William S. Verplanck,2 John W. Cotton,3 and George H. Collier4
In a recent paper (4) the authors presented the results of an investigation of the statistical properties of responses to repeated visual stimulations. On each of four days, 16 human Ss were stimulated 300 successive times by a patch of light of fixed duration, dimensions, position, and brightness. The data were analyzed for the statistical independence of successive responses by means of a serial-correlation technique, and sequential dependencies were found. Hence, assumptions that successive responses are randomly arranged and statistically independent (often implicit in psychophysical measurements and in theories of the visual threshold) cannot be made for any specific situation in the absence of a strict empirical test. The probabilities of a “yes” following a “yes” and of a “no” following a “no” are greater than would be expected from the values estimated from the over-all relative frequencies of “yeses” or “noes” in the whole series of trials.
In the above experiment Ss were given five days of preliminary training on a method-of limits procedure that presents stimuli in intensity-ordered series. A brief method-of-limits session also preceded and followed the single-brightness series on the four experimental days. This procedure necessarily leads Ss to give long groups or sequences of “yeses” to stimuli of the higher intensities followed by similar groups or sequences of “no” responses to those of lower intensity. Some might infer that such repetitions of “yes” and “no” in series produce a tendency to give similar sequences of “yeses” and “noes” when all stimuli are of the same brightness.
To test this possibility, we have employed two methods of threshold determination in the preliminary training and in the short sessions which precede and follow single-brightness stimulations. The first method is the same as that used previously; it necessarily produces sequences of “yeses” and “noes.” The second method employs a series of stimuli randomly varied in brightness so that few, if any, long sequences of the same responses may be expected to occur. Evidence of nonindependence is then sought in the responses to 300 single-brightness stimulations of two groups of Ss, of which one has been trained according to the first procedure, and the other according to the second.
It may be argued that a “set” for, or “expectancy” of, stimuli of graded intensity will be set up in members of the first group and hence that they will tend to repeat responses and so exhibit serial dependencies. No such expectancy should be set up by the training procedure in members of the second group, and hence they should not exhibit serial dependencies.
Apparatus and Procedure
Apparatus.–The apparatus used in this experiment was the Model III Hecht-Shlaer Adaptometer (2), modified as described in a preceding paper (4).
Subjects.–Six paid Indiana University male students, two graduates and four undergraduates, served as Ss. All met or exceeded Service visual standards as tested by the Orthorater. All were previously inexperienced in visual experimentation and were assigned at random to one or the other of the two experimental groups.
Dark adaptation and instructions.–The Ss were run fully dark-adapted. The dark-adaptation procedure and the instructions in this experiment were the same as those already described (4).
Procedure.–The experimental procedure may be outlined as follows: A one-day session of threshold determination served as a training period and preceded four days in which Ss of each group were given repeated single-brightness stimulations in order to test for randomness of response. The treatments of two groups of three Ss each differed in the methods of stimulation followed during the training session and in the procedures that preceded and followed the single-brightness series on the other four days.
In the preliminary session (Day A) members of Group I received 20 series of intensity-ordered stimulations as described elsewhere (5). These gave each Sapproximately 225-250 stimulations. The members of Group II received 240 randomized stimulations with brightnesses ranging from 1.9 to 3.0 log FFL in .1 log-unit steps. The value of the 50% threshold obtained for each S on this first day was used as the single brightness employed in subsequent sessions. These values are presented in Table 1.
On Days 1 and 2 the members of Group I received three pairs of descending and ascending series of intensity-ordered stimulations, followed by 300 stimulations at the 50% threshold. The experimental day ended with six more series of intensity-ordered stimulation. On Days 1 and 2, members of Group II received 60 stimulations with randomized brightness before 300 single-brightness stimulations at their 50% thresholds. Another 60 randomized stimulations followed the single-brightness series. A comparison of the incidence of sequences on Days 1 and 2 for the two groups should indicate the relative effects of the two training procedures.
The procedures followed on Days 3 and 4 were exactly like those on Days 1 and 2, except that the treatments of the two groups were interchanged. Group I received the random series of brightnesses before and after the single-brightness tests, and Group II received the six intensity-ordered series before and after single-brightness tests.
Table 1 presents the relative frequencies of “yes” for each series of single-brightness stimulations. The mean relative frequency (PR) of “yes” is .49, and the SD of these is .19. These values are not significantly different from those reported earlier in a larger group of
50% Threshold, Relative Frequencies of “Yes” (PR), and CRR‘s by Subjects and Days
(300 presentations in each cell)
Ss (mean relative frequency = .55; SD = .25) and, as before, are not significantly different from the .50 predicted from the first day’s results.
The analysis of results to determine whether or not the responses within the blocks of single-brightness stimulations are statistically independent of one another was performed by the test for serial correlations described by Hoel (3). The sue of this method has been discussed elsewhere (4). Table 1 presents the critical ratios (CRR) of the serial correlation coefficient for each series of single-brightness stimulations. Of the six CRR‘s of the Ss of Group I on Days 1 and 2, three exceeded the 5% level of significance and two exceed the 1% level. Of the six comparable tests for Group II, four exceeded the 1% level and two failed to reach the 5% level. Both groups demonstrated nonrandom arrangements during Days 1 and 2, although Group II had received training calculated to minimize the possibility of development of a response set towards repetition of responses from trial to trial. On Days 3 and 4 when the antecedent stimulation procedures are reversed, five of the six serial-correlation values of Group I are significant at the 1% level, with the other exceeding the 5% level. The groups do not differ, but there is the suggestion of a trend towards increased serial correlations with successive days.
The possibility of changes within a group attendant upon the change of procedure beginning on Day 3 must be considered. The number of significant correlations found in each of the two-day periods remained constant for members of Group I. Group II shows an increase from four significant values on Days 1 and 2 to six on Days 3 and 4. These findings suggest that a change from intensity-ordered to random training procedures in Group I did not produce independence of successive responses while a change from random to intensity-ordered training procedures may have increased the degree of dependence of successive responses. Thus the hypotheses with which we are concerned may seem partially verified; although an appropriate history of responses is not at all necessary for the appearance of the sequential effect, it may increase it in magnitude. A further analysis was performed.
The experimental design employed is such that a crossover analysis of variance (1) may be made on the data. This analysis closely resembles the latin square, and is suitable to the present data since our number of treatments is two. A test of training effects was therefore performed using a modified crossover-design analysis of variance. The data were first grouped into blocks of two days each. Following the crossover-design method, the data were then broken down into sums of squares for subjects, blocks of days, treatments, and error. A further breakdown of the sum of squares for Ss into a sum of squares for groups (differing in the order of treatments) and a sum of squares for Ss within groups was made. Thus the possibilities that training and the history of training affected the data could both be tested. Analysis-of-variance tables based on these sums of squares are presented in Table 2 for both the relative frequencies and the CR ‘s. Only the difference between Ss’ frequencies of response proved significant. Groups, treatments, and blocks of days apparently have no effect upon either the relative frequency or the CR‘s. One qualification must be stated. In our experimental design, the training procedure during the preliminary sessions and the temporal order of treatments are combined into one estimate of variance, that for groups. If both variables were effective, but operated in opposite directions, the same results might be obtained. We must conclude that either the confounded variables have no effect or that their effects are equal and opposite. The latter is unlikely.
Analyses of Variance of Relative Frequency of “Yes” Responses (FPR) and of CRP‘s
A possibility that must be considered is that nonindependence of responses may arise as a result of slow temporal trends in the relative frequency of response (PR). If, for example, S gives almost all “yeses” during the first portion of the session and almost all “noes” during the second portion, a significantCRR will result. This follows from the facts that the serial correlation test is sensitive to trend, as well as to statistical nonindependence of successive responses, and that it does not distinguish between the two effects.
This possibility was tested by the use of Pearsonian correlation. The number of “yeses” to each successive set of ten stimulations was correlated with the ordinal position of the set in the series of 300. Three Ss yielded low (about .30) negative correlations that net the 5% level of significance, between PR and ordinal position; a small decrease in PR occurred throughout the experimental period. The other three Ss showed near-zero correlations that were not significant. However, highly significant sequential effects occurred not only in the data of Ss showing this decrement, but also in those of the Ss who did not.5 A trend in time may be operating, but it is not adequate to account for the nonrandomness observed.
The results of this experiment lend little credence to the hypothesis that the lack of randomness found in responses to invariant visual stimuli in our previous experiment (4) can be attributed to the training procedures used in that experiment. In the present study, nonrandomness characterized by long series of similar responses (“yeses” or “noes”) was obtained following training either by randomly ordered or intensity-ordered stimuli.
A second factor, that of trends within sessions, may be present in our experimental data. Again, the absence of sizable correlations between the number of “yeses” per group and the ordinal position of the group, shows that such trends cannot account for the nonrandomness found. This nonrandomness is evidence for nonindependence of successive responses.
This experiment consisted of five experimental sessions, one devoted to an original threshold determination for each of six Ss and the other four devoted to 300 successive single-brightness stimulation periods per session. The six Ss were divided into two groups of three Ss each. Group I was given intensity-ordered stimulations in all threshold determinations on the first three days of the experiment; Group II was given randomized stimulus brightnesses at the comparable times. On the last two days of the experiment the procedures for Groups I and II were reversed.
Despite the differences in training procedures for the two groups, both groups exhibited nonindependence of responses in the single-brightness series.
Some evidence was found for a linear trend toward decreased frequency of seeing late in each experimental session. Although this factor alone might produce nonrandomness of response, there is strong evidence that other mechanisms are operative.
1. Cochran, W. G., & Cox, G. M. Experimental Designs. New York: Wiley, 1950.
2. Hecht, S., & Shlaer, S. An adaptometer for measuring human dark adaptation. J. Opt. Soc. Amer., 1938, 28, 269-275.
3. Hoel, P. G. Introduction to Mathematical Statistics. New York: Wiley, 1947.
4. Verplanck, W. S., Collier, G. H., & Cotton, J. W. Nonindependence of successive responses in measurements of the visual threshold. Journal of Experimental Psychology, 1952, 42, 273-282.
5. Verplanck, W. S., & Cotton, J. W. The dependence of frequencies of seeing on procedural variables: I. Direction and length of series of intensity-ordered stimuli. Journal of General Psychology, in press.
(Received November 17, 1952)
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 ONR Contract N5ori-07639.
2 Now at Harvard University
3 Now at Northwestern University.
4 Now at Duke University.
5 In other, as yet unpublished, data we have evaluated CRR on blocks of 50 trials–a set short enough so that no trend can be observed; CRR‘s are still significant statistically, although less so than in the present data, as they must be owing to the decreased sample size.