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

**Abstract**

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.

**Introduction**

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|
\_/ \_/
| |
| |
| |
| __ |
| / \ |
\---->|AB|<----/
\__/

__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.

**Experiment**

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.

**Methods**

__Subjects__
Subjects were fifteen

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

**Procedure**
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.

**Results**

**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*** http://www.students.uiuc.edu/~serotkin/fig3.jpg *for Figure 3: Percent correct in the experiment as a function of number of repetitions for Test 1 and Test 2.

**See ***http://www.students.uiuc.edu/~serotkin/fig4.jpg* 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.

**Discussion**

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.

**References**

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.