1 — number of “yeses” minus the number of “noes.”

12. 12 Again, this test assumes a normal distribution of CRR‘s. Again, there are insufficient degrees of freedom to make a reliable test of this hypothesis for the present data. however, evidence from other experiments indicates that the distribution of CRR‘s is normal, with a mean different from zero and a sigma different from one in cases where significant serial correlation is present. A nonparametric sign test of the same hypothesis on the present data yields a probability between 5 and 10%.
13. 13 This statistical tool exhibits serial dependencies in a single session’s data rapidly. We used it extensively in preliminary analyses.
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