APPLICATION OF THE BOWER AND TRABASSO

Acta Psychologica 38,215-233. © North-Holland Publishing Company 1974
APPLICATION OF THE BOWER AND TRABASSO THEORY TO
F O U R - C A T E G O R Y CONCEPT L E A R N I N G WITH PROBABILISTIC
FEEDBACK 1
Jens WANDMACHER and Dirk VORBERG2
Universitiit Konstanz, 775 Konstanz, W-Germany
Four-category conjunctive concept problems were learned under nine different conditions
with the number of irrelevant dimensions and the percentage of correct feedback as independent variables. Concept identification performance could be adequately predicted by assuming
that the two relevant dimensions defining the conjunctive problems were learned concurrently
and independently. A generalized version of the Bower-Trabasso theory of concept identification, encompassing a distinct response learning phase, was applied to the data. The theory
adequately described performance under conditions with consistently correct feedback. Several
alternative models for learning under misinformative feedback conditions were developed from
the theory. Using parameter estimates from data under correct feedback conditions, these
models were found to predict performance under misinformative feedback fairly well.
1. Introduction
The purpose of this investigation is to test extensions of the concept
learning theory proposed by Bower and Trabasso (Bower and Trabasso
1964; Trabasso and Bower 1968) to some rather complicated concept
identification tasks. Bower and Trabasso's theory, called B - T theory
hereafter, postulates two processes going on during concept identification, namely, a selection o f stimulus dimensions and a learning process
by which responses are assigned to the values of a selected stimulus
dimension; b o t h processes are assumed to be all-or-none. More specifically, for simple two-category concept problems with binary stimulus
dimensions to which the theory has been mainly applied, it is assumed
that on each error trial S chooses a dimension at random from the set
1 Paper presented at the annual meeting of Gruppe Mathematische Psychologie, Bern, Switzerland, 1969. The authors wish to thank Micha Razel for helpful comments.
2 Now at New York University, Department of Psychology.
216
J. Wandmacher, D. Vorberg, The Bower and Trabasso theory
of all stimulus dimensions. A simplifying assumption is that the response assignment phase is completed with the selection of the relevant
dimension. The S is assumed to accomplish this by behaving with 'local
consistency' (Trabasso and Bower 1968); whenever he selects a new
dimension he assigns responses such that they are consistent with the
stimulus and feedback given on the last trial. Since S is assumed to
respond always according to the dimension he currently holds and to
resample only on error trials, these assumptions lead to a finite Markov
process with an absorbing state which corresponds to the solution state
of the problem in which only correct responses are given, and two
transient states in which correct and incorrect guesses occur, respectively.
The B - T theory was supported in a series of experiments (Bower
and Trabasso 1963, 1964; Trabasso and Bower 1964b, 1966, 1968) and
was shown to provide an excellent quantitative fit to data from simple
concept identification experiments. On the other hand, not all process
assumptions of the theory, particularly the assumption of exclusive
learning after errors, have been confirmed in direct experimental tests
(Chumbley 1967 ; Dodd and Bourne 1969; Falmagne 1970).
The concept identification problems employed in this study were
more complicated in two respects. First, the problems to be learned
involved four conjunctive concepts with two relevant dimensions. Second, under some experimental conditions probabilistic feedback was
given, so that on some percentage of the trials the feedback was misinformative and false. An application of the B - T theory to fourcategory concept identification was described by Trabasso and Bower
(1964a). Experiments with concept learning under misinformative feedback conditions have been performed by Morin (1955), Pishkin (1960)
and Bourne (1963); however, a successful application of a mathematical
theory to concept learning under this condition is not known to the
authors. It would be of interest and theoretical significance if the B - T
theory, which was developed to account for concept identification performance in a highly artificial and simple experimental paradigm, could
cope with the learning of more complicated problems under noisy feedback conditions. In spite of the limitations mentioned above, the B - T
theory seems to be a good starting point for applications to more
realistic conceptual problems.
J. Wandmacher, 19. Vorberg, The Bower and Trabasso theory
217
2. Experiment
2.1. Method
Stimulus material
The stimuli to be classified were pictures of geometric figures which could vary in up to five
dimensions. The dimensions were:
a
b
c
d
e
form:
size:
number:
position:
outline:
triangle or square;
large or small;
one or two figures;
figures could be located in the right or the left half of the picture;
figures could be drawn with a single or double black outlines.
The background of the pictures was a white rectangle, 30 cm in width and 21 cm in height. The
stimuli were projected with a slide projector on a screen which was at a distance of 1 m from
the S.
Apparatus and procedure
The four combinations of the values of two relevant dimensions defined the four concept
categories; to each category, a color (red, blue, green, or yellow) was assigned as a class name.
Thus, a possible conceptual problem could be 'large triangle - red; small triangle - blue; large
square - green; small square - yellow'.
A trial started with the presentation of a stimulus picture the duration of which was
controlled by the S's response. When S had named a color, a feedback light was turned on for 1
sec whose color indicated the class name of the stimulus presented. After an intertrial interval
of 4 sec during which the stimulus slide was removed, the next trial started. The experiment
was finished when S had reached a learning criterion of 16 successive correct responses or after
208 trials, whichever was first.
Design
Two independent variables, number of irrelevant dimensions (factor A) and probability of
correct feedback (factor B) were varied orthogonally. Both factors had three levels: either one,
two or three dimensions were irrelevant, denoted by A 1, A 2 and A a, respectively; the probability of a correct feedback could be 1.00, 0.844, or 0.688, denoted by B1, B2 and B3, respectively. Each of the nine conditions AiB] was replicated twelve times, with each of the six pairs
of the four dimensions a, b, c and d used twice as relevant dimensions under each condition.
Dimension e was kept constant except under the A 3 conditions, where it was only used as an
irrelevant dimension.
The presentation order of the stimuli was random with the restriction that all the stimuli of
the stimulus set had to be presented before a stimulus could be repeated.
Under the misinformative feedback conditions (B2 and B3) incorrect feedback occurred
with equal frequency for each of the four concept categories; on a misinformative feedback
trial, each of the remaining three incorrect feedbacks was equiprobable.
J. Wandmacher, D. Vorberg, The Bower and Trabasso theory
218
Table 1
Mean total o f errors (T) and mean trial n u m b e r of the last error (L) of the C subjects who
reached the learning criterion.
B1
B2
B3
Al
T
L
C
13.7
25.5
12
25.0
62.0
12
47.8
109.2
12
A2
T
L
C
18.6
39.5
12
37.8
92.5
12
79.8
188.3
6
A3
T
L
C
42.3
94.2
12
75.4
152,2
12
84.5
191.5
2
Sub/ects
Forty-two female and 66 male students of the University of Hamburg served as Ss. The Ss
were randomly assigned to the nine experimental conditions; each S solved one conceptual
problem.
Instruction
After explaining and demonstrating the variable stimulus dimensions, Ss were told that they
had to learn h o w the stimuli could be classified into the four concept categories. The Ss were
informed that the classification was to be done only according to the values of the dimensions
of a stimulus; no hint was given about the structure of the concepts to be learned, i.e., that the
concepts were conjunctive. Under the misinformative conditions B2 and B3, Ss were warned
that E could occasionally give an incorrect feedback, b u t that nevertheless a unique classification rule existed which they should try to discover and according to which they should
respond.
2.2. Results
Each response inconsistent with the classification rule was scored as an error even when it
coincided with a given incorrect feedback. Table 1 presents a gross s u m m a r y of the results; for
each of the nine conditions AiBj the average total number of errors, the average trial number of
the last error and the n u m b e r of Ss reaching the learning criterion are given. As can be seen
from inspection o f the data, n u m b e r of irrelevant dimensions and probability of correct feedback markedly influenced concept identification performance.
3. Theoretical analysis of the data
3.1. Component learning in the four-category concept problem
Bourne and Restle (1959) have proposed that the four-category problem with its two
J. Wandmacher, D. Vorberg, The Bower and Trabasso theory
219
relevant dimensions is learned as two independent subproblems, where each subproblem corresponds to a simple concept identification problem with one relevant dimension. If the two
subproblems are in fact learned independently of each other, the theoretical task can be
simplified considerably by applying the B - T theory directly to subproblem learning, as has
been suggested by Trabasso and Bower (1964).
Let {Xn } be a sequence o f r a n d o m variables which indicates whether the response on a given
trial was correct, i.e., X n = 0 if the response on trial n was correct, and X n = 1 otherwise. In the
same way we construct two sequences {Xn,i~, i = 1,2, which indicate whether the response
given on trial n was correct with respect to subproblem i, i = 1,2. An example should clarify the
definitions and the scoring procedure employed: assume that S has to learn the problem 'large
triangle - red; small triangle - blue; large square - green; small square -- yellow', and on trial n
he responds 'blue' to a stimulus showing a large triangle. The correct response would have been
'red', therefore X n = 1. With respect to the relevant dimension o f form, the response was
correct, since all triangles either belong to the categories 'blue' or 'red'; however, with respect
to the relevant dimension o f size, the response is incorrect, since large figures are either 'red' or
'green', b u t never 'blue'. If subproblem 1 corresponds to learning that form is relevant, and
subproblem 2 to learning that size is relevant, it follows that Xn, 1 = 0 and Xn, 2 = 1.
If the two subproblems are learned independently,
P(X n = l) = P(Xn, l = l ) + P(Xn, 2 = 1 ) - P(Xn, 1 = 1)P(Xn, 2 = 1).
(1)
Since the difficulty of the problems was not significantly affected by which dimensions were
relevant, it will be assumed in the sequel that P(Xn, 1 = 1) .--P(Xn, 2 = 1), and (1) becomes
P(X n = l ) = 2 P ( X n , 1 = 1 )
P(Xn, 1 = 1 ) 2 .
(2)
In order to test the independence a s s u m p t i o n for each condition AiBj, the expected error
frequencies in the original response sequence was predicted from the error frequencies in the
subproblem sequences using (2). The expected and predicted error frequencies agreed fairly
well and a comparison of the predicted and observed frequencies resulted in insignificant
chi-square statistics for each condition (p > 0.3). However, this test is not very stringent since
the observed and predicted frequencies were not independent o f each other. A further check of
the independent learning a s s u m p t i o n was conducted by using the B - T theory to predict the
total n u m b e r o f errors in the original four-category problem from the total n u m b e r of errors
observed in the subproblem response sequences.
S u m m i n g (2) from n = 1 to n = oo, we obtain
P(X n = 1 ) = 2 ~ 1 P(Xn, l = 1 ) n=t
n=l
~ P(Xn,1 = 1)2;
n=l
it follows that
E ( T ) = 2E(T1) - ~ P(Xn, 1 = 1) 2,
n=l
(3)
where T is the total n u m b e r o f errors in the four-category problem and Ti is the total n u m b e r
220
J. Wandmacher, D. Vorberg, The Bower and Trabasso theory
of errors in the subproblem i. By using the B - T theory and estimating the parameters from the
subproblem data only (cf. 3.2) E(T) was predicted from (3) for each of the conditions AiB 1
with errorless feedback. T a b l e 2 presents the observed m e a n number of errors together with the
corresponding predictions. Of course, the predictions are not completely independent of the
observed data since part of the data had to be used to estimate the parameters. In order to see
h o w strongly this dependence restricts the relationship between the expected n u m b e r of errors
in the four-category response sequence and the subproblem response sequences, we derive
b o u n d s for E(T).
In general, it is true that
P(X n = 1 ) = P(Xn,1)+ P(Xn, 2 = 1) - P ( X n , 2 = l tXn, 1 = 1)P(Xn, 1 = 1),
which reduces to
P(X n = 1) =-P(Xn, 1 = 1)[2 - P ( X n , 2 = 1 IXn, 1 = 1)],
if the two subproblems are equally difficult. Since the conditional probability m u s t be between
zero and one, we find
P(Xn, 1 = 1) < P(X n = 1) < 2P(Xn, 1 : 1).
S u m m i n g from n = 1 to n = ~ , the result is
E ( T 1 ) < E(T) ~< 2E(T1) ,
(4)
which means that the dependence between the four-category response sequence and the subproblem response sequences restricts the total n u m b e r of errors in the original sequence, E(T),
to be at least as large as E(T1), the total n u m b e r of errors in the subsequence, b u t not larger
t h a n twice E(T1). Table 2 presents the b o u n d s for each prediction. It is evident that compared
with the rather large possible ranges the predictions are surprisingly close to the data.
Analyses of t h e data obtained under the misinformative conditions give the same results.
Under every condition, the agreement between predicted and observed error frequencies in the
original sequences is excellent, using t h e chi-square criterion m e n t i o n e d above. Since the derivation o f the expected n u m b e r of errors and the corresponding b o u n d s is m u c h more involved for
the misinformative conditions, we do n o t present the detailed results.
We conclude that there is good evidence in favor of the assumption of independent subproblem learning. In the following, all the analyses will therefore only be concerned with the
subproblem response sequences. It should be noted that with respect to subproblem learning
the n u m b e r o f irrelevant dimensions in the A 1, A2 and A3 conditions is two, three and four,
respectively; the n u m b e r of response sequences on which the analyses will be based is 24 per
condition AiB j (12 Ss X 2 subproblems).
3.2. Application o f the B - T theory to subproblem learning under
correct feedback conditions
The B - T theory implies stationarity of the error probability before the last error, if the
usual a s s u m p t i o n is made that the response assignment phase is finished as soon as the relevant
J. Wandmacher, D. Vorberg, The Bower and Trabasso theory
221
Table 2
Observed and predicted m e a n totals o f errors in the original response sequences and ranges o f
predictions. The predictions and ranges are based on the subproblem sequences and c o m p u t e d
from (3) and (4).
Conditions
Observed
Predicted
Range
A1B1
A2BI
A3BI
13.7
13.8
8.3 - 16.6
18.6
19.0
10.9 - 21.8
42.3
41.9
24.2 - 48.4
dimension has been selected. Analyses of the subproblem response sequences with Jonckheere
and Bower's (1967) test for stationarity revealed strong non-stationarity under all three conditions with correct feedback; this agrees with Trabasso and Bower's (1964a) finding. A possible
explanation for this is that in the four-category problem response assignment with local consistency is not sufficient to learn the correct responses to both values of the relevant dimension in
one trial; if S has assigned a response to one value of the relevant dimension, he needs at least
one additional trial o n which a stimulus with the remaining value of the relevant dimension is
presented in order to learn which of the remaining three responses to attach to it. For this
reason, the B - T theory was generalized to encompass a paired-associate learning phase which
accounts for the learning o f the response assignments, Following Bower and Trabasso's (1964)
suggestion, the paired-associate phase is assumed to be all-or-none.
The generalized B - T theory starts from the following assumptions. As long as the S holds
an irrelevant dimension, a new dimension is sampled on every error trial. The probability of
sampling the relevant dimension equals c. A response is assigned to the newly sampled dimension with local consistency, i.e., the response assigned is consistent with the feedback received.
If a stimulus is presented to whose value o f the relevant dimension a response has not yet been
assigned, S forms the second response assignment with probability a; with l-a, nothing is
learned. On a n y ~rial, S gives his responses according to what he has learned or guesses if
necessary.
These a s s u m p t i o n s lead to a four-state Markov chain whose transition matrix, starting vector
and response vector are given by (5).
L2
L1
F
o
o
l-0.5a
0
F
c
(l-c)q0
(1 - c ) ( 1 - q 0
R
0
q0
l-q0
L1
0 a
R
P(error)
(5)
-J
SV= (0, O, qo, 1 - q o )
The states o f the chain have a simple interpretation: F and R are states in which S is on a trial
before the relevant dimension is selected and on which he guesses incorrectly or correctly,
respectively; in L1, S has sampled the relevant dimension but learned only one of the two
response assignments; in L2, b o t h response assignments are formed. Before the relevant dimen-
222
J. Wandmacher, D. Vorberg, The Bower and Trabasso theory
sion is sampled, S guesses incorrectly with probability q 0; in L 1, he responds correctly on those
trials on which stimuli are presented to which a response is learned, and guesses on the remaining trials. In L2, all responses are correct. It is easy to show that q0 = 0.5 and q 1 = 0.25, if the
stimuli are presented with equal frequency. The model thus predicts that S should start with an
initial error probability of 0.5 which decreases to 0.25 on the trial on which the relevant
dimension is sampled and to zero w h e n the second response assignment is learned. Therefore,
before the last error no stationarity of the error probability is expected.
In order to see h o w well the model describes the detailed features of the data, predictions
for a n u m b e r of statistics were derived using the standard techniques for Markov learning
models (see e.g., Atkinson et al. 1965; or Vorberg 1970) and compared to the observed data.
For each of the three conditions AiB 1 with correct feedback, the two model parameters a and c
were estimated separately by using a modified m e t h o d of m o m e n t s which minimized the
discrepancy between the observed and the predicted mean number of errors, T, m e a n trial
number of the last error, L, and the mean n u m b e r of adjacent error pairs, U s. T h e statistic used
to measure the discrepancy was
s 2 = ( T - E ( T ) ) 2 / E ( T ) + (L - E(L))2/E(L) + (U 2 - E(U2))2/E(U2).
Table 3 presents the values of the observed statistics together with the corresponding predictions. The agreement predictions and data is excellent. It should be noted, that the predicted
standard deviations of the n u m b e r of errors and the trial n u m b e r of the last error are quite close
to the observed standard deviations, thus lending support to the implicit assumption of parameter h o m o g e n e i t y between Ss. We conclude that the modified B - T theory adequately describes
four-category concept identification under correct feedback conditions. Notice in passing that
all the parameters seem to be dependent on the condition, i.e., the n u m b e r of irrelevant
dimensions. According to the B - T theory, the parameter c should be proportional to the
relative saliency of the relevant dimensions and thus decrease as the n u m b e r of irrelevant
dimensions increases; however, the theory does not include a m e c h a n i s m which accounts for
the dependency of the response assignment parameter a on the n u m b e r of irrelevant dimensions. 3 This problem will be taken up in the discussion.
3.3. Application o f the B - T theory to subproblem learning under raisinformative feedback conditions
Let f ' be the (experimentally controlled) probability that a given feedback is correct. Since
exactly one of the three incorrect feedback alternatives in the four-category problem is correct
with respect to a subproblem, the probability of a correct feedback for each subproblem is
f = f ' + l / 3 ( 1 - f ) . Hence, the probability f equals 0.896 and 0.792 for the B~ and B3 conditions,
respectively.
An incorrect feedback converts an objectively correct response into a subjective error and vice
versa. Therefore, according to t h e B - T theory, resampling of a new dimension takes place if the
S's response was incorrect and a correct feedback was given or if the response was correct b u t
was followed by an incorrect feedback. Thus, a transition from state F to state L1 occurs with
probability re. With probability (1--f)c S samples the relevant dimension after a correct re3 The differences between the obtained estimates of a seem to be reliable and not simply due to
sampling error. Estimation of one c o m m o n parameter a for all three conditions results in a
markedly reduced goodness of fit.
,1. Wandmacher, D. Vorberg, The Bower and Trabasso theory
223
sponse in state R followed by an incorrect feedback. If this happens, S attaches an incorrect
response to the value of the relevant dimension, since response assignment is assumed with local
consistency; S thus reaches a new state L 1' characterized by attention to the correct dimension
but with an incorrect response assignment. The error probability in this state is q'l = 1 - q l =
0.75.
A feedback inconsistent with the S's current hypothesis can occur if S is in L1 and an
incorrect feedback is given or if S is in L I ' and a correct feedback is given. We investigate three
different assumptions about transitions from states L1 and L I ' when inconsistent feedback
occurs.
3. 3.1. Response Reversal (RR).
if an inconsistent feedback occurs with probability a S simply reverses his response assignment, i.e., replaces the old response assignment with one that is consistent with the feedback
given; with probability 1 - a the old response assignment remains unchanged. Response reversal
thus causes transitions from L 1 to L 1' or vice versa. For reasons of simplicity it is assumed that
the probability of a response reversal is equal to the probability of assigning a new response.
The RRV model is characterized by the following transition matrix:
L2
L1
LI'
F
R
L2
1
0
0
0
0
L1
0.5fa
1-(1-0.533a
(1-f)a
0
0
1-fa
0
0
0
(1-fc)q 0
(1 - f c ) ( l - q o )
(1 - f ) e
(l-c+fc)qo
(1-e+fc)(1 qo
LI'
0
fa
F
0
f
R
0
0
(6)
3. 3.2. Resarnpling with R eplacern en t (R WR)
An incorrect feedback in state L1 or LI' according to this model causes S to dismiss his
hypothesis and to resample. With probability e the relevant dimension will be sampled again
and the response now consistent with the current feedback will be assigned to it. The transition
matrix of this model is given by
LI
LI'
L2
L1
LI'
F
R
.
f(1-0.5a)
(1 - f ) c
(1 f)(1 - c ) q 0
(1-f)(1-c)(1-q 0
fc
l-f
f(1-e)q 0
f ( 1 - c ) ( l - q O)
~
F
0
fe
0
(l-fc)q 0
R
L_ 0
0
(1-De
(1-c+fe)q 0
(1-fe)(1-qo)
(7)
d
(1-c+fc)(1-q O) j
3. 3. 3. Resampling with Delayed Replacement (RDR)
After an inconsistent feedback in L1 or L I ' , S resamples but this time it is assumed that he
can not immediately resample the previously held dimension, i.e., the last dimension is assumed
J. Wandmacher, D. Vorberg, The Bower and Trabasso theory
224
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c~
0
~
o o
,
0
z
~
,.0
,.Q
,.~
¢3
~
~
~
~
,
°
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Z Wandmacher, D. Vorberg, The Bower and Trabasso theory
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~ t t 3
¢qCq~
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225
J. Wandmacher, D. Vorberg, The Bower and Trabasso theory
226
to be available for resampling only after some delay. This assumption implies that every inconsistent feedback in L1 or LI' causes a transition to F or R, since S can only sample an irrelevant
dimension. The transition matrix for model R D R is as follows:
L2
L2
rl
L1
LI'
F
R
0
0
0
0
0.5J(2-a)
0
(1-jOq 0
0
l-f
fqo
0
(1-fc)q 0
(1 -fe)(1-qo )
(1-De
(1-e+fc)qo
(1-c+fe)(l-ql)]
!
L1
LI'
F
R
0.5fa
[~
|0
fc
0
( 1 - f ) ( 1 - q 0)
(8)
f(1-q0)
For all three models it is assumed that once S enters state L2 he is absorbed in that state,
i.e., he ignores all further incorrect feedback. Of course, this assumption can only be approximately correct; the B - T theory does not provide a m e c h a n i s m by which S could decide
whether he has sampled the correct dimension and receives only incorrect feedback or whether
inconsistent feedback is due to the fact that his currently held dimension is irrelevant. A
possible decision rule would be to keep a dimension if it has led to k consistently reinforced
8
trial n
t0
56
24
I
I
~
Q
i
I
t
t
I
I
7f
I
t
--
I
I
I
I
i
I
I
T~
predicted
observed
I
•4
96
I
I
I
I
I
AIB3
7
2
0
0
°0~
L
0
I
16
[
/
48
I
I
~0
I
I
112
I
[
144
I
Z ~
176
trial n
Fig. 1. Predicted and observed learning curves in blocks of 8 and 16 trials in c o n d i t i o n s A 1B2
and A 1B3, respectively.
Z Wandmacher, D. Vorberg, The Bowerand Trabasso theory
trials
16
4~
I I
i
80
l
r
l
227
n
112
l
1zi4
l
l
l
l
176
,4
.7
0 observed
.2
.I
.0
I
I
I
I
I
I
f
I
I
i
I
f
.5
.4 ~ 0 ~ 3 ~
0
o
A2B3
o
o
t
I
144
.1
°o
I
16
r
I
48
f
I
80
i
I
112
trials
J It76 i
n
Fig. 2. Predicted and observed learning curves in blocks o f 16 trials in conditions A2B 2 and
A2B3.
responses, where k is a parameter reflecting S's willingness to tolerate inconsistent information;
this possibility was not investigated further.
The three alternative assumptions for handling inconsistent information are not mutually
exclusive; it might be for example that S sometimes uses response reversal and sometimes
resampling when inconsistent feedback is encountered, or it might be that S sometimes samples
with and sometimes w i t h o u t replacement. In order to investigate this possibility, mixed models
were tested in addition to the three pure models (6), (7) and (8) in which it was assumed that
the different assumptions are effective only with a given probability. The following four special
cases were considered: ( R R - R W R ) , ( R R - R D R ) , ( R W R - R D R ) with each pure model effective
with probability 0.5 and ( R R - ( R W R - R D R ) ) , where RR model had a probability of 0.5 and
both resampling models had a probability o f 0.25 each.
For each condition AiB i predictions for the three pure and the four mixed models were
numerically derived by using the parameter estimates from the corresponding AiB 1 condition
and the experimentally controlled value of f for that condition. Thus, no parameters were
estimated from any data under misinformative feedback conditions; performance under misinformative conditions was predicted solely from performance under conditions with consistently correct feedback.
Table 4 presents the observed and predicted m e a n s of several statistics. Since under some o f
228
J. Wandmacher, D. Vorberg, The Bower and Trabasso theo~T
the conditions n o t all Ss reached the criterion within 192 trials, the predictions are based on the
first 192 trials only. Since none of the models was consistently superior for all conditions, the
predictions of the best fitting model are displayed for each condition. In order to test the
discrepancies between the observed and predicted means, t-tests were c o m p u t e d using the
corresponding observed standard deviations. Three out of 24 differences between observed and
predicted m e a n s turned out to be significant (p ~< 0.05, cf. table 4). Considering the fact that
these tests underestimated the error variance since the predicted values were treated as errorless,
the results seem quite impressive. However, it should be noted that the predicted standard
deviations are consistenly larger than the observed ones.
Figs. 1, 2 and 3 display the predicted and observed learning curves for the six conditions
with misinformative feedback. Again the fit is remarkable, especially so if we take into account
that the predictions were done from conditions with widely different error rates.
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J. Wandmacher, D. Vorberg, The Bower and Trabasso theory
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4. Discussion
The B T theory was originally developed to account for concept
identification performance in situations involving simple concepts with
a single relevant dimension and correct feedback on every trial. We have
found that the theory can be successfully extended in two respects: to
learning of concept problems with four conjunctive concepts and to
learning under misinformative feedback. Let us comment upon each in
turn.
Our finding that four-category concept learning can be broken down
into the learning of two concurrent and independent subproblems confirms Trabasso and Bower's (1964a) results who found strong evidence
in favor of the independence assumption. They also observed that in
order to account for subproblem learning at least a two-stage process
had to be assumed because of the marked non-stationarity in precriterion performance; this again was replicated by our results. Since
the assumption of a separate response assignment learning stage seems
necessary upon a logical analysis of the problem, as was noted earlier,
these findings seem promising for a generalization of the B - T theory,
especially so if we take into account that contrary to the usual practice
the Ss were not informed about the structure of the concepts to be
learned in the present experiment.
Still, some puzzles remain. As Chumbley (1972) has pointed out, the
independence assumption is somewhat counterintuitive; even if the subproblems are dealt with independently by the Ss, we should expect that
after the first subproblem is solved learning on the second problem
should proceed faster since Ss can eliminate the relevant dimension of
the first subproblem from consideration. It is not clear whether this
deviance from independence is strong enough to be detected by the
presently employed tests.
A second related problem is that the estimated parameters do not
show the dependence upon conditions predicted by the theory. The
parameter a o f the paired-associate phase should be independent of the
number of irrelevant dimensions; however, under condition A3B 1 it is
considerably smaller than under the remaining conditions. Moreover,
according to the theory the estimates of c should be proportional to the
number of irrelevant dimensions if dimensions are assumed to be of
equal salience and background cues are ignored (Trabasso and Bower
1968: 27) i.e., the values of c for conditions A I B 1, A2B l and A3B 1
232
J. Wandmacher, D. Vorberg, The Bower and Trabasso theory
should be in ratio 13 . 41 . 51 to each other. Since the obtained estimates
differ markedly from this prediction, we are forced to conclude that
some of the process assumptions of the theory are incorrect.
Quite surprising to us was the finding that performance under misinformative feedback conditions can be accurately predicted from performance under correct feedback conditions. Considering the large increase in the number of trials necessary to reach criterion with increasing probability of incorrect feedback, it is quite impressive that the
order of magnitude of this increase is adequately predicted by the
model. It should be remembered that the model traces the additional
number of trials to criterion to the state L l' which is a logical consequence of the original assumptions. There seems to be no evidence for a
qualitative difference between concept identification under correct and
under misinformative feedback conditions, which could have been ex.pected since the Ss under the misinformative feedback conditions were
informed about the possibility of incorrect feedback.
Again, there remain some problems. First, as pointed out before, the
model gives only an approximately correct description of the on-going
processes since it cannot explain how Ss manage to ignore inconsistent
feedback. Second, the observed variances were smaller than the predicted ones; this indicates that the underlying processes are more complicated than those assumed. If a Markov model is adequate at all, the
model needed in order to remedy the variance prediction would have
more states than the proposed model. Third, the fact that none of the
pure and the mixed models was consistently superior demonstrates again
a kind of parameter invariance. We could conceive of a general model in
which two parameters determine probabilistically whether response
reversal or resampling with or without resampling occurs on a given
trial. We have considered only a few special cases of this general model.
From the comparison of these models with the data it is clear that the
two additional parameters will not be invariant over conditions; instead,
resampling seems to occur more often on trials with inconsistent feedback as the probability of incorrect feedback increases. No explanation
for this can be offered from the theory.
In summary, we conclude that the B - T theory is flexible enough to
permit quite accurate quantitative predictions in much more complex
concept identification situations and certainly deserves further study.
On the other hand, since the underlying process assumptions of the
theory are doubtful, they should be investigated in more detail in
future application of the theory.
Z Wandmacher, D. Vorberg, The Bower and Trabasso theory
233
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