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A developmental study of sequence structure in binary prediction
JOURNAL
OF EXPERIMENTAL
CHILD
PSYCHOLOGY
7,255-2&t
0969)
A Developmental Study of Sequence in Binary Prediction1 NANCY
Structure
A. MYERS AND JEROME L. MYERS University
of Massachusetts
Sixteen groupsof 20 Ss eachwere run individually in a binary prediction situation with each alternative occurring equally often. Four age groups were employed: Ss were either in kindergarten, second,fourth, or sixth grades.For half the 8s at each age level the events occurredin runs of either 1 or 5; the other Se observedeventswhich occurredin runs of either 4 or 5. The probability of occurrenceof the long run (/?I wasalsomanipulated: for half the Ss receiving each set of run lengthsthere were equal numbersof short and long runs (p = .5); for the other 8s p = .2. The results indicated that all groups showed some sensitivity to run length characteristics.However, errors of all types decreasedasage increased,and 45 sequences were followed more accurately than l-5 sequences. The results were discussed in terms of their implicationsfor a generalmodelof binary choice behavior.
The binary prediction paradigm requires the S to predict which of two events will occur on each of a series of trials. Recent developments in the binary prediction literature have been in the direction away from noncontingent probabilistic sequences and toward the study of more carefully controlled sequential structures. With adults, a wide variety of both partially and completely learnable sequences have been employed in attempts to isolate the effective stimuli and response tendencies in binary prediction (e.g., Gambino and Myers, 1967; Restle, 1966). With children, too, there is increasing evidence of some sensitivity to at least some characteristics of sequences.Thus, Craig and Myers’ (1963) kindergarten Ss alternated more with a 60:40 than with an 80:20 sequence, and Bogartz’ (1966) 4- and 5-year-olds followed single alternations quite well, at least in early trials, and were obviously affected by changes in
sequential structure. Jones (1967) found that both nursery and kindergarten 8s responses reflected the pattern of highly discriminable event sequences.She exposed Ss to sequenceswith event repetition probabilities ‘This research was supported by funds from a University of Massachusetts Faculty Research Grant and NIMH Grant MH-O3803-06 and NSF Grant GS-386. The authors thank the Research Computing Center. University of Massachusetts fol assistance in analyzing data. 255
256
MYERS
AND
MYERS
of either .l or .9 for 100 trials .and for the second 100 trials transferred half the Ss to the opposite event repetition probability, while continuing the others at the same repetition probability level. When the repetition probability level changed, responses showed only limited dependency upon the new event sequence. However, kindergarten Ss followed any sequence better than nursery Ss and both age groups followed predominately repeating sequences more accurately than predominately alternating sequences. The present study was designed to explore further those characteristics of event sequences to which children may respond. In particular, we were interested in sensitivity to run length, because Myers, Butler, and Olson (in press) had observed such sensitivity in adults. Following the Myers et al. procedure, the events always appeared in runs of two standard lengths, either 4 and 5 or 1 and 5. For some Ss the two run lengths occurred equally often; for others, the shorter run length occurred more frequently. In addition, four age levels were employed, to permit evaluation of developmental aspects of performance. METHOD Apparatus. The event box consisted of two 11/2 v. red event lights mounted in parallel on the front of an 8- X 12- X a-inch grey box, one 3 inches from the left, the other 3 inches from the right edge. Below each light was a 31/2- X $&inch slot through which one of 200 3- X 2-inch playing cards, blank on both sides, could be placed to make a response. A grey panel attached to the back of the event box concealed the control unit from S. The control unit contained a buzzer that was used to indicate when choices should be made. One switch on the control unit sounded the buzzer and two other switches turned on the event lights. Design. Ss were randomly assigned to one of sixteen experimental groups differing with respect to age and two characteristics of the event sequences employed, in a 4 X 2 X 2 factorial design. There were 20 Ss in each group, half males and half females. Four age groups were run: Ss were either in kindergarten, second, fourth, or sixth grades. For half the Ss at each age level the events occurred in runs of either one or five ; the other Ss observed events which occurred in runs of either four or five. The other aspect of event sequences which was manipulated was @, the probability of occurrence of the long run. For half the Ss receiving each set of run lengths, ~3 was 5, i.e., there were equal numbers of short and long runs. For the other Ss, fi was .2, and thus short runs predominated. Event sequences were generated in two blocks of approximately I00 trials each, for each of four run length+ combinations, with the one additional requirement that each light occur equally often.
SEQUENCE
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257
Procedure. The Ss were tested one at a time in what they were told was a “guessing game.” After E signaled the beginning of a trial with the buzzer, S predicted which one of the two lights would come on by taking a card from the stack of 200 in front of him and placing it in the slot under the appropriate light. For the first five trials, X was required to point to the light and then put the card in the slot. This procedure was used in order to ensure that S understood the instructions. The game was played at X’s own rate, and S was required to look at the lights before E ended the trial. It was emphasized in the instructions that the order of the lights was predetermined and that the card did not turn on the lights. Each session lasted approximately 30 minutes. Subjects. The Ss were 320 children attending public school classes in Northampton, Massachusetts.2 Eighty children were run at each of four grade levels: kindergarten, second, fourth, and sixth grades. No child who had repeated a grade was employed as S. Tile mean age, and the age range, in years and months was respectively: kindergarten, 5-7, 4-10 to 6-l; second grade, 7-8,6-S to S-3; fourth grade, 9-6,8-g to 10-I ; sixth grade, 11-7,10-g to 12-3. RESULTS
Three measures will be considered in addition to the usual sequential analysis of repetition responses as a function of the preceding run of events. Anticipatory errors are failures to predict the preceding event when the probability of an event repetition is one, and perserverative errors are failures to predict the alternative event when the probability of an event switch is one. The proportion of repetition responses at the uncertainty point represent repetitions after runs of length one in l-5 groups and repetitions after runs of length four in 4-5 groups. These measures are of particular theoretical import because of their sensitivity to sequence structure (Gambino and Myers, 1967) and because previous models which viewed runs as the critical stimuli were unable to predict the occurrence of such errors (Restle, 1966). Anticipatory errors. An anticipatory error is defined as the prediction of the alternative event when the probability of a run continuing is one. Such errors can occur following runs of length 2, 3, and 4, in l-5 groups and following runs of length 1, 2, and 3, in 45 groups. The proportion of anticipatory errors made by each of the experimental groups at each grade level may be seen in Fig. 1, and the results of an analysis of vari*The authors thank John Buteau, Superintendent of Schools, and Robert Moriarty, Elementary School Supervisor, Northampton, Mass., for their cooperation in providing facilities and Ss for this study, and Mrs. Florence Mador for assistance in
running Ss.
258
MYERS
AND
-o- - -7-B. CO-k4--0.5,4-5
Iz .I0Y E a
MYERS
“0.2.4-5
or K
2
6
4 GRADE
FIQ. 1. Percentage combination.
anticipatory
errors
as a function
TABLE
of
grade,
for
each
/3-RL
1
SUMMARY OF ANALYSES OF VARIANCE OF THREE DIFFERENT MEASURES Anticipatory errors Source
df
MS
F
1
.1918
5.7”
3 1 1 3 1 3 1 3 1 3 3 1 3 304 3 304
.6667 3.7130 1.1391 .0513 .9048 .0523 .0214 .0224 .0017 .0581 .0081 .0013 .0136 .0337 .0058 .0057
19.8C 110.1” 199.8” 1.5 26.9” 1.6 3.8 3.9
B (Proportion of long G (Grade) R (Run length) T (Trial block) PG PR GR PT GT RT @GR PGT BRT GRT S/N.= BGRT S x T/j3GR a Significant * Significant c Sign&ant
Perseverative errors MS
F
1.3995
15.5”
1.3331 4.4051 .0328 .1877 .4456 1.0750 .0023 .0046 .0767 .5768 .0054 .0722 .0545 .0904 .0045 .0320
14.7c 48.7” 1.0 2.1 4.9” 11.9”
Repetitions uncertainty
at point
MS
F
6.3090
114.5c
runs)
at the .0.5 level. at the .Ol level. at the .OOl level.
.4880 .1545 .2125 .4690 .1445 .0772 .1686 .0014 .0004 .2492 .0364 .0688 .0332 .0551 .0163 .0190
8.8” 2.8 11.2” 8.5” 2.6 1.4 8.9*
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IN
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PREDICTION
259
ante performed on these data may be found in Table 1.3 Many more anticipatory errors were made by the l-5 run length (RL) groups than by the 4-5 RL groups, F (1,304) = 110.1, p < .OOl. There are also significantly more anticipatory errors in the .2 than in the .5 groups, F (1,304) = 5.7, p < .05. This effect is due primarily to the l-5 groups, and is actually reversed in the younger 4-5 groups; this p X RL interaction is significant at the .OOl level, F (1,304) = 26.9. Anticipatory errors also decreased with age [F (3,304) = 19.8, p < .OOl], and with trial block [F (1,304) = 199.8; p < .OOl]. Perseverative errors. A perseverative error is defined as the prediction of the preceding event (or the failure to predict the alternative event) when the probability of a run continuing is zero, i.e., after a run of five events. The proportion of perseverative errors made by each of the experimental groups at each grade level may be seen in Fig. 2, and the .so r .70
-
w----
f5 .60
-
*
5 l= .50
-
:z
2 y u .40 v) t5 Q .30
/* Y----\\ \
2, l-5 \ .5. l-5 -I. . . .-.n 5.4-5
-
I \
-
w s
.20-
3 z it
.lOk
‘,.r.4-5 K
2
GRADE
4
6
FIG. 2. Percentage perseverative errors as a function of grade for each /?-RL combination.
summary of the appropriate analysis in Table 1. The l-5 RL groups made many more perseverative errors than the P5 groups, F (1,304) = 48.7, p < .OOl. It is interesting to note that, with the tot.al number of trials equated at approximately 200, l-5 groups observed considerably more runs of length 5 than did 65 groups, and therefore, if practice was a factor, should have shown less, not more perseverative errors. ‘All number
error proportions reported in the Results were computed by dividing the of such errors by the number of times such an error could have occurred.
260
MYERS
AND
MYERS
Averaging over all grade and RL conditions, perseverative errors decreased with a decrease in the proportion of long runs from .5 to .2; F (1,304) = 15.5, p < .OOl. A significant /3 x RL interaction, F (1,304) = 4.9, p < .05, supports the interpretation that the fl effect is due primarily to the large decrease in perseverative errors for the .2, 4-5 groups compared to the .5, &5 groups. Perseverative errors decreased significantly with age, F (3,304) = 14.7, p < .OOl. The decrease with age is primarily in the 4-5 groups, particularly the .2, 4-5 groups; the RL x Grade and RL X Grade X /3 interactions are both significant; F (3,304) = 11.9, p < .OOl, and F (3,304) = 6.4, p < .05, respectively. Repetitions at the uncertainty point. The proportion of repetition responses at the uncertainty point (i.e., following runs of length one in the l-5 groups and of length four in the P5 groups) was also analyzed. The appropriate analysis is summarized in Table 1 and the relevant results may be seen in Fig. 3 which shows the percentage of repetitions at the t
.eo I
K
2
4
6
GRADE Fro. 3. Percentage of repetition responsesat the uncertainty point for each experimental group at each grade level.
uncertainty point for each experimental group at each grade level. As the proportion of long run lengths decreased, the proportion of repetitions at the choice point decreased, F (1,304) = 114.5, p < .OOl. There was an overall decrease in uncertainty point repetitions with age, F (3,304) = 8.8, p < .Ol. This was due mainly to a decrease with age in repetitions among the .2 groups, and particularly in the .2, 45 RL groups; the &I X Grade and /3 X Grade X RL interactions were both significant, F (3,304) = 8.5, p < .Ol, and F (3,304) = 4.5, p < .05. Not visible in the figure is a slight increase in uncertainty point repetitions in the second block of trials, F (1,304) = 11.2, p < ,001. This effect was almost entirely due to the increase in repetitions shown by the .5
SEQUENCE
STRUCTURE
IN
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261
PREDICTION
groups in block 2, while the .2 groups made almost the same proportion of repetition responses at the uncertainty point in the two trial blocks; the p x Trial Block interaction was significant at the .Ol level, F (1,304) = 8.9. At all ages, the proportion of repetitions was always greater than the values of p, indicating a tendency to predict the repetition of an event more often than it happens. Run curves. The run curves in Fig. 4 show the percentage of repetition responses as a function of the length of the preceding run of events for each grade level, for each of the four /3-RL combinations. The measures I .oo 90 30
7b “, : z ,” 8; ar
60 so .40 .30
i I LENGTH
OF
PRECEDING
2
3
4
5
RUN
4. Percentage of repetition responses as a function of the length of the preceding run of events, for each grade level, for each of the P-RL combinations. FIG.
analyzed earlier partially determine these curves, but Fig. 4 serves as a quick visual summary of the differences in sensitivity of the four age groups to the regularities of the sequences, and their differential sensitivity to different sequences. In both top panels, all grade levels show a decline in repetition responses following runs of length four, and a further decline following runs of length five. This finding becomes more marked with age, and it is even more obvious in the .2, 4-5 RL con-
262
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AND
MYERS
dition than in the 5, 4-5 RL condition. The l-5 RL conditions shown in the bottom half of Fig. 4 reveal much less sensitivity to these sequences at all grade levels, and this is particularly true in the .2, l-5 RL condition, where runs are of length one 80% of the time. DISCUSSION
The results of this experiment demonstrate that children are sensitive to more than just the last event in the binary prediction situation. Even the youngest age group was clearly picking up some information about run lengths in some sequences; witness the negative recency (decrease in repetition responding with increasing run length) in the .2, 4-5 sequence, and the much lower rate of repetitions after one event in the .2, l-5 group than in the .5, l-5 group. Errors of all types decreased as age increased; the older children were better able to process information about the event sequences. Indeed, the sixth graders demonstrated almost the same proportion of anticipatory errors as did college students run on similar sequences (Myers, Butler, and Olson, in press). Sixth graders made 733% more perseverative errors, however, and about 15% more repetition responses at uncertainty points, than did the college groups. It is equally obvious that certain characteristics of event sequences are more salient than others. In general, there were more perseverative than anticipatory errors. Furthermore, all age groups were better able to follow 4-5 sequences than l-5 sequences, and improvement with age was most clear with 4-5 sequences. Runs of length one are apparently difficult to process for Ss of all ages, especially when they predominate in a sequence. Even the sixth graders, presented the .2, l-5 sequence, showed little knowledge about the length of the long run, or the preponderance of single alternations. These effects of RL and /3, and differences in anticipatory and perseverative errors are also typical of college students (Myers, Butler, and Olsen, 1969). Thus it appears that children operate in the same fashion as adults, if less effectively. Gambino and Myers (1967) have recently proposed a model for binary prediction behavior which predicts many of the effects observed above. They assume that runs of events are discriminative stimuli but that Ss cannot discriminate perfectly. Because of this imperfect perception, reinforcement of the prediction that a run length will continue or break off may generalize and serve as reinforcement for the prediction of other run lengths. The degree of generalization is determined by the absolute difference between the reinforced run length and the particular other run length of concern. This view can account for the
SFWJENCE
SMWCTURE
IN
BINARY
PREDICTION
263
greater number of anticipatory errors made by the l-5 groups. There is one basic structural difference between the two sequences: In l-5 sequences the anticipation error points are sandwiched between two break-off points (i.e., they can occur following runs of length 2, 3, or 4)) while in 45 sequences runs of length 5 are at least two points removed from the closest anticipatory error point (anticipatory errors can occur following runs of length 1, 2, or 3). Therefore, according to the generalization model, there should be more switching at anticipatory points in l-5 groups than in 4-5 groups. The concept of a generalization gradient over run lengths would also predict more perseverative errors in the l-5 groups ; the breaking off of a run of length one yields less generalized switching at runs of length five than does the breaking off of a run of length four. The fact that differences in the error rates of l-5 and 4-5 groups are more marked at p I=: .2 is consistent with this account. The results would be equally well accounted for by the assumption that the primary source of confusion is miscounting, that is, that the subject misperceives the current run length and that only the probability associated with a run of the perceived length is influenced by the tria1 outcome. If it is assumed that the subject’s count is most likely to be off by a small amount, miscounting would produce results very similar to those that were attributed to generalization. Since it is also highly probable that counting accuracy improves with age, this interpretation would account equally well for the developmental changes. In summary, the ability to process structured sequential information improves continuously with age, and shows the same relative effects of experimental manipulation at all age levels, including adulthood. A model which views event runs as imperfectly discriminated stimuli, and attributes age level differences to differences in capacity to discriminat,(: among run lengths, provides a reasonabIe first account of the data. REFERENCES R. S. Variables influencing alternation prediction by preschool children: I. Previous recurrent, dependent, and repetitive sequences. Journal of &per& mental Child Psychology, 1966,3,4056. BOQARTZ, R. S., AND PEDERSON, D. R. Variables influencing alternation prediction by preschool children: II. Redundant cue value and intertrial interval duration. Journul of Experimental Child Psychology, 1966,4, 211-216. Ca~rc, G. J., AND Mvnas, J. L. A developmental study of sequential two-choice decision making. Child Development, 1963, 34, 483-494. GAMBINO, B., AND MYERS, J. L. The role of event runs in probability learning. Psvchological Review, 1967, 74,410419. BOQARTZ,
264
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AND
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S. The effect of repetition probabilities on children’s two-choice learning, Unpublished doctoral dissertation, University of Massachusetts, July, 1967. MYERS, J. L., BUTLER, P., AND OLSON, D. Run lengths and probabilities in binary prediction. Journal of Mathematical Psychology, 1969,6, in press. RESTLE, F. Run structure and probability learning: disproof of Restle’s model. Journal of Experimental Psychology, 1966,72,382-389.
JONES,
OF EXPERIMENTAL
CHILD
PSYCHOLOGY
7,255-2&t
0969)
A Developmental Study of Sequence in Binary Prediction1 NANCY
Structure
A. MYERS AND JEROME L. MYERS University
of Massachusetts
Sixteen groupsof 20 Ss eachwere run individually in a binary prediction situation with each alternative occurring equally often. Four age groups were employed: Ss were either in kindergarten, second,fourth, or sixth grades.For half the 8s at each age level the events occurredin runs of either 1 or 5; the other Se observedeventswhich occurredin runs of either 4 or 5. The probability of occurrenceof the long run (/?I wasalsomanipulated: for half the Ss receiving each set of run lengthsthere were equal numbersof short and long runs (p = .5); for the other 8s p = .2. The results indicated that all groups showed some sensitivity to run length characteristics.However, errors of all types decreasedasage increased,and 45 sequences were followed more accurately than l-5 sequences. The results were discussed in terms of their implicationsfor a generalmodelof binary choice behavior.
The binary prediction paradigm requires the S to predict which of two events will occur on each of a series of trials. Recent developments in the binary prediction literature have been in the direction away from noncontingent probabilistic sequences and toward the study of more carefully controlled sequential structures. With adults, a wide variety of both partially and completely learnable sequences have been employed in attempts to isolate the effective stimuli and response tendencies in binary prediction (e.g., Gambino and Myers, 1967; Restle, 1966). With children, too, there is increasing evidence of some sensitivity to at least some characteristics of sequences.Thus, Craig and Myers’ (1963) kindergarten Ss alternated more with a 60:40 than with an 80:20 sequence, and Bogartz’ (1966) 4- and 5-year-olds followed single alternations quite well, at least in early trials, and were obviously affected by changes in
sequential structure. Jones (1967) found that both nursery and kindergarten 8s responses reflected the pattern of highly discriminable event sequences.She exposed Ss to sequenceswith event repetition probabilities ‘This research was supported by funds from a University of Massachusetts Faculty Research Grant and NIMH Grant MH-O3803-06 and NSF Grant GS-386. The authors thank the Research Computing Center. University of Massachusetts fol assistance in analyzing data. 255
256
MYERS
AND
MYERS
of either .l or .9 for 100 trials .and for the second 100 trials transferred half the Ss to the opposite event repetition probability, while continuing the others at the same repetition probability level. When the repetition probability level changed, responses showed only limited dependency upon the new event sequence. However, kindergarten Ss followed any sequence better than nursery Ss and both age groups followed predominately repeating sequences more accurately than predominately alternating sequences. The present study was designed to explore further those characteristics of event sequences to which children may respond. In particular, we were interested in sensitivity to run length, because Myers, Butler, and Olson (in press) had observed such sensitivity in adults. Following the Myers et al. procedure, the events always appeared in runs of two standard lengths, either 4 and 5 or 1 and 5. For some Ss the two run lengths occurred equally often; for others, the shorter run length occurred more frequently. In addition, four age levels were employed, to permit evaluation of developmental aspects of performance. METHOD Apparatus. The event box consisted of two 11/2 v. red event lights mounted in parallel on the front of an 8- X 12- X a-inch grey box, one 3 inches from the left, the other 3 inches from the right edge. Below each light was a 31/2- X $&inch slot through which one of 200 3- X 2-inch playing cards, blank on both sides, could be placed to make a response. A grey panel attached to the back of the event box concealed the control unit from S. The control unit contained a buzzer that was used to indicate when choices should be made. One switch on the control unit sounded the buzzer and two other switches turned on the event lights. Design. Ss were randomly assigned to one of sixteen experimental groups differing with respect to age and two characteristics of the event sequences employed, in a 4 X 2 X 2 factorial design. There were 20 Ss in each group, half males and half females. Four age groups were run: Ss were either in kindergarten, second, fourth, or sixth grades. For half the Ss at each age level the events occurred in runs of either one or five ; the other Ss observed events which occurred in runs of either four or five. The other aspect of event sequences which was manipulated was @, the probability of occurrence of the long run. For half the Ss receiving each set of run lengths, ~3 was 5, i.e., there were equal numbers of short and long runs. For the other Ss, fi was .2, and thus short runs predominated. Event sequences were generated in two blocks of approximately I00 trials each, for each of four run length+ combinations, with the one additional requirement that each light occur equally often.
SEQUENCE
STRUCTURE
IN
BIKARY
PREDICTION
257
Procedure. The Ss were tested one at a time in what they were told was a “guessing game.” After E signaled the beginning of a trial with the buzzer, S predicted which one of the two lights would come on by taking a card from the stack of 200 in front of him and placing it in the slot under the appropriate light. For the first five trials, X was required to point to the light and then put the card in the slot. This procedure was used in order to ensure that S understood the instructions. The game was played at X’s own rate, and S was required to look at the lights before E ended the trial. It was emphasized in the instructions that the order of the lights was predetermined and that the card did not turn on the lights. Each session lasted approximately 30 minutes. Subjects. The Ss were 320 children attending public school classes in Northampton, Massachusetts.2 Eighty children were run at each of four grade levels: kindergarten, second, fourth, and sixth grades. No child who had repeated a grade was employed as S. Tile mean age, and the age range, in years and months was respectively: kindergarten, 5-7, 4-10 to 6-l; second grade, 7-8,6-S to S-3; fourth grade, 9-6,8-g to 10-I ; sixth grade, 11-7,10-g to 12-3. RESULTS
Three measures will be considered in addition to the usual sequential analysis of repetition responses as a function of the preceding run of events. Anticipatory errors are failures to predict the preceding event when the probability of an event repetition is one, and perserverative errors are failures to predict the alternative event when the probability of an event switch is one. The proportion of repetition responses at the uncertainty point represent repetitions after runs of length one in l-5 groups and repetitions after runs of length four in 4-5 groups. These measures are of particular theoretical import because of their sensitivity to sequence structure (Gambino and Myers, 1967) and because previous models which viewed runs as the critical stimuli were unable to predict the occurrence of such errors (Restle, 1966). Anticipatory errors. An anticipatory error is defined as the prediction of the alternative event when the probability of a run continuing is one. Such errors can occur following runs of length 2, 3, and 4, in l-5 groups and following runs of length 1, 2, and 3, in 45 groups. The proportion of anticipatory errors made by each of the experimental groups at each grade level may be seen in Fig. 1, and the results of an analysis of vari*The authors thank John Buteau, Superintendent of Schools, and Robert Moriarty, Elementary School Supervisor, Northampton, Mass., for their cooperation in providing facilities and Ss for this study, and Mrs. Florence Mador for assistance in
running Ss.
258
MYERS
AND
-o- - -7-B. CO-k4--0.5,4-5
Iz .I0Y E a
MYERS
“0.2.4-5
or K
2
6
4 GRADE
FIQ. 1. Percentage combination.
anticipatory
errors
as a function
TABLE
of
grade,
for
each
/3-RL
1
SUMMARY OF ANALYSES OF VARIANCE OF THREE DIFFERENT MEASURES Anticipatory errors Source
df
MS
F
1
.1918
5.7”
3 1 1 3 1 3 1 3 1 3 3 1 3 304 3 304
.6667 3.7130 1.1391 .0513 .9048 .0523 .0214 .0224 .0017 .0581 .0081 .0013 .0136 .0337 .0058 .0057
19.8C 110.1” 199.8” 1.5 26.9” 1.6 3.8 3.9
B (Proportion of long G (Grade) R (Run length) T (Trial block) PG PR GR PT GT RT @GR PGT BRT GRT S/N.= BGRT S x T/j3GR a Significant * Significant c Sign&ant
Perseverative errors MS
F
1.3995
15.5”
1.3331 4.4051 .0328 .1877 .4456 1.0750 .0023 .0046 .0767 .5768 .0054 .0722 .0545 .0904 .0045 .0320
14.7c 48.7” 1.0 2.1 4.9” 11.9”
Repetitions uncertainty
at point
MS
F
6.3090
114.5c
runs)
at the .0.5 level. at the .Ol level. at the .OOl level.
.4880 .1545 .2125 .4690 .1445 .0772 .1686 .0014 .0004 .2492 .0364 .0688 .0332 .0551 .0163 .0190
8.8” 2.8 11.2” 8.5” 2.6 1.4 8.9*
SEQUENCE
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IN
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259
ante performed on these data may be found in Table 1.3 Many more anticipatory errors were made by the l-5 run length (RL) groups than by the 4-5 RL groups, F (1,304) = 110.1, p < .OOl. There are also significantly more anticipatory errors in the .2 than in the .5 groups, F (1,304) = 5.7, p < .05. This effect is due primarily to the l-5 groups, and is actually reversed in the younger 4-5 groups; this p X RL interaction is significant at the .OOl level, F (1,304) = 26.9. Anticipatory errors also decreased with age [F (3,304) = 19.8, p < .OOl], and with trial block [F (1,304) = 199.8; p < .OOl]. Perseverative errors. A perseverative error is defined as the prediction of the preceding event (or the failure to predict the alternative event) when the probability of a run continuing is zero, i.e., after a run of five events. The proportion of perseverative errors made by each of the experimental groups at each grade level may be seen in Fig. 2, and the .so r .70
-
w----
f5 .60
-
*
5 l= .50
-
:z
2 y u .40 v) t5 Q .30
/* Y----\\ \
2, l-5 \ .5. l-5 -I. . . .-.n 5.4-5
-
I \
-
w s
.20-
3 z it
.lOk
‘,.r.4-5 K
2
GRADE
4
6
FIG. 2. Percentage perseverative errors as a function of grade for each /?-RL combination.
summary of the appropriate analysis in Table 1. The l-5 RL groups made many more perseverative errors than the P5 groups, F (1,304) = 48.7, p < .OOl. It is interesting to note that, with the tot.al number of trials equated at approximately 200, l-5 groups observed considerably more runs of length 5 than did 65 groups, and therefore, if practice was a factor, should have shown less, not more perseverative errors. ‘All number
error proportions reported in the Results were computed by dividing the of such errors by the number of times such an error could have occurred.
260
MYERS
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Averaging over all grade and RL conditions, perseverative errors decreased with a decrease in the proportion of long runs from .5 to .2; F (1,304) = 15.5, p < .OOl. A significant /3 x RL interaction, F (1,304) = 4.9, p < .05, supports the interpretation that the fl effect is due primarily to the large decrease in perseverative errors for the .2, 4-5 groups compared to the .5, &5 groups. Perseverative errors decreased significantly with age, F (3,304) = 14.7, p < .OOl. The decrease with age is primarily in the 4-5 groups, particularly the .2, 4-5 groups; the RL x Grade and RL X Grade X /3 interactions are both significant; F (3,304) = 11.9, p < .OOl, and F (3,304) = 6.4, p < .05, respectively. Repetitions at the uncertainty point. The proportion of repetition responses at the uncertainty point (i.e., following runs of length one in the l-5 groups and of length four in the P5 groups) was also analyzed. The appropriate analysis is summarized in Table 1 and the relevant results may be seen in Fig. 3 which shows the percentage of repetitions at the t
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GRADE Fro. 3. Percentage of repetition responsesat the uncertainty point for each experimental group at each grade level.
uncertainty point for each experimental group at each grade level. As the proportion of long run lengths decreased, the proportion of repetitions at the choice point decreased, F (1,304) = 114.5, p < .OOl. There was an overall decrease in uncertainty point repetitions with age, F (3,304) = 8.8, p < .Ol. This was due mainly to a decrease with age in repetitions among the .2 groups, and particularly in the .2, 45 RL groups; the &I X Grade and /3 X Grade X RL interactions were both significant, F (3,304) = 8.5, p < .Ol, and F (3,304) = 4.5, p < .05. Not visible in the figure is a slight increase in uncertainty point repetitions in the second block of trials, F (1,304) = 11.2, p < ,001. This effect was almost entirely due to the increase in repetitions shown by the .5
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groups in block 2, while the .2 groups made almost the same proportion of repetition responses at the uncertainty point in the two trial blocks; the p x Trial Block interaction was significant at the .Ol level, F (1,304) = 8.9. At all ages, the proportion of repetitions was always greater than the values of p, indicating a tendency to predict the repetition of an event more often than it happens. Run curves. The run curves in Fig. 4 show the percentage of repetition responses as a function of the length of the preceding run of events for each grade level, for each of the four /3-RL combinations. The measures I .oo 90 30
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4. Percentage of repetition responses as a function of the length of the preceding run of events, for each grade level, for each of the P-RL combinations. FIG.
analyzed earlier partially determine these curves, but Fig. 4 serves as a quick visual summary of the differences in sensitivity of the four age groups to the regularities of the sequences, and their differential sensitivity to different sequences. In both top panels, all grade levels show a decline in repetition responses following runs of length four, and a further decline following runs of length five. This finding becomes more marked with age, and it is even more obvious in the .2, 4-5 RL con-
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dition than in the 5, 4-5 RL condition. The l-5 RL conditions shown in the bottom half of Fig. 4 reveal much less sensitivity to these sequences at all grade levels, and this is particularly true in the .2, l-5 RL condition, where runs are of length one 80% of the time. DISCUSSION
The results of this experiment demonstrate that children are sensitive to more than just the last event in the binary prediction situation. Even the youngest age group was clearly picking up some information about run lengths in some sequences; witness the negative recency (decrease in repetition responding with increasing run length) in the .2, 4-5 sequence, and the much lower rate of repetitions after one event in the .2, l-5 group than in the .5, l-5 group. Errors of all types decreased as age increased; the older children were better able to process information about the event sequences. Indeed, the sixth graders demonstrated almost the same proportion of anticipatory errors as did college students run on similar sequences (Myers, Butler, and Olson, in press). Sixth graders made 733% more perseverative errors, however, and about 15% more repetition responses at uncertainty points, than did the college groups. It is equally obvious that certain characteristics of event sequences are more salient than others. In general, there were more perseverative than anticipatory errors. Furthermore, all age groups were better able to follow 4-5 sequences than l-5 sequences, and improvement with age was most clear with 4-5 sequences. Runs of length one are apparently difficult to process for Ss of all ages, especially when they predominate in a sequence. Even the sixth graders, presented the .2, l-5 sequence, showed little knowledge about the length of the long run, or the preponderance of single alternations. These effects of RL and /3, and differences in anticipatory and perseverative errors are also typical of college students (Myers, Butler, and Olsen, 1969). Thus it appears that children operate in the same fashion as adults, if less effectively. Gambino and Myers (1967) have recently proposed a model for binary prediction behavior which predicts many of the effects observed above. They assume that runs of events are discriminative stimuli but that Ss cannot discriminate perfectly. Because of this imperfect perception, reinforcement of the prediction that a run length will continue or break off may generalize and serve as reinforcement for the prediction of other run lengths. The degree of generalization is determined by the absolute difference between the reinforced run length and the particular other run length of concern. This view can account for the
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greater number of anticipatory errors made by the l-5 groups. There is one basic structural difference between the two sequences: In l-5 sequences the anticipation error points are sandwiched between two break-off points (i.e., they can occur following runs of length 2, 3, or 4)) while in 45 sequences runs of length 5 are at least two points removed from the closest anticipatory error point (anticipatory errors can occur following runs of length 1, 2, or 3). Therefore, according to the generalization model, there should be more switching at anticipatory points in l-5 groups than in 4-5 groups. The concept of a generalization gradient over run lengths would also predict more perseverative errors in the l-5 groups ; the breaking off of a run of length one yields less generalized switching at runs of length five than does the breaking off of a run of length four. The fact that differences in the error rates of l-5 and 4-5 groups are more marked at p I=: .2 is consistent with this account. The results would be equally well accounted for by the assumption that the primary source of confusion is miscounting, that is, that the subject misperceives the current run length and that only the probability associated with a run of the perceived length is influenced by the tria1 outcome. If it is assumed that the subject’s count is most likely to be off by a small amount, miscounting would produce results very similar to those that were attributed to generalization. Since it is also highly probable that counting accuracy improves with age, this interpretation would account equally well for the developmental changes. In summary, the ability to process structured sequential information improves continuously with age, and shows the same relative effects of experimental manipulation at all age levels, including adulthood. A model which views event runs as imperfectly discriminated stimuli, and attributes age level differences to differences in capacity to discriminat,(: among run lengths, provides a reasonabIe first account of the data. REFERENCES R. S. Variables influencing alternation prediction by preschool children: I. Previous recurrent, dependent, and repetitive sequences. Journal of &per& mental Child Psychology, 1966,3,4056. BOQARTZ, R. S., AND PEDERSON, D. R. Variables influencing alternation prediction by preschool children: II. Redundant cue value and intertrial interval duration. Journul of Experimental Child Psychology, 1966,4, 211-216. Ca~rc, G. J., AND Mvnas, J. L. A developmental study of sequential two-choice decision making. Child Development, 1963, 34, 483-494. GAMBINO, B., AND MYERS, J. L. The role of event runs in probability learning. Psvchological Review, 1967, 74,410419. BOQARTZ,
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S. The effect of repetition probabilities on children’s two-choice learning, Unpublished doctoral dissertation, University of Massachusetts, July, 1967. MYERS, J. L., BUTLER, P., AND OLSON, D. Run lengths and probabilities in binary prediction. Journal of Mathematical Psychology, 1969,6, in press. RESTLE, F. Run structure and probability learning: disproof of Restle’s model. Journal of Experimental Psychology, 1966,72,382-389.
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