My 1998 Westinghouse Science Talent Search Semifinalist paper. The contest has since been renamed the Intel Science Talent Search.  Please excuse my poor ASCII representations of the original diagrams.

Associative Symmetry: An Examination of the Representation of Associations in Human Memory


    This paper presents a theoretical and empirical analysis of the representation of associations in human memory.  In order to distinguish between the model of associative symmetry and the model of independent associations, an experimental investigation of the correlation between forward and backward associations was conducted.  The findings of this experiment demonstrated that the strength of forward and backwards associations exhibit a near-perfect correlation.  These findings provide strong support for the model of associative symmetry.


    There are currently two major theories concerning the representations of associations in human memory.  Given a pair of informational elements, A and B, the independent association theory claims that the strength of the associations A-B and B-A are independent of one another.  In this model, each association is believed to be separate and uni-directional (Figure 1).  

/ \ ----------------------------->/ \
|A|                               |B|
\_/<-  -  -  -  -  -  -  -  -  -  \_/

Figure 1

A representation of the independent association
model, with separate associations for forward and
backward retrieval.

    Belief in this model stems from the late nineteenth century, when forward associations were postulated to be stronger than their reciprocals, suggesting asymmetric storage and retrieval (Ebbinghaus, 1885).  This idea has been incorporated into models of spreading activation (e.g. Anderson, 1983) and in production system models used in artificial intelligence.
    An alternate theoretical representation, referred to as the principle of associative symmetry (e.g. Asch and Ebenholtz, 1962) claims that associations between meaningful elements result in the formation of a singular, higher order representation combining features of both elements. A good model of associative symmetry shows associations from A and B creating a single representation AB (Figure 2).  

/ \            / \
|A|            |B|
\_/            \_/
 |              |
 |              |
 |              |
 |      __      |
 |     /  \     |

Figure 2

A model showing the associative symmetry theory, wherein
the two halves of the pair are stored together in memory as
the conglomerate “AB”.

    Since the two words are stored together in memory, either half of the pair can be recalled equally well when prompted with the other.  Kahana (in press-a) has shown that, under certain conditions, symmetric models can predict asymmetric retrieval, just as some models of independent association can sometimes predict symmetric retrieval.
    A and B could be representative of anything—patterns, numbers, objects, ideas, or anything else that can be stored into memory.  Through what is known as reductionism in science, it is necessary to choose one representation that can be “life” on a smaller scale—an artificial memory phenomena that can be controlled easily by the experimenter.  For the purposes of the experiment that follows, A and B are represented as words.  Words are everyday devices; subjects use them every day of their lives—language has meaning.  When other media are used, such as complex patterns, it has been found that subjects code them as words anyway (defining to themselves what the pattern looks like, what color it is, how big it is, the pattern’s shape).  Therefore, it makes sense to use the medium that is the basis by which subjects interpret most other mediums.

    The correlation between forward and backward recall is essential to the question of the representation of associations.  This is the area in which the model of associative symmetry and the model of independent associations most drastically differ in their predictions.  The associative symmetry model yields a perfect positive correlation between forward and backward associations, while the independent associations model predicts that there will be no correlation whatsoever.

    There has been no experiment to date that measured the correlation between forward and backward recall associations.


    The study that follows was designed to fill in this gap in the scientific literature, and thereby allow us to distinguish between the model of associative symmetry and the model of independent associations.  It is an empirical investigation of the correlation between forward and backward associations.  It was conducted in order to distinguish between the model of associative symmetry and the model of independent associations.  To examine the correlation between forward and backward associations, we employ the method of successive tests (see Kahana in press-b for a review).  In this technique, subjects study a list of word pairs and are then given two phases of testing: some pairs of words are tested in the same manner in both phases, whereas other pairs are tested differently in the two phases.  A visual pattern-matching test was used to separate the two testing phases from each other.


    Subjects were fifteen Brandeis University summer students who participated for payment in the summer of 1997.

    Trials in this experiment consisted of the study of a list of word pairs followed by two successive cued recall tests of all the pairs presented in the study list.  There were four test conditions representing the combinations of forward and backward cues in Test 1 and Test 2 (i.e., forward-forward, forward-backward, backward-forward, and backward-backward).
    Lists consisted of high frequency nouns sampled from the Toronto word pool (Friendly, Franklin, Hoffman, & Rubin, 1982).  Study lists consisted of 12 unique word pairs.  These 12 pairs were divided into three repetition conditions (one, three, and five repetitions) each consisting of four unique pairs.  Each of these four pairs was assigned to one of the four test conditions described previously.  During study, each pair was displayed on the screen for 2 seconds.  The order of the pairs in each study list was randomized, subject to the constraint that successively presented pairs were always unique.  In addition to the within-list randomization, the entire word pool was randomized separately for each subject to ensure complete randomization of all the materials across experimental conditions.
    Following the study list, two successive cued recall tasks were administered.  In each cued recall task, each of the pairs from the study list were tested individually and in a randomly determined order.  When a cue word was shown on the computer screen, subjects attempted to vocally recall the word with which it was paired.  Responses were digitally recorded and response times were determined using a computerized voice-key algorithm.  If subjects could not recall a target item they were instructed to say “pass”.
    A pattern-matching distractor task was used to separate the study list and the two successive test lists of each experimental trial.  This was done to eliminate the possibility that a given pair would be tested immediately after it was seen in the study list or in the prior list.  The distractor task consisted of multiple trials of pattern matching.  On each trial of the distractor task, two 4x4 matrices consisting of filled and empty cells were shown.  Subjects were instructed to make same-different judgments on these matrices.  The pattern-matching task continued until subjects correctly identified 30 consecutive pairs of matrices.  A complete experimental trial thus consisted of studying word pairs, a distractor task, Test 1, another distractor task, and finally Test 2.  During a single session, subjects were given six trials with different word lists.


Percent Correct
    Figure 3 illustrates the effect of repetitions on recall accuracy for Test 1 and Test 2.  The probability of correct recall rises slightly from the first test to the second, suggesting that the first test acts as a secondary study phase.  This can be seen more clearly in Figure 4 where performance for each of the four cueing conditions are plotted for Test 1 and Test 2.  A paired t-test analysis of the accuracy of all Test 1 responses as compared to all Test 2 responses reveals that the increased accuracy on Test 2 is statistically significant (t(14) = -2.66, p < 0.01).

See for Figure 3: Percent correct in the experiment as a function of number of repetitions for Test 1 and Test 2.

See for Figure 4: Percent correct in the experiment as a function of different cueing conditions.

Response Times
    Mean response times for each of the experimental conditions are given in Tables 1a and 1b.  While the mean response times for three of the conditions (Test 1/Same Cues, Test 1/Different Cues, and Test 2/Different Cues) are very close to each other, each approximately 2100 milliseconds, the mean response time on Test 2 with the same cues drops to 1826 (see Table 1b).  This difference is statistically significant, based on a paired t-test of Test 2/Same Cues and Test 2/Different Cues (t(14) = -1.98, p < 0.05).  The mean times for both of the Test 1 conditions are expected to be similar, since there is no difference in how they are tested at this time.  This expectation holds true (a paired t-test of Test 1/Same Cues and Test 1/Different Cues yields p > 0.05, which is not statistically significant).  During Test 2, the subject has already been tested on the prompt word for those pairs with the same cue.  The previous test strengthens the association for that pair, lowering the amount of time necessary for the subject to respond.

Number of Repetitions:       One Rep           3 Reps             5 Reps___
Mean Test 1 RT:                  2462 (± 309)    2170 (± 171)    1827 (± 102)
Mean Test 2 RT:                  1966 (± 200)    1911(± 242)     1972 (± 190)

Table 1a: Mean Reaction Times in the experiment for prompts answered correctly as a function of number of repetitions.  Numbers in parentheses represent 95% confidence intervals centered around the means.

Test and Cue Condition:    Test 1, Same Cues    Test 1, Diff. Cues    Test 2, Same Cues    Test 2, Diff. Cues
Mean RT:                                  2223 (± 283)              2058 (± 154)             1826 (± 216)              2106 (± 168)

Table 1b: Mean reaction times in the experiment for prompts answered correctly as a function of cue type.  Numbers in parentheses represent 95% confidence intervals centered around the means.

Correlation Between Successive Tests
    The relevant measure for the correlation between two dichotomous variables is Yule’s Q, given by the formula Q = ((ad – bc) / (ad + bc)).  In this equation, a is the number of instances where the subject answered correctly for a pair during both tests, b is the number of incorrect followed by correct responses, c is correct followed by incorrect responses, and d is the number of instances where the subject answered incorrectly during both tests.  Yule’s Q values are bounded by –1 and 1, where 1 is a perfect positive correlation and –1 is a perfect negative correlation.  This means that if Q is exactly –1, then success on one test invariably leads to failure on the other (and vice-versa).  If Q is exactly 0, then there is no correlation between responses on the two tests.  If Q is exactly 1, then success or failure on one test always leads to a like result on the other test (the prediction of the symmetric model).
    Yule’s Q did not vary significantly between the Test 1-Test 2 conditions (see Table 2).

Cue Type:             Same Cues       Different Cues
Yule's Q Value:    0.951 (± 0.015)    0.943 (± 0.017)

Table 2: Yule’s Q calculated for pairs cued the same and differently over the two tests.  Numbers in parentheses represent 95% confidence intervals centered around the means.

    In addition, the correlation between successive cued recall tests was determined using reaction time data.  This was done in the following manner.  The four cells used in determining Yule’s Q (a, b, c, and d) were considering by considering responses faster than the mean for that subject and the given test (1 or 2) as positive responses and responses slower than the mean for that subject as negative responses.  For example, if a Test 1 response was faster than the mean for Test 1 responses and the response for the same pair at Test 2 was faster than the mean for Test 2 responses, we would increment cell a.  Two separate Q values were obtained for each subject: one for pairs in which the cues were the same and one for when they were different.  A paired t-test of these two conditions indicates that the difference is not statistically significant (p > 0.05).

Cue Type:    Same Cues      Diff. Cues_
Mean:           0.33 (± 0.16)    0.44 (± 0.14)

Table 3: Yule’s Q in the experiment based on response time data.  Numbers in parentheses represent 95% confidence intervals centered around the means.

    The discrepancy found between the near-perfect correlation in the accuracy data (see Table 2) and the relatively small but statistically significant correlation in the latency data (see Table 3) may seem to be a cause for concern.  However, it is well known that reaction times are highly inherently variable (Luce, 1986).  For this reason, it is not surprising that the latency correlations are quite low. The point to note is that there is no evidence of a significant difference between correlations of similar and different cues.  If anything, the different cues are numerically higher.


    The experiment’s Yule’s Q data is the most convincing evidence of associative symmetry by providing an accurate measure of the correlation between forward and backward associations.  The Q value for cues tested identically during both tests is expected to be close to 1 in both the associative symmetry and independent association models (with a return of 0.951, this expectation is upheld), but the two models differ drastically in what they predict for cues tested differently between the two tests.  The model of independent associations predicts no correlation between the forward and backward correlation (Q = 0), while the associative symmetry model predicts a perfect forward correlation (Q = 1).  With a Q value of 0.943, the experiment shows strong evidence in support of the associative symmetry model.
    The experiment’s percent correct data also suggests associative symmetry.  Since the correlation between forward and backward associations is expected to be perfect in this model, it holds that the percentages correct for cues repeated identically between the two tests would be the same as the percentages for cues repeated differently.  This is exactly what occurred.  In both the first and second tests, the results for same and different cues vary by no more than 5 percentage points.  The independent association model predicts a larger difference between the two values.  One might expect that the order in which one studies influences the facility in which the pair is retrieved.  In fact, the order of study seems to have no effect on how well the pair is recalled.  Pairs tested in the forward direction during the first test (the second test is partially dependent on the first and therefore will not be considered) were recalled correctly 56.2% of the time, and pairs tested in the backwards direction were recalled correctly 55.6% of the time.


Anderson, J. R. (1983). A spreading activation theory of memory. Journal of Verbal Learning and Verbal Behavior, 22, 261-295.

Asch, S.E. & Ebenholtz, S.M. (1962). The Principle of Associative Symmetry.  Proceedings of the American Philosophical Society, 106, 135-163.

Ebbinghaus, H. (1913 Reprinted by Dover, 1964).  Memory: A contribution to Experimental Psychology.  New York: Teachers College, Columbia University.

Friendly, M., Franklin, P.E., Hoffman, D., & Rubin, D.C. (1982).  Norms for the Toronto Word Pool.  Behavior Research Methods and Instrumentation, 14, 375-399.

Horowitz, L. M., Norman, S. A., & Day, R. S.  (1966).  Availability and associative symmetry.  Psychological Review, 73, 1-15.

Kahana, M. J. (in press-a). Associative Symmetry and Memory Theory.  Psychonomics Bulletin and Review.

Kahana, M. J. (in press-b).  An analysis of the recognition-recall relation in four distributed memory models.  Psychological Review.

Levy, C. M., & Nevill, D. D. (1974).  B-A learning as a function of degree of A-B learning.  Journal of Experimental Psychology, 102, 327-329.

Tedford, W. H., Jr., & Hazel, J. S. (1973)  Stimulus location as a factor in associative symmetry.  Journal of Experimental Psychology, 101, 189-190.

Wollen, K. A., Allison, T. S., & Lowry, D. H. (1969).  Associate symmetry versus independent association.  Journal of Verbal Learning and Verbal Behavior, 8, 283-288.

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