The Psychological Record, 2000, 50, 79-104
PROBABILITY OF STIMULUS EQUIVALENCE AS A FUNCTION
OF CLASS SIZE VS. NUMBER OF CLASSES
ERIK ARNTZEN and PER HOLTH
University of as/a, Norway
Two experiments were conducted to study stimulus
equivalence as a function of class size and number of classes. In
the first experiment, equivalence was tested in 50 normal adult
subjects following a linear series training structure. Subjects were
successively assigned to either of 10 groups, exposed to a specific
stimulus material. For subjects in which number of classes
increased, up to six, B-stimuli served as pictures, while A-, and Cstimuli were Greek letters, and "equivalence" was tested in CA
tests. The A-, B-, and C-stimuli were the same for all subjects in
whom class members increased up to six, where D-, E- and Fstimuli were Greek letters. FOllowing AB, BC, CD, DE, and FE
training , FA, EA, FB, FC, EB, DA, FD, EC, DB, and CA
"equivalence" tests were run. In the second experiment, a manyto-one training structure was used to study equivalence as a
function of increasing class size without increase number of
nodes. The results indicate that the probability of equivalence
decreased more as function the number of nodes than as a
function of number of classes. Reaction times, particularly to the
comparison stimuli, generally increased initially during tests,
possibly indicating precurrent problem solving behavior prior to the
response to a comparison stimulus.
Equivalent stimulus classes include relations not explicitly trained.
The equivalence relation is defined by reflexivity, symmetry, and
transitivity (e.g., Sidman & Tailby, 1982). Three different training structures
have been used, linear series, many-to-one (comparison as node) and
one-to-many (sample as node) to study equivalence (e.g., Green &
Saunders, 1998). Stimulus equivalence has been consistently
demonstrated in language-able human subjects, using a variety of
stimulus materials (e.g., Barnes, McCullagh, & Keenan, 1990; Devany,
Thanks to Lanny Fields for comments on parts of an earlier version of the manuscript
and we also thank Dermot Barnes and an anonymous reviewer for helpful suggestions.
Correspondence and reprints requests may be sent to Erik Arntzen, Department of
Psychology, University of Oslo, PO Box 1094 Blindern, 0317 Oslo, Norway. (E-mail:
Erik.Arntzen@psykologLuio.no).
80
ARNTZEN AND HOLTH
Hayes, & Nelson, 1986; Fields, Reeve, Rosen, Varelas, Adams, Belanich,
& Hobbie, 1997; Sidman, Kirk, & Willson-Morris, 1985; Sidman & Tailby,
1982; Sidman, Willson-Morris, & Kirk, 1986).
However, equivalence classes in adults are unlikely to emerge following
a linear series training structure when all baseline relations are trained
concurrently and all probes for emergent relations are then introduced
concurrently (Fields et aI., 1997). Similar results with 3 three-member
stimulus classes were found by Arntzen and Holth (1997) when comparing
percentages of subjects responding in accord with equivalence following
linear series, many-to-one, and one-to-many training structure. Following a
linear series training structure, only 2-3 of 10 subjects responded in accord
with a positive equivalence test. Because the probability of a positive
equivalence outcome is low after linear series training, such a structure is
suitable for identification of historical variables that enhance equivalence
class formation by adults (Fields et aI., 1997).
The use of familiar stimulus material could obviously affect the
probability of equivalence outcome. One study (Holth & Arntzen, 1998a) was
concerned with effects of familiar stimuli on the rates of equivalence
formation in a linear series training structure. The results showed that the
probability of equivalence was low when the stimulus material consisted only
of Greek letters. For the remaining groups, the probability of equivalence
varied considerably depending upon whether the A-, B-, and/or C-stimuli
were pictures, and 10 of 10 subjects responded in accord with equivalence
when pictures served as B-stimuli. The results indicated that such a training
structure may be suitable for studying effects of class size and number of
classes on the formation of stimulus equivalence.
Fields, Verhave, and Fath (1984) have predicted that there is an
inverse relation between number of nodes and measures of response
strength (accuracy and speed), and that the number of nodes may
differentially affect the order in which derived relations come to exert
stimulus control. Furthermore, derived relations would emerge
differentially depending upon the number of nodes between stimuli during
training. Similarly, Kennedy (1991) argued that derived relations emerge
only after stimulus control by other derived relations with fewer nodes has
been established , and less complex derived relations could be
prerequisites for more complex derived relations.
Saunders, Wachter, and Spradlin (1988) and Sidman et al. (1985)
have both reported disruption in the emergence of equivalence classes
that seems to be a function of the number of nodes separating stimuli.
Furthermore, the results indicated that for some subjects control by
stimuli separated by several nodes tended to emerge only after untrained
relations separated by fewer nodes were established. Similarly, Haring,
Breen, and Laitinen (1989) and Kennedy and Laitinen (1988) have
reported disruptions in subject performances that may be a function of
nodal number. Kennedy (1991) showed that in 2 seven-member stimulus
classes the number of nodes separating stimuli influenced the
emergence of derived relations in the formation of equivalence classes.
CLASS SIZE VS. NUMBER OF CLASSES
81
Also, the same effect was obtained with 3 seven-member stimulus
classes. However, Sidman (1994) pointed out that with a higher number
of comparison stimuli, the influence of nodal number on probability of
equivalence class formation is less obvious, and he called for
experiments with three or more comparison stimuli per trial.
Spencer and Chase (1996) argued that measures of response speed
are important because they provide a more thorough understanding of
the substitutability of stimuli in an equivalence class than do accuracy
measures alone.
Class formation is influenced by the training structures, the structure
of the class as governed by the relations trained as the prerequisites for
the classes, the familiarity of stimuli that are members of the potential
classes, and the nodal structure of the class. A variable that has not yet
been explored is the effect of number of classes on class formation. In the
current experiment, the combined effects of number of classes and class
size will be explored. This will be done by using a linear series training
structure with familiar stimuli as B-stimuli. In addition to an index of
equivalence, measures relevant to the readiness of class formation
include reaction times to comparison stimuli and a measure of the
consistency of responding during testing.
Experiment 1
Method
Subjects
Fifty adults served as subjects, university or college students, and
were successively assigned to 10 different experimental groups. The
subjects were familiar with neither the stimulus equivalence research nor
the stimulus material presented, except the pictures.
Apparatus
The apparatus was the same as described in Arntzen and Holth (1997).
Procedure
Stimulus material. Visual stimuli were displayed on the monitor. Greek
letters and Cyrillic symbols were used in the experiment (see Table 1).
Furthermore, the pictures used as B-stimuli are shown in Table 2. The
presentation of the sample stimulus was always in the left-hand key (7 x 7
cm) of the monitor. Six comparison stimulus keys (4 x 4 cm) were arranged
in two columns and three rows on the right-hand side of the monitor.
As class size increases in a linear series training structure, the
number of nodes in the class also increase as a function of class size. If
class size is N, number of nodes will be N-2 (Fields & Verhave, 1987;
Fields et aI., 1984).
General information to the subjects. When asked to join the
experiment, the subjects were told that the experiment was within the field of
learning and was concerned with tasks presented on a computer with a touch
ARNTZEN AND HOLTH
82
Table 1
Stimuli Used in Experiment 1
Classes
Class members
A
0
2
3
4
5
6
<p
A
1K
eJ
3
B
Pictures
C
'P
(
8
V
~
D
P
a
11
3
.!l
E
Y
11
~
~
F
.!l
ts
~
Table 2
Pictures Used as B-stimuli
--. ,
<~~
~--
;
Note. The number of pictures was dependent upon how many classes were used, but
always included the three pictures in the upper row.
83
CLASS SIZE VS. NUMBER OF CLASSES
screen. They were also told that the experiment would last for approximately
60 min, depending on how rapidly and correctly they responded.
Instruction. The experimenter gave the following instruction:
When you touch the left-hand stimulus, one or more stimuli will
appear on the right. A touch on the correct stimulus will be followed
by music from the cassette player, and incorrect responses will be
followed by the blanking of the screen for 5 s before a stimulus in
the left-hand key is presented again. Each part of the training
requires a certain number of correct responses before proceeding
to the next part. The training will be followed by tests, in which
there will be no different consequences for correct and incorrect
responses - no music and no blank screen.
Training and test. Each trial started with the presentation of a sample
stimulus. A touch on the sample stimulus was followed by the
presentation of comparison stimuli in the keys on the right side of the
monitor. The sample remained until a comparison stimulus was touched.
To minimize the number of errors initially during training, the conditional
discrimination tasks were introduced step by step: When a sample
stimulus appeared for the first time during training, a touch on the sample
stimulus was followed by the presentation of the correct comparison
stimulus only. Next, each correct comparison was presented together with
one incorrect comparison, then with the second incorrect comparison,
and this was gradually increased until all comparison stimuli were
presented. The comparison stimuli appeared in a random position from
trial to trial. Both AB training and BC training required the successive
correct completion of the randomly intermixed seven trials of each type
before testing. For the subjects in which class size increased also CD, DE,
and EF training required the successive correct completion of the
Table 3
Stimulus Relations Trained for Each Class Size and Number of Classes
Classes
Members
4
3
4
5
A1B1 , A2B2, A.'lB3
B1C1, B2C2, B3C3
A1B1 , A2B2, A.'lB3, A4B4 A1B1 , A2B2, A.'lB3, A4B4, A5B5
B1C1, B2C2, B3C3, B4C4, B1C1, B2C2, B3C3, B4C4, B5C5
A1B1 , A2B2, A.'lB3
B1C1, B2C2, B3C3
C1D1 , C2D2, C3D3
A1B1 , A2B2, A.'lB3, A4B4 A1B1 , A2B2, A.'lB3, A4B4, A5B5
B1C1, B2C2, B3C3, B4C4 B1C1, B2C2, B3C3, B4C4, B5C5
C1D1 , C2D2, C3D3, C404 C1D1 , C2D2, C3D3, C404, C5D5
A1B1, A2B2, A.'lB3
B1Cl, B2C2, B3C3
C1 01 , C2D2, C3D3
01E1 , D2E2, D3E3
A1B1, A2B2, A.'lB3, A4B4
B1C1, B2C2, B3C3, B4C4
C1D1 , C2D2, C3D3, C404
01E1 , D2E2, D3E3, D4E4
A1B1 , A2B2, A.'lB3
B1C1, B2C2, B3C3
C1D1 , C2D2, C3D3
01E1 , D2E2, 03E3
E1 Fl , E2F2, E3F3
6
A1B1 , A2B2, A.'lB3, A4B4, A5B5, A6B6
B1C1, B2C2, B3C3, B4C4, B5C5, B6C6
84
ARNTZEN AND HOLTH
randomly intermixed of seven trials of each type before testing. The
equivalence tests were organized as two consecutive blocks with four of
each trial type in each test block. For subjects in groups in which the class
size increased the test with the largest number of nodes was given first,
and tests which included pictures at the end.
The experimental conditions for the 10 groups are summarized in Table 3.
Dependent measures. Key presses on the touch screen in front of the
monitor, reaction times, and the number of trials to criterion were
recorded. An index of equivalence was calculated for each subject on
each test half by dividing number of "correcf' responses by the total
number of trials during each phase of the test. Equivalence was defined
as an index of 0.9 or 1.0.
For each subject who did not respond in accord with equivalence,
consistency of responding was evaluated throughout the equivalence
test. Similar to a "moving average," consistency indices were calculated
for each block of six consecutive trials as described in Holth and Arntzen
(1998a). In the group with 3 three-member stimulus classes, each time a
sample occasioned a comparison selection consistent with a previous
choice within the six-trial block, the consistency index would increase by
0.33. In the group with 4 four-member stimulus classes the consistency
index was calculated for each block of eight consecutive trials. For each
time a sample occasioned a comparison selection consistent with a
previous choice within the eight-trial block, the consistency index would
increase by 0.25. Furthermore, in the group with 6 three-member stimulus
classes the consistency index was calculated for each block of 12
consecutive trials. For each time a sample occasioned a comparison
selection consistent with a previous choice within the 12-trial block, the
consistency index would increase by 0.17.
Results
The probability of equivalence-class formation was significantly more
disrupted when the class size increased than when number of classes
increased, as shown in Figure 1.
In Groups 1, 2, and 3, with 3, 4, and 5 three-member stimulus classes
respectively, all subjects responded in accord with equivalence (see Table
4), while in Group 4, with 6 three-member stimulus classes, 2 of 5
subjects responded in accord with equivalence in the first half and 1 more
in the second half.
In Group 5 with 3 four-member stimulus classes, 3 of 5 subjects
responded in accord with equivalence in the DA test (two nodes), as
shown in Table 5. In addition 1 subject responded with a positive CA test
(one node) and additionally 1 subject in the DB test (one node). When
number of classes increased to four (Group 6), 1 subject responded in
accord with equivalence in the DA test. All 5 subjects responded in accord
with equivalence in the DB test, and 1 of them (#785) did not in the CA
test. When the number of classes further increased to five (Group 7),3 of
CLASS SIZE VS. NUMBER OF CLASSES
85
percent
3
members
3
classes
Figure 1. The figure shows the number of subjects who responded in accord with
equivalence when either number of classes or nodes increased or both.
Table 4
Total Number of Training Trials to Criterion and Indices for
Equivalence Tests for Each Subject in the Different Groups
No.
classes
Subj. #
Total number of
training trials
Equivalence indices in the
CA test
Second half
First half
3
321
322
323
324
325
132
117
121
100
130
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
4
751
752
753
754
755
187
240
206
155
163
0.9
0.9
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
5
761
762
763
764
765
284
429
521
309
321
0.8
1.0
1.0
1.0
1.0
0.9
0.9
1.0
1.0
1.0
6
771
772
773
774
775
453
320
402
272
981
0.8
0.3
1.0
0.9
0.7
0.9
0.4
1.0
0.9
0.7
Note. Number of classes are Indicated by the numbers in the left column.
ARNTZEN AND HOLTH
86
5 subjects responded in accord with equivalence in the DA test, and
furthermore, 1 subject responded in accord with equivalence in the two 1node tests.
Table 5
Total Number of Training Trials to Criterion and Indices for Different Equivalence Tests for
Each Subject in Groups with Four Members with an Increasing Number of Stimulus Classes
#Equivalence tests
3 members
DB
Subj.
4 members
DA
Number
of classes
3
4
5
731
732
733
734
735
CA
1
2
1
2
1
2
0.9
1.0
1.0
1.0
0.4
0.5
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.8
1.0
1.0
0.3
0.3
781
782
783
784
785
0.7
0.8
0.6
0.7
0.8
0.4
1.0
0.7
0.9
1.0
1.0
1.0
0.9
1.0
0.9
806
807
808
809
810
0.2
0.2
0.2
1.0
0.9
0.9
1.0
0.9
0.9
0.3
0.4
1.0
1.0
0.9
0.9
0.3
0.3
0.9
0.9
0.9
1.0
1.0
1.0
0.9
0.8
0.8
0.2
0.1
0.1
1.0
1.0
1.0
1.0
1.0
1.0
0.9
1.0
1.0
1.0
1.0
0.9
1.0
1.0
Note. Because of a programming error Subjects #731 and #781 received only one CA test.
As shown in upper panel of Table 6, the group with 3 five-member
stimulus classes, 1 (#744) of 5 subjects responded in accord with
equivalence. Furthermore, 2 subjects responded with positive CA tests in
Table 6
Indices for Different Equivalence Tests for Each Subject in Groups
with Five Members with an Increasing Number of Stimulus Classes
Subj#
5 members
EA
No. of
classes
1
3
4
2
4 members
EB
DA
2
Equivalence tests
3 members
EC
DB
2
2
1
2
2
741
742
743
744
745
0.3 0.6
0.6 0.3
0.2 0.3
0.5 0.3
0.0 0.5
0.1 0.0
0.7 0.6
0.5 0.4
0.2 0.2
0.2 0.2
0.4 0.2
0.1 0.5
1.0 1.0
1.0 1.0
1.0 1.0
1.0 1.0
1.0 1.0
1.0 1.0
0.3 0.4
0.3 0.3
0.3 0.2
0.3 0.3
0.3 0.3
1.0 1.0
0.8 0.9
1.0 1.0
1.0 1.0
0.8 1.0
801
802
803
804
805
0.5 0.5
1.0 1.0
1.0 1.0
0.8 0.8
0.5 1.0
1.0 1.0
1.0 1.0
1.0 1.0
1.0 1.0
1.0 1.0
1.0 1.0
0.3 0.3
0.3 0.3
0.3 0.5
0.0 0.0
0.6 0.5
0.3 0.3
0.4 0.4
0.5 0.4
0.3 0.3
0.3
0.5 0.5
0.4 0.5
0.9 1.0
0.0 0.0
0.4 0.3
0.5 0.6
Note. #805 refused to complete the experiment.
0.9 1.0
CA
0.5 0.5
O.
87
CLASS SIZE VS. NUMBER OF CLASSES
Table 7
Indices for Equivalence Tests for Each Subject in Group with 3 Six-Member Classes
Subj. #
6 members
FA
Equivalence tests
4 members
FC
EB
DA
5 members
EA
FB
12
FD
EC
3 members
DB
CA
21212121212121212
791 1.0 1.0 0.3 0.3 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.7 0.7 1.0 1.0 1.0 1.0 1.0 1.0
792 1.0 1.0 0.9 0.9 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
793 0.8 1.0 0.9 0.8 1.0 1.0 0.8 0.9 1.0 0.9 0.3 0.4 0.4 0.4 0.9 1.0 0.3 0.3 0.9 1.0
794 0.4 0.1 0.3 0.3 -
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
795 0.0 0.0 0.0 0.1 0.3 0.3 0.3 0.3 1.0 1.0 0.0 0.1 0.3 0.3 0.3 0.3 1.0 1.0 0.0 0.1
Note. Subject #794 refused to complete the experiment.
the first test half, and also the last two in the second test half. Two subjects
responded with positive DB-tests. In Group 9, the bottom panel, 1 (#805)
of 5 subjects responded in accord with equivalence in all tests, and
subject (#801) responded in accord with equivalence in one of the 2nodes test (EB) and during the three 1-node tests.
As shown in Table 7, in Group 10, with 3 six-member stimulus
classes, 2 of 6 subjects (#792 and 796) responded in accord with
equivalence during all tests. Subject #791 responded with indices of 1.0
6 three-member stimulus classes
CA test
OJ)
1.0
0
0-
0.83
c::
:.ec::
~
Q)
I-;
.....
......
(.)
~
S
0.67
Q)
......
~
>.
~
......
0
B# 771 +# 772
... #775
0.5
><
Q)
'"0
,s
0.0
0
5
10
15
20
25
30
35
Moving 12-trial blocks
Figure 2. Individual curves for each subject who did not respond in accord with equivalence
in the group with 6 three-member stimulus classes.
88
ARNTZEN AND HOLTH
or 0.9 in 8 of the 10 tests, and Subject #793 had indices of 1.0 or 0.9 in
7 of the 10 tests, and Subject #795 had indices of 1.0 or 0.9 in 2 of the
10 tests. Subject #794 failed in the two first tests and refused to
participate further in the experiment.
For the group with 6 three-member stimulus classes, the data for
consistency indices throughout testing for subjects who did not respond
in accord with equivalence showed that 1 subject (#771) responded
consistently throughout the CA test, and the 2 others responded partly
consistently (see Figure 2). (The data for 5 four-member, 4 five-member,
and 3-six-member stimulus classes are not presented).
3 four-member stimulus classes
:\-••0 "07,0.,
•• j V
DA test
1.0
•••••••
0.67
0.33
+# 733
0.0
1-6
5-10
-A
10-15
# 735
15-20
19-24
CA test
1.0
o
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.67
0.33
B#735
0.0
1-6
5-10
10-15
15-20
19-24
Moving 6-trial blocks
Figure 3. Individual curves for each subject who did not respond in accord with equivalence
in the group with 3 three-member stimulus classes.
89
CLASS SIZE VS. NUMBER OF CLASSES
As shown in Figure 3, for the group with 3 four-member stimulus
classes, 1 subject responded consistently during the whole DA test, and
the other subject responded partly consistently. The only subject, who did
not respond in accord with equivalence, did respond consistently during
the CA test.
In the group with 4 four-member stimulus classes, subjects
responded more consistently during the test, and 1 subject responded
consistently during the whole CA test, as shown in Figure 4.
4 four-member stimulus classes
DA test
1.0
+#781
.,. # 782
0.75
0.5
+#783
B#785
0.25
0.0
o
5
10
15
25
20
CA test
1.0
000000000000000000000000
0.75
B#785
0.5
0.25
0.0
o
5
10
15
20
25
Moving 8-trial blocks
Figure 4. Individual curves for each subject who did not respond in accord with equivalence
in the group with 4 four-member stimulus classes.
The group with 3 five-member stimulus classes showed that
consistency of responding was higher in the DA test than in the EA, EC,
EB, and DB test (see Figure 5). Furthermore, a pattern of more consistent
responding during the test was seen only in the DA test.
90
ARNTZEN AND HOLTH
3 five-member stimulus classes
EA test
*#741
·10#742
*#743
B#745
1-6
5-10
10-15
15-20
19-24
DA test
1 .0
0 . 67
0 . 33
*#741
l~::::~
·10#742
*#743
B#745
0 .0
1-6
5-10
10-15
19-24
15-20
EC test
*#741
·10#742
*#743
B#745
1-6
10-15
5-10
19-24
15-20
EB test
*#741
1.0
·10#742
0 . 67
*#743
B#745
0 . 33
0.0
1-6
5-10
10-15
15-20
19-24
DB test
1 .0
DDDD.iII,D\
0 . 67
1o.·IoA
10
·10#742
100000
B#745
-0000-
.10.
0 . 33
0 .0
1-6
5-10
10-15
15-20
19-24
Moving 6-trial blocks
Figure 5. Individual curves for each subject who did not respond in accord with equivalence
in the group with 3 five-member stimulus classes.
.
CLASS SIZE VS. NUMBER OF CLASSES
91
3 three-member stimulus classes
16
CAtest
Training
sample
...... comparison
00
12
8
~
4
0
~
4 three-member stimulus classes
16
Training
CA test
12
8
en
-C
C
oC,.)
~
~.IiI " ~ "1iI
4
O~~~~L-~--~~~--~~~~
5 three-member stimulus classes
CD
en
CA test
Training
o
6 three-member stimulus classes
16
Training
CA test
12
8
4
0
~
Last five
~
~
1 234 6
42
0"
"0
44
46
trials
Figure 6. Mean reaction times for subjects in all groups when increasing number of classes.
In each graph the last five training trials followed by the first five test trials and the last five
test trials. Reaction times to the sample stimuli is shown as open squares and reaction times
to the comparison stimuli are shown as filled circles.
92
ARNTZEN AND HOLTH
3 four-member stimulus classes
18
15
12
9
6
3
0
Tr
DA
DB
eN
\4
CA
,'-
4 four-member stimulus classes
Tr
en
"0
I=:
0
(.)
(1)
CI)
18
15
12
9
6
3
0
DB
DA
CA
.....
eJ-
5 four-member stimulus classes
Tr
DB
DA
CA
18
15
12
9
6
3
0
....
Last five
V1 5 Last five
~
1m .....
1 5 Last five
1 5 Last five
Figure 7. Mean reaction times for subjects when increasing number of classes in the fourmember groups. In each graph the last five training trials followed by the first five test trials
and the last five test trials. Reaction times to the comparison stimuli is shown as filled circles
in the different "equivalence tests."
93
CLASS SIZE VS. NUMBER OF CLASSES
In all groups there was a marked increase in reaction times to
comparison stimuli from the final phase of training to the initial phase of the
test, as shown in Figures 6, 7, 8, and 9. There was also a significant decrease
in reaction times from the initial five test trials to the last five test trials for all
groups. There was no difference in reaction time when comparing the final
phase of training and the final phase of the test within each group.
3 five-member stimulus classes
16
Tr
EA
EB
DA
EC
DB
CA
12
en
"d
s::
u
0
8
~
CI,)
4
0
~
1':
---':-.~~~_\-'
4 five-member stimulus classes
16
Tr
EA
EB
DA
EC
DB
CA
12
8
4
0
~
Last
five
\\.:
1 5 Last 1 5 Last 1 5 Last
1 5 Last
five
five
five
five
-~~
1 5 Last
five
1 5 Last
five
Figure 8. Mean reaction times for subjects when increasing number of classes in the fivemember groups. In each graph the last five training trials followed by the first five test trials
and the last five test trials. Reaction times to the comparison stimuli is shown as filled circles
in the different "equivalence tests."
94
ARNTZEN AND HOLTH
Tr FA
16
14
12
~ 10
§ 8
~
~
6
4
2
o
EA
FB
FC
EB
DA
FD
EC
DB
CA
!
~li)I~I .uv.~~~~~~
Tests
Figure 9. Mean reaction times for subjects in the 3 six-member group. In each graph the last
five training trials followed by the first five test trials and the last five test trials. Reaction
times to the comparison stimuli is shown as filled circles in the different "equivalence tests."
Discussion
The probability of equivalence class formation was significantly more
disrupted when class size increased than when number of classes were
increased. Consistency indices showed that most of the subjects tended
to respond more systematically throughout testing whether or not they
responded in accord with equivalence. However, even patterns of
consistent responding that were not consistent with the predicted
equivalence pattern were more disrupted when class size increased than
when number of classes increased. Since consistency within trial types
must be easier with fewer choices, the lower consistency with fewer
choices, but more members, indicates that the tendency for consistency
extends across nodal numbers. It should be noted that in two-choice
matching-to-sample tasks, a consistent choice of a different comparison
for each sample would, by chance, produce responding in accord with
equivalence in 50% of the cases. This percentage is lower as the number
of comparisons increases. In all groups in which there are three classes
there is a 0.17 probability for equivalence to emerge by chance; while for
instance, in 6 three-member stimulus classes the probability of
'equivalence'-by-chance is little more than one per thousand.
To explain the differences in equivalence outcome in the present
experiment, five different variables could be set forth.
First, Sidman (1994) suggested that the change from simultaneous to
successive discrimination or vice versa (e.g. , from training to test) as
relevant in explaining the diverging outcomes of the different training
structures, in terms of the probability of class formation. However, the change
CLASS SIZE VS. NUMBER OF CLASSES
95
Table 8
Number of Successive or Simultaneous
Discriminations as Function of Class Size and Number of Classes
No. of
classes
No. of
members
Total no. of
discriminations
3
4
5
6
3
3
3
3
36
66
105
153
3
4
5
4
4
4
3
4
3
No. of
discriminations
required
Equivalence
outcome in
percent
Ditt. between
total and
required
discriminations
No. of
discriminations
during each test
21
42
70
105
15
24
35
48
6
8
10
12
100
100
100
60
66
120
190
30
60
100
36
60
90
6
8
10
60
20
60
5
5
105
190
39
78
66
112
6
8
20
20
6
153
48
105
·6
20
Note. The number of stimuli which must be discriminated (successively and .3jmultaneously)
needed in each condition were calculated according to the following formula: M~/C-i) + C(C-1)
(M-1). C is number of classes and M is number of members. The first part of the formula is for
calculating the number of discriminations between classes, and the last part is for calculating
number of discriminations within each stimulus set. Total number of discriminations in each
condition is calculated according to the following formula fY. x Ivf2 - CM2.
from training to test with respect to the switch in discrimination could not be a
critical variable in explaining the results, because the equivalence outcome is
highest when the number of classes increases, in which also the number of
stimuli discriminated is much higher, compared to the number of stimuli
discriminated when class size increases (see Table 8).
Second, the total number of stimuli presented in each condition cannot
explain the differences either, because the rate of equivalence is higher in the
6 three-member stimulus classes than in the 3 six-member stimulus classes.
Third, increasing the number of classes may produce lower yields of
equivalence outcome than a comparable increase in the number of
members simply as a function of the higher discrepancy between the
number of discriminations required to discriminate all the stimuli from
each other stimulus and the number of discriminations actually required
during training. This is in accord with the discrimination analysis set forth
by Saunders and Green (1999).
Fourth, Fields and coworkers (Fields, Adams, & Verhave, 1993) have set
forth the idea that nodal number could also be an essential variable affecting
the equivalence outcome. In the current experiment, number of nodes could
be an important factor in explaining the differences in disruption of equivalence
when either increasing size or classes. In the Kennedy (1991) study the
greatest differences with respect to equivalence outcome seemed to occur
between one and two nodes. In contrast, Sidman (1994) maintained that, with
repeated testing, equivalence has eventually been demonstrated regardless of
96
ARNTZEN AND HOLTH
nodal number and this might rather be a question of delayed emergence.
However, as shown in the present study and elsewhere (Holth & Arntzen,
1998a, in press), subjects sometimes respond consistently with a pattern other
than the predicted equivalence pattern.
Fifth, number of familiar stimuli could be an important factor in explaining
these results, because increasing number of classes in Experiment 1 involves
an increasing number of familiar stimuli, while when increasing class size the
number of familiar stimuli is constant, and as can be shown in Table 8 the
equivalence outcome is highest in the case of increasing class number. This
variable is linked to the fact that when increasing either the number of classes
or the class size, the number of stimuli that are simultaneously or successively
discriminated are increased differently with respect to training sequences. If
the number of classes is increased by one, for example from a 4 to 5 threemember class, the number of stimuli that has to be simultaneously
discriminated is increased by one in each trial type. On balance, if the number
of members is increased by one, for example from a 4- to 5-member class, the
number of stimuli that are simultaneously discriminated is constant.
As in all stimulus equivalence experiments when a linear series training
structure is used, the effect of number of nodes and class size are entangled.
To separate the effects of these variables and to see if the difference between
class size and number of classes is dependent on number of familiar stimuli,
a second experiment included a many-to-one training structure, in which class
size is increased. Obviously, increasing class size in a many-to-one training
structure does not influence the number-of-nodes variable.
Experiment 2
Method
Subjects
The subjects were 16 staff-members from a treatment center for retarded
and autistic children, and teachers from a school for'youths with behavioral
problems. They were successively assigned to four different experimental
groups. The subjects were familiar neither with stimulus equivalence research
nor with the stimulus material presented, except for the pictures.
Apparatus and Procedure were the same as in the preceding experiment.
The experimental conditions for the four groups are summarized in
Table 9.
Table 9
Stimulus Relations Trained for Group Depending on Class Size
Members
2
3
4
Stimulus relations
A1B1 , A2B2, A3B3, C1B1 , C2B2,C3B3
A1B1 , A2B2, A3B3, C1B1 , C2B2, C3B3,D1B1,D2B2,D3B3
A1B1 , A2B2, A3B3, C1B1 , C2B2, C3B3, D1B1,D2B2,D3B3,E1B1 , E2B2, E3B3
A1B1 , A2B2, A3B3, C1B1 , C2B2, C3B3, D1B1,D2B2, D3B3, E1B1 , E2B2,E3B3, F1B1,F2B2,F3B3
97
CLASS SIZE VS. NUMBER OF CLASSES
Results
The probability of an equivalence outcome decreased as a function of
increasing class size, as shown in Figure 10. In the groups with three and
four members, all subjects responded in accord with equivalence, and when
the number of members was increased to five and six, 75% and 50%,
respectively, of the subjects responded in accord with equivalence.
100
80
60
40
20
o-'---'----3
4
5
6
Number of members
Increasing class size
Figure 10. The figure shows the percentage of subjects who responded in accord with
equivalence following the many-to-one structure.
ARNTZEN AND HOLTH
98
As shown in Tables 10 and 11, all subjects who were exposed to
three- and four-member classes responded in accord with equivalence.
Table 10
Indices for Equivalence Test for Each Subject in Group with Three Members in Experiment 2
Subject Number
Equivalence test
CA
2
811
1.0
1.0
812
0.3
1.0
813
1.0
1.0
814
1.0
1.0
Table 11
Indices for Different Equivalence Tests for Each Subject
in Group with Four Members in Experiment 2
Equivalence tests
DC
2
Subject Number
DA
1
2
816
1.0
1.0
1.0
CA
1
2
1.0
1.0
1.0
817
1.0
1.0
0.9
1.0
1.0
1.0
818
1.0
1.0
1.0
1.0
1.0
1.0
819
1.0
1.0
1.0
1.0
1.0
1.0
When class size was increased to five members, 3 of 4 subjects
responded in accord with equivalence, as shown in Table 12. Subject
#821 had indices of 1.0 on all tests except for the ED test, where he had
indices of 0.8 and 0.6.
Table 12
Indices for Different Equivalence Tests for Each Subject
in Group with Five Members in Experiment 2
Subj#
EA
1
2
DA
1
2
Equivalence tests
EC
ED
1
2
2
1
DC
2
CA
1
2
821
0.9 1.0
1.0 0.8
822
1.0 0.9
0.8 1.0
1.0 1.0
1.0 1.0
1.0 1.0
1.0 0.9
823
1.0 0.9
1.0 0.9
1.0 1.0
1.0 0.9
1.0 0.9
1.0 1.0
824
1.0 1.0
0.9 0.9
0.8 1.0
1.0 1.0
0.9 1.0
1.0 0.8
1.0 0.9
0.8 0.6
1.0 1.0
1.0 1.0
As shown in Table 13, 2 of 4 subjects which were exposed to training
for the establishment of six-member classes responded in accord with
equivalence. Subject #828 had indices of 1.0 on 9 of the 10 tests, and on
the DA-test the indices were 0.8 and 0.7. Subject #827 had indices of 1.0
on all but two tests, the EA and DA tests.
99
CLASS SIZE VS. NUMBER OF CLASSES
Table 13
Indices for Equivalence Tests for Each Subject in Group with 3 Six-Member Classes
Subj. #
FA
EA
FC
DA
Equivalence Tests
FD
EC
FE
ED
DC
CA
22222122222
826 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
827 0.8 0.9 0.7 0.8 1.0 1.0 0.8 0.8 0.9 1.0 0.8 1.0 1.0 1.0 1.0 0.8 1.0 1.0 1.0 1.0
828 1.00.9 0.9 1.0 1.00.9 0.8 0.7 1.0 1.0 0.8 1.0 0.9 1.0 1.0 0.9 0.9 1.0 1.0 1.0
829 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.9 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
I:~o
100
I
0\
o
I-<
o
~
......
t)
.-""
80
~
(.)
'"d
.S
60
.....
""(.)
:S
::l
40
""
4-<
o
~
5
I::::
20
~
~
tl..
o
FA EA FC DA FD EC FE/ ED/ DC/ CA
FB EBDB
"Equivalence tests"
Figure 11. The figure shows percentage of subjects with indices of 1.0 and 0.9 across the
different "equivalence tests:' Subjects tested following the linear series structure are marked with
filled bars and subjects tested following the many-to-one structure are marked with gray bars.
ARNTZEN AND HOLTH
100
CA
Bsl
9
6
3
~
0
9
rJ1
~
~
5
Last five
Bsl
DA
CA
DC
6
0
()
(l,)
Last five
:-J
3
r./'J
j.. :
0
Last
five
1 5
Bsl
EA
Last
five
DA
1 5
EC
1\ J
5
Last
five
ED
Last
five
DC
CA
9
6
3
o
Last 1 5 Last 1 5 Last 1 5 Last 1 5 Last 1 5 Last 1 5 Last
five
five
five
five
fIVe
five
five
Figure 12. Mean reaction times for subjects when increasing number of members. In each graph
the last five training trials followed by the first five test trials and the last five test trials. Reaction
times to the comparison stimuli is shown as filled circles in the different "equivalence tests."
101
CLASS SIZE VS. NUMBER OF CLASSES
The percentage of subjects who had indices of 1.0 or 0.9 on the different
tests for equivalence where higher across all tests for subjects trained with
the many-to-one structure with six-member classes compared to subjects
trained with linear series structure, as shown in Figure 11 .
Reaction times to comparison stimuli increased from the five last training
trials to the first five test trials, as shown in Figures 12 and 13. Furthermore,
there was no systematic decrease in reaction times across successive tests.
12
Bsl
FA
EA
FC
DA
FD
EC
ED
FE
DC
CA
10
00
8
"'0
~
0
U
6
~
V)
4
Jv~~ W
.
2
0
Last 1 5 Last 1 5 Last 1
five
five
five
.
5 Last 1 5 Last 1 5 Last 1 5 Last 1 5 last 1 5 Last
five
five
five
fIVe
five
five
!v.v
1 5 Last
five
1 5 last
five
Figure 13. Mean reaction times for subjects when increasing number of members. In each
graph the last five training trials followed by the first five test trials and the last five test trials.
Reaction times to the comparison stimuli is shown as fi lled circles in the different
"equivalence tests."
Discussion
When stimulus equivalence was tested in subjects after conditional
discrimination training according to a many-to-one structure, all subjects did
respond in accord with equivalence in the three- and four-member classes,
and 3 and 2 of 4 subjects, respectively responded in accord with equivalence
in the five- and six-member classes. The results indicate that the number of
familiar stimuli could not be a critical factor in explaining the differences in
equivalence outcome following either increasing number of classes or class
size, because the equivalence outcome was about the same for the subjects
following the many-to-one training structure as for the subjects following the
linear series structure when number of classes was expanded.
Although the test procedures used in the current experiments are
sensitive to occasional errors, the percentage of subjects who responded
in accord with equivalence was higher on all tests following the many-toone structure than following the linear series structure (see Figure 11).
102
ARNTZEN AND HOLTH
General Discussion
The purpose of the current study was to investigate the likelihood of
equivalence class formation as a function of class size and number of classes.
Because class size and nodal number are confounded in a linear series training
structure, a many-to-one structure was included to separate these variables.
The results from the current experiments showed that the equivalence outcome
was decreased as a function of both number of classes and nodes, but the
decrease was much more pronounced as a function of nodes. Furthermore, as
shown in Experiment 2, in which class size was increased in a many-to-one
structure, the equivalence outcome did not differ from the equivalence outcome
when the number of classes increased as shown in Experiment 1. This
indicates that equivalence class formation was not so much influenced by
increasing class size as by an increasing number of nodes.
The number of singles linked to a node is termed "nodal density" and is
determined by the distribution of singles (Fields & Verhave, 1987). Therefore,
increasing the number of singles linked to a nodal stimulus might inhibit or
enhance the formation of emergent relations (Fields & Verhave, 1987). In a
many-to-one training structure, nodal number is constant while the "nodal
density" increases as a function of increasing class size. The differential
outcome with respect to probability of class formation in the 6 three-member
classes in Experiment 1 and 3 six-member classes in Experiment 2, could be
a function of such "nodal density" with respect to lower equivalence outcome
when increasing the nodal density.
Although as suggested by Sidman (1994) equivalence might eventually
emerge regardless of nodal number, this is clearly not always the case. The
present finding of a gradual change in responding to become consistent with
some pattern other than the predicted equivalence pattern was also evident in
Holth and Arntzen (1998a). A history of differential reinforcement for consistent
responding might explain the finding of improved performances during testing,
that is, delayed emergence of stimulus equivalence and increased consistency
of responding even when not in accord with experimenter-predicted equivalence
performances. Bush, Sidman, and deRose (1989) found that the emergence of
new conditional relations came about abruptly with repeated testing. The
comparison selection on every trial becomes consistent with expected
equivalence classes because the test trials provide no other basis for choice that
is consistently available (Sidman, 1994). The gradual vs. abrupt emergence
could be related to the fact that Sidman's analyses were based on testing for
emergent relations in which the tests were arranged as test probes and not as
separate test blocks as in the present study. The differential effects of conducting
tests for the properties of equivalence arranged in test probes or separate test
blocks are explored relatively little (Green & Saunders, 1998), and experiments
should be designed to study these differences in test arrangements.
The current study has replicated earlier findings in which an increase in
reaction time from training to test, and a decrease during the test has been
shown (Arntzen & Holth, 1997; Holth & Arntzen, 1998a, 1998b). Fields and
Verhave (1987) have postulated that as nodal distance increases, accuracy of
CLASS SIZE VS. NUMBER OF CLASSES
103
responding decreases. Reaction times are sensitive to different variables even
when the accuracy of responding is not (Holth & Arntzen, in press; Spencer &
Chase, 1996). In the Spencer and Chase (1996) study, response speed was
inversely related to the number of nodes on which the tested relations were
based. Furthermore, differences in accuracy across nodal distance and trial
type were significant only on the first tests for equivalence, whereas
differences in speed were significant even after extended testing. For example,
during combined testing all subjects responded faster to one-node than to fivenode combined relations, and data from half of the subjects indicated an
inverse relation. All subjects responded faster on one-node than on five-node
transitivity relations. The results of the current experiment showed similar
effects as a function of increasing nodes in which reaction times decreased
across successive tests. Furthermore, such effects were not found following
the many-to-one training structure in which class size was increased.
During recent years Fields and coworkers have been concerned with
variables influencing the likelihood of emergent stimulus classes. First, the
formation of stimulus classes has been shown to be a direct function of
number of nodes; the formation of stimulus classes is less likely as number of
nodes increases (Fields et aL, 1993). Second, pretraining enhances the
emergence of new stimulus classes, and the enhancement effect is a direct
function of class size trained (Buffington, Fields, & Adams, 1997). Third, the
number of stimuli in the pretrained classes enhance the emergence of new
equivalence classes (Fields et aL, 1997). Finally, emergence of new
equivalence classes have been shown to be a function of three training
protocols, simple-to-complex, complex-ta-simple, and simultaneous protocols
(Fields et aL, 1997). The current experiments have extended these findings,
showing that the emergence of stimulus equivalence was more disturbed by
number of nodes than both class size and number of stimulus classes. The
results indicate that an increasing number of nodes, and not just increasing
class size, inhibits an equivalence outcome to a greater extent than does an
expanding number of classes.
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