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The development of hierarchically complex equivalence classes.

Research on equivalence classes and categorization has generally been bereft of a developmental analysis and has never to our knowledge addressed performances that only adults make. This paper will attempt to introduce some ideas and findings from post-Piagetian developmental research to equivalence-class theory and research. Typically, Piagetian (Armon, 1984a, 1984b; Colby & Kohlberg, 1987; Kegan, 1982; Kohlberg, 1981a, 1981b, 1984, 1987, 1990; Kohlberg & Armon, 1984; Selman, 1980, 1981) and neo-Piagetian (for examples, see the review by Alexander, Druker, & Langer, 1990) research is mentalistic in that it judges stage of performance by inferring developmental structures of thought. The various responses are analyzed independent of the stimuli. For the most part, researchers do not correlate changes in responses with changes in the stimuli because, typically, stimuli are not varied. In our post-Piagetian work, we have divided the assessment of the developmental sequence in two ways. First, we look at how tasks themselves can be hierarchically arranged. Second, we examine how organisms perform such tasks. Hence the responses are categorized by what order of hierarchical complexity task they adequately solve.

In the case presented here, we propose that equivalence classes can be arranged according to their order of hierarchical complexity. The order of hierarchical complexity is an analytical description of the developmental difficulty of a task. The equivalent developmental stage of performance derived from Inhelder and Piaget (1958)--without their mentalistic assumptions--is an empirical measure of the power of subjects' problem-solving skills. To see how well subjects detect equivalence classes of various hierarchical complexities, we introduce a task that requires subjects to identify equivalence relationships in a real world situation, the doctor-patient relationship.

Traditionally, the identification of equivalence relations depends on particular physical stimuli, as in the standard conditional discriminations used in all other papers in this special issue. Here, the equivalence relations depend on relations among relations of actions embedded within seven doctor-patient vignettes. The Doctor-Patient Task was designed so that the equivalence relations at the highest order of hierarchical complexity were constructed out of the next lower order equivalence classes.

The seven vignettes in the Doctor-Patient task are about doctors performing medical treatment in "another country." Each vignette consists of a number of statements that describe doctors with different perspectives for treating patients. The Doctor-Patient Task was constructed so that the doctors' approaches to their patients differ in developmental stages. Doctors' developmental stage is one abstract stimulus dimension. It is partially determined by the stage of the way the doctors inform patients. Do they dictate what the patient should do and fail to consider any input from the patient? Doctors' developmental stage is also determined by whether the informed-consent process organizes the systems of informing and of consenting into a "metasystem." Whether or not informing and consenting systems are organized in the vignette is a second stimulus dimension. For example, in the informed-consent procedure, informing is a system of relations among lower order actions. At "systematic" conceptual relational order, some doctors successfully inform their patients about the relations among alternative treatments and their possible benefits and side effects, but they fail to obtain consent.

ln the informed-consent procedure, consenting is another system of relations among actions at the equivalent "systematic" conceptual relational order as informing. For example, other doctors try to obtain consent by offering the patient choices but without informing them. Such doctors offer treatment options in relation to the doctors' recommendations. In other words, the doctors ask for the patient's preference in the context of the doctors' preferences. When doctors fail to inform or gain consent they engage in two kinds of punishment: On the one hand, when they do not inform their patients about treatment options, patients are threatened with the prospect that they will not be treated if they do not accept the treatment offered. On the other hand, when doctors do not afford patients the opportunity to give consent, they impose a treatment that the patient may not wish to have administered.

Finally, the consenting system may be related to the informing system at the next conceptual relational order, "metasystematic." In order to demonstrate performance at the metasystematic order, subjects must identify doctors that are similar in how they engage in the informed-consent process, which coordinates both the informing system of statements and the consent system. Subject performance on this Doctor-Patient Task is reported here.

Equivalence Relation

To begin, James and James (1976) define equivalence in the following manner.

An equivalence relation is a relation between elements of a given set which is a reflexive, symmetric and transitive relation and which is such that any two elements of the set are either equivalent or not equivalent.

Hence, I is an equivalence relation if the relation is

(a) reflexive:

I(a) = a, and in simple algebra, a = a

(b) symmetric:

I(ab) if I(ba), in algebra, a = b if b = a

(c) transitive:

If I(ab) and I(bc), then I(ac); in algebra, if a = b and b = c then a = c.

A nonequivalence relation (-I)is the negation of an equivalence relation:

I(ab) [is not equal to] I(ba); in algebra, a [is greater than] b [is not equal to] b [is greater than] a.

James and James (1978) define an equivalence class as follows:

If an equivalence relation is defined on a set, then the set can be separated into classes by the convention that two elements belong to the same class if and only if they are equivalent. These classes are equivalence classes. Two equivalence classes are identical if they have an element in common.

In the past, there has been much work on equivalence classes in animals and people. From a developmental perspective, a problem with much of this work is that it does not address the reasons why species at different phylogenetic levels show differences in what equivalence class may be detected. Even within species, developmental differences have not been studied. To address this deficiency, this paper introduces the notion of hierarchical complexity of equivalence classes. The notions of stage and of hierarchical complexity will be introduced next.

Hierarchical Complexity and Stage

Inhelder and Piaget (1958) postulated that one of the fundamental actions of people was to classify objects and actions. These classifications could be grouped together on the basis of the number of actions that each coordinates as described below. We (Commons, Krause, Fayer, & Meaney, 1993; Commons & Rodriguez, 1990; Commons, Trudeau, Richards, & Stein, in preparation) call this number the order of hierarchical complexity when it refers to task demands and the stage when it refers to performance. Historically, Piaget's model confounds the hierarchical complexity of the task with performance on the task. One would like to know the order of hierarchical complexity of the task independent of how some subject performs on that task. Commons and Richards (1984a) solved this problem by separating two conceptually different but related issues: (a) the hierarchical complexity of the task to be solved; and (b) the psychology, sociology, and anthropology of how such task performance develops.

Hierarchical Complexity

The General Stage Model (GSM) of Commons and Richards (1984a, 1984b) is a system that classifies development in terms of a task-required hierarchical organization of required responses. That model was derived in part from Piaget's (lnhelder & Piaget, 1958) notion that the higher-stage actions coordinate lower stage actions by organizing them into a new more hierarchically complex pattern. The stage of an action is found by answering the following two questions: (a) What are the organizing actions? (b) What are the stages of the elements being organized?

Specifically (Commons, Sonnert, Gutheil, & Bursztajn, 1991), hierarchical complexity refers to the number of recursions that the coordinating actions must perform on a set of primary elements. Actions at a higher order of hierarchical complexity (a) are defined in terms of the actions at the next lower order of hierarchical complexity, (b) organize and transform the lower order actions, and (c) produce organizations of lower order actions that are new and not arbitrary, and cannot be accomplished by those lower order actions alone. After meeting these conditions, we say the higher order action coordinates the actions of the next lower order. Stage of performance is defined as the highest order hierarchical complexity of the task solved.

For example, multiplying 3 X (9 + 2) requires a distributive action at the concrete order of hierarchical complexity. The distributive action is as follows: 3 X (9 + 2) = (3 X 9) + (3 X 2) = 27 + 6 = 33. That action coordinates (organizes) adding and multiplying by uniquely organizing the order of those actions. The distributive action is therefore one order more complex than the acts of adding and multiplying alone. Although someone who simply adds can arrive at the same answer, being able to do both addition and multiplication in a coordinated manner indicates a greater freedom of thought and action. Through such task analysis, the hierarchical complexity of a task may be determined.

Fourteen orders of hierarchical complexity have been analyzed (e.g. Commons, 1991; Commons et al., in preparation). Here we present examples from one domain to show how the identity relation can be used to define such equivalence classes, and that such equivalence classes have a different specific definition at each stage and perhaps with each domain.


In the General Stage Model, stage is defined as follows. An action is at a given stage when it successfully completes a task of a given hierarchical order of complexity. When people successfully perform a task at a given order of hierarchical complexity, the stage of their performance is considered to be of the equivalent order. These stages are also described by Case (1985), Fischer, Hand, and Russell (1984) and Pascual-Leone, (1976, 1980, 1984). In a most simplistic sense, at each stage in the sequence, a more complex equivalence relation may be exhibited. Such equivalence operations fall into the epistemological domain and inform the other developmental dimensions (Commons & Rodriguez, 1990).

The Hierarchy of Equivalence Relations

Many complex forms of human behavior are based on an individual's proficiency in identifying the relatedness of new stimuli, events, and representations of events (names, labels), even though these stimuli do not resemble one another. Fields (1992a, 1992b) uses as an example one such equivalence class that develops rather late in humans. In becoming multilingual in a broad sense at the primary operational stage, people come to understand that the alphabetically written word in English "five" or in Spanish "cinco" symbolically forms "5," "V," "0101," and the quantity (* * * * *) all mean the same thing. Children discriminate when there are an equal number of randomly placed objects within a small space. Piaget (Inhelder & Piaget, 1958) uses this conservation of number as one of his prototypical early concrete operational or primary stage 3a examples as shown in Table 1.

Equivalence tasks are used to assess adult stages of performance that called abstract, formal, systematic, and metasystematic stages. The required groups of actions in such tasks form a hierarchical sequence of equivalence classes. These stages are hierarchical because each builds on equivalence classes that were formed by the previous equivalence relation. Subjects detect which sets of actions are equivalent and thereby form equivalence classes. The following line of reasoning extends the way Piaget (Inhelder & Piaget, 1958) used the "INRC group" to describe formal operations. The extension shows that the Doctor-Patient Task measures postformal stages.

Actions at the abstract stage (4a) form variables, such as x, y, black people-white people, good people-bad people, true propositions-false propositions, and so forth. Two 2-valued examples from the Doctor-Patient Task used here are the "there is treatment"-"there is no treatment" variable and the "it has side effects"-"it does not have side effects" variable.

From Piaget's perspective, the formal operational stage (4b) actions are the INRC (identity, negation, reciprocation, correlation) group actions. From our Doctor-Patient example, the "telling" statement is formal operational, "for any x:(x) ff x is a treatment ([T.sub.x]) then it has side effects ([E.sub.x]), (x) ([T.sub.x][right arrow][E.sub.x])." Inhelder and Piaget (1958, p. 297) state that "implication, p[right arrow]q which expresses the combination (p&q) or (-p&q) or (-p&-q) is employed by subject every time a cause, expressed by proposition p, produces an effect, expressed by q, but is not the only cause which can produce the same effect."

The INRC group actions transform different events and actions into formal operational equivalence classes. These equivalence relations are formed from abstract stage propositions such as variables. These mathematical equivalence relations include identity, equality, and logical equivalence. These equivalence relations are formal operational because they are formed from abstract stage elements such as variables. Organizing multiple formal operational relational statements into a system is a systematic stage action. The particular axioms in simple algebra that define the properties of equivalence are systematic-stage elements. Testing for isomorphism and homomorphism of systems are metasystematic stage actions. Isomorphisms and homomorphisms are metasystematic forms of equivalence because they relate systems. Nonequivalence relations include "greater than," its psychological instantiation "more similar to," "preferred over," and "implication," and a somewhat related empirical notion of causation.

The logical argument made here is that the identity relation as used by Piaget (Inhelder & Piaget, 1958) is an equivalence relation. For Piaget, the equivalence relation was between abstract propositions. For example, the following propositions are roughly equivalent forms of offering treatment: "Doctor Adams offers a treatment the hospital has studied and prefers" [Approximately equal to] "Doctor Brown offers a treatment preferred by fellow doctors." For us, the propositions are equivalent when they have the "same meaning." Without accepting that people reason using the INRC group operations described by Piaget, or without accepting the INRC group of operations as a definition of formal operations, we illustrate why Piaget considered the identity (I) operation to be a formal-operational coordination of propositions. The point is that such a coordination is one stage above the abstract stage propositions that are coordinated.

The INRC group of operations describes the logical relations between actions that subjects reasoning at the formal-operational stage should be able to do. The operation, I, is the identity transformation of a proposition. Applying I to a proposition yields that proposition, l(p) = p. That action, I, identical (identity) is related to a combination of three other actions: l = NRC (negation, reciprocation, correlation) transformations. Consider the causal statement A, written in implicative form (Suppes, 1958):

A:p[right arrow]q

I(A) = A

I(p[right arrow]q) I(p[right arrow]q]) = p[right arrow]q.

For example, "The doctor describes different treatment options and their side effects" is equivalent under the I transformation to "The doctor describes all the side effects of these treatments."

The N action is a negation or an inversion of the statement p[right arrow]q.

N(A) = -A

N(p[right arrow]q) = -(p[right arrow]q) = p and -q.

The negation of "if it is a treatment, then it has side effects" is "it is not the case that if it is a treatment, then it has side effects" which in turn is the same as "it is a treatment and it has no side effects."

Piaget's (Inhelder & Piaget, 1958) point is that the subject compares and contrasts propositional statements using INRC actions of a formal-operational order of hierarchical complexity. The abstract propositions p, q are at the abstract (beginning formal operational) stage. The coordinations of the abstract propositions are the 16 combinations of I, N, R, C applied to A.


In summary, the l action forms formal operational equivalence classes out of abstract stage propositions. By definition, then, "coordination" means identifying the equivalence class of same hierarchically ordered actions.

The General Stage Model builds postformal equivalence classes from Piaget's basic definition of the INRC. Each postformal stage requires the same group of operations on elements of the previous stage. Therefore, formal operations performs the idendity action I on abstract propositions, systematic operations performs Ion formal operational causal statements, and metasystematic operations performs I on systems of formal operational causal statements.

Measuring Equivalence Detection

The Doctor-Patient Task (Rodriguez, 1992; Rodriguez, Commons, & Hill, 1990) assesses how adults perform on an equivalence task. In the problem, seven vignettes describe doctors performing medical treatment in "another country." Each vignette illustrates the actions of a doctor with a different perspective for treating patients. The problem requires subjects to compare groups of actions taken by doctors. The Doctor-Patient Task belongs to a class of problems called multisystems tasks (Commons, Richards, & Kuhn, 1982; Richards & Commons, 1984). Such tasks include multiple stories or vignettes that represent variations of a number of variables embodied in an incident. The multisystems tasks here were constructed using the method developed by Commons (Commons, Miller, & Kuhn, 1982) and extended from formal to postformal problems by Commons, Richards, & Kuhn (1982).

The vignettes contain five orders of hierarchical elements: concrete statements of an act, abstract stage propositions, formal operational statements relating the propositions, systematic stage systems organizing those relations, and finally metasystematic relations among the system. When subjects assert that two doctors' actions are similar or dissimilar, they are identifying an equivalence class, and therefore, they are performing a hierarchical operation. The experimenter then determines if the assertion is a hit, miss, false alarm, or correct rejection for each order of hierarchical complexity.

The metasystematic equivalence task requires subjects to determine how similar the ten doctor pairs are. Inhelder and Piaget (1958) found that by the formal operational stage, people make transitive inferences with propositions--if A = B, B = C, then A = C. Richards and Commons (1984) have found that if systems A and B are given a particular similarity rating, then B and A are given the same rating. They also always give A the same rating, so that A = A. To perform the task correctly all the subjects have to do is differentiate all the perspectives represented in the vignettes, and to order their degree of similarity on an 8-point scale.

The Hierarchical Equivalence Classes

Formal-Operational Equivalence Classes

To identify a formal operational equivalence class, the subject has to detect two abstract propositions in the vignettes that are similar and relate them. For example, in one abstract proposition ([T.sub.x]), "the doctor tells the patient about treatment options." In another abstract proposition ([E.sub.x]), "the doctor tells the patient about side effects." Together these two propositions form a statement, [F.sub.1], that "tells" the patient about the treatment. Telling the patient about the treatment is the new equivalence class, a formal element. Logicians call this a statement.

At the formal operational order of hierarchical complexity, for this task, the relations among these abstract order propositions may be roughly formalized as an implicative sequence. For what the subject perceives, "implicative" is used here in the psychological, not logical or physical, sense.

For the "telling" variable, if there is a treatment then it has side effect

(x)([T.sub.x][right arrow][E.sub.x]).

The two specific abstract propositions that form the formal statement "telling" are: "Doctor Adams offers a treatment preferred by the hospital," and "describes different treatment options, and their side effects,"

([T.sub.Adams][right arrow][E.sub.Adams]). A = (x) ([T.sub.x][right arrow][E.sub.x]). l[(x)([T.sub.x][right arrow][E.sub.x])] = (x)([T.sub.x][right arrow][E.sub.x]) = l([T.sub.Adams][right arrow][E.sub.Adams]) = ([T.sub.Flynn][right arrow][E.sub.Flynn])

The identity transformation, when applied to all the cases that preserve the relation between [T.sub.x] and [E.sub.x], forms the formal order equivalence class.

Similarly for the "understanding" statement, the implicative statement is, if the patient relates the treatments' effects back to the doctor then the patient understands ([E.sub.x][right arrow][U.sub.x]).

For Dr. Adams, the two specific abstract propositions that form the formal relational statement "understanding" are: "the doctor describes different treatment options...side effects," and "the patient is asked to explain these things back to the doctor," ([E.sub.Adams][right arrow][U.sub.Adams],). If Adams describes side effects (E), and asks the patient to explain that back (U), then the doctor confirms understanding.

Systematic Stage Equivalence Classes

Informing. When the two formal statements, "telling" and "understanding" are coordinated, they form a systematic equivalence class, informing. To identify a systematic equivalence class, subjects have to detect two similar formal statements. For example, in a formal operational statement, telling the doctor "tells the patient about treatment options" ([T.sub.x]) and their corresponding "side effects" ([E.sub.x]). This statement [F.sub.1] is that [T.sub.x] is correlated with [E.sub.x]. In the formal statement [F.sub.2], understanding, "the doctor tells the patient about side effects" ([E.sub.x]) and "asks the patient to relate back that information" ([U.sub.x]). Together, statements [F.sub.1], with [F.sub.2], describe the system, [S.sub.I], of the doctor informing the patient about the treatment.

All doctors' accounts of so informing their patients belong to that equivalence class. Note that there is only one doctor pair that is similar in this way, Adams and Flynn.

The informing system coordinates telling with understanding. It takes the form, if 'the doctor tells about all treatments' side effects,' and then 'the patient demonstrates an understanding by relating back that information,' then informing has taken place:

([T.sub.x][right arrow][E.sub.x])[right arrow]([E.sub.x][right arrow][U.sub.x]) = ([F.sub.1][right arrow][F.sub.2]) = [S.sub.I] = Informing.

Consenting. Two other formal order statements, in the vignettes, form a second systematic order equivalence class, consent. The first formal statement [F.sub.I], obtaining the agreement, consists of two abstract propositions. For example, the doctor indicates that the offered treatment is a recommendation and not a command by saying "experts found other treatments to have less favorable results," [R.sub.1], and then leaving the decision up to patient by asking "if the patient accepts the suggested treatment," [L.sub.1]. The second formal statement [F.sub.2], preparing for treatment, similarly consists of two abstract propositions, "feeling doctor knows best," [J.sub.1], and then "the patient prepares for treatment," Pa' Together, statements [F.sub.1], and [F.sub.2], describe the system [S.sub.C] of the doctor obtaining consent for administering the treatment.

The consent system coordinates obtaining the agreement with preparing for the treatment. It takes the form, if 'the doctor tells how the treatment decision will be made,' and 'asks the patient for agreement' and then the patient engages in the judgment process by 'feeling the doctor knows best,' and agrees with the doctor by 'preparing for the treatment,' then consent has taken place:

([H.sub.x][right arrow][O.sub.x])[right arrow]([J.sub.x][right arrow][U.sub.x]) = ([F.sub.1][right arrow][F.sub.2]) = [S.sub.C] = consenting.

All doctors' accounts of so obtaining consent from their patients belong to that equivalence class. Note that two vignette pairs are similar in how the doctors perform these actions.

The hierarchical complexity of the equivalence class is the same for "informing" and "consenting." However, subjects performing systematically identify informing as a single class of statements, and consenting as another single class of statements. In doing so, they find Adams and Flynn to be similar with respect to informing. They also find two other doctors, Casey and Greys, to be similar with respect to consent. Hence, systematic performance yields up to two doctor pairs that appear similar. Subjects performing systematically will identify these equivalence classes and rate those pairs "similar."

The Metasystematic Stage Equivalence Classes

Finally, the metasystematic equivalence class is formed by coupling the two systems [S.sub.I] with [S.sub.C]. Metasystematically consenting is not independent of informing. Statements that systematically belonged to the informing system, now also belong to the consent system. For example, the elements that belong to informing, [S.sub.I], telling and understanding are part of consenting, [S.sub.C], obtaining agreement and preparing for treatment. [S.sub.I] with [S.sub.C], are two systematic elements that form a metasystematic equivalence class, informed-consent. All the positive forms of the statements belong to the informed-consent equivalence class. Subjects that detect this relationship are performing metasystematically. Only one vignette pair is similar in this way, Casey and Greys.

Because the metasystematic supersystem, [M.sub.IC], informed-consent, consists of two systematic elements, informing and consenting and their alternatives, it is possible to test whether subjects use the supersystem (informed, not informed; consent, no consent) to detect instances of the metasystem of informed-consent.

Casey and Greys fail the same part, informing (I), and succeed at the same part, consenting (C).

A = [M.sub.IC] [M.sub.IC] = ([S.sub.C][right arrow][S.sub.I]). I[(x)([S.sub.C][right arrow][S.sub.I])] = (x)([S.sub.C][right arrow][S.sub.I]) = I([S.sub.C, Adams][right arrow][S.sub.I, Adams]) = ([S.sub.C, Flynn][right arrow][S.sub.I, Flynn])

The identity transformation, I, when applied to all the cases that preserve the relation between [S.sub.C] and [S.sub.I] forms the metasystematic order equivalence class, [M.sub.IC].

Failing to obtain informed-consent would be:

N(I) = ([S.sub.C] & -[S.sub.I]) = -([S.sub.C] [right arrow] [S.sub.I]) = -([S.sub.C,Casey][right arrow][S.sub.I,Casey]) = -([S.sub.C,Greys][right arrow][S.sub.I,Greys]) Casey = Greys

The Doctor-Patient Task is designed so that at the metasystematic order of hierarchical complexity, the subject must detect that two doctors have identical values of the informed system and consent system. To detect that they are identical, the I transform is applied to Casey's action and yields Greys's action. Both Casey and Greys try to obtain consent, but fail to inform ([S.sub.C] & [-S.sub.I]). Hence,

l([M.sub.-IC]) = [M.sub.-IC]

I([M.sub.-I, Casey; C,Casey]) = [M.sub.-I,Greys; C,Greys]

which are the identical metasystems consisting of not informed-consent.

Signal Detection of Similarity Shows Metasystematic Class Equivalence

The vignettes in the Doctor-Patient Task illustrate actions taken by different doctors that form hierarchical equivalence classes. Subjects must identify which doctors' actions are equivalent, thus belonging to a particular equivalence class. Two doctors who perform a set of actions that form a systematic equivalent class of actions are called a systematic pair. Two doctors who perform a metasystematic equivalent set of actions are called a metasystematic pair, and so forth. The higher the order of the equivalence class the greater the degree of similarity between the doctor pair. A person detecting metasystematic equivalence detects a greater degree of similarity between a metasystematic pair than a systematic pair. When such pairs were rated on a scale, the metasystematic pair is rated higher than the systematic pair.

Note that each higher order equivalence class consists of reordered lower order equivalence classes. That means that a metasystematically similar pair also contains systematic similarities, formal similarities, and so forth. Therefore, to persons with lower stage equivalence class skills, more pairs appear equally similar. For example, if presented with two pairs, one with systematic similarity and one with metasystematic similarity, a person performing at the systematic stage will detect the systematically similar but fail to detect the metasystematically similar. That subject will assert that both pairs are the same. As stage decreases, similarity appears to increase.

Subjects rated the degree of similarity in method between each of 10 pairs of doctors on a scale of 0 to 7. Because the scale runs from 0 to 7, there are 8 possible criterion rating levels to determine sensitivity to the metasystematic and systematic pairs. A rating at or above the criterion rating level is called a positive assertion. Lower ratings are called negative assertions. The details of this process are explained in the coding section of the results. There is only 1 doctor pair out of 10 pairs whose similarities form a metasystematic equivalence class. Therefore, the correct metasystematic rating pattern is one particular positive assertion, Casey & Greys, and nine negative assertions. In contrast, there are 2 pairs of doctors out of 10 who are systematically similar (two positive and eight negative assertions). Thus, metasystematic detection produces a different rating pattern (one and nine) from the pattern for systematic detection (two and eight). Therefore, it is impossible for a subject to derive the metasystematic rating pattern using systematic logic. The rating scheme in and of itself differentiates between systematic and metasystematic performance. The Fisher Exact test with Overall's (1980; Overall, Rhoades, & Starbuck, 1987; Rhoades & Overall, 1982; Rosenthall & Rosnow, 1991) correction tests the likelihood that the metasystematic rating pattern occurs by chance.

Expected Performance

Researchers have generally assumed that by adulthood, some people have become proficient at formal operational reasoning, and none experience further cognitive development. This view has been increasingly challenged (Commons, Richards, & Armon, 1984). With formal-operational complexity of reasoning, logical and organized experimental means are used to establish truth. Here we examine the possibility that postformal equivalence class problems exist and that some adults do solve them.

Identifying the relatedness of all of these disparate representations would probably not be a consequence of memorizing all combinations of two representations. Rather, after learning a small number of combinations, all of the remaining pairs would automatically be recognized without the benefit of additional explicit training.

Condition of Assessment in Equivalence Studies

The degree of support. In most traditional studies of equivalence, each part of the equivalence relation is tested separately. The relation must be shown to be reflexive, symmetric, and transitive. Here, we test for a performance that requires the use of equivalence relations as discussed above. Because the three properties are embedded in the task, there is a lower degree of support. As mentioned above, subjects performing at the systematic stage already see identity relations as reflexive, symmetric, and transitive. Hence, those properties are not tested here. Organisms quite often do better when directly tested on the three properties than indirectly.

Degree of training. In most traditional studies of equivalence, some properties of the equivalence relation are trained and then the other properties are tested. Here, no training whatsoever was given, including pretraining.

Degree of reflection. In most traditional studies of equivalence, the behavior evidencing equivalence is directly measured. Here, ratings that require equivalence and verbal descriptions were obtained.

Degree of feedback. During training of equivalence, feedback and reinforcement are quite often given. Here, neither were given.



The 18 subjects of varied ethnicity and income consisted of 14 females and 4 males with a mean age = 30.28, SD = 7.87, and range of 23 to 47. In this "purposeful" sample (Patton, 1990), a criterion of having postgraduate education increased the likelihood of finding subjects whose performance would meet the higher stage task requirements. Subjects' educational backgrounds ranged from entry level graduate school to doctoral education.


This psychophysical descriptive study broadly explores the existence of higher-stage equivalence classes in a particular social setting, the doctor-patient relationship. This study assesses subjects' responses to hierarchically complex stimuli embedded in a metasystematic equivalence class task. The doctor-patient encounter is used because it represents a common occurrence in adult life that places complex decision making, perspective-taking (Selman, 1980) and equivalence class formation demands on the individuals involved. The situation also lends itself readily to investigations of equivalence classes. All adults are patients at some time or other, and, therefore, will be forced to face the issues addressed in this study.


Subjects were asked to read seven vignettes in the Doctor-Patient Task. Subjects then rated the different vignettes and gave reasons for their ratings. Further probes were then used to help extract all of the subjects' reasons for their ratings. The interviews lasted between 1 hour and 40 minutes to 4 hours, averaging about 2 1/2 hours.


The data show that the seven metasystematic-reasoning individuals use all the information as to informing, consenting, and stage of the doctors' justification of information sources. The six people who were in transition between systematic and metasystematic reasoning used less of the information. The four systematic-stage-reasoning subjects used even less. This showed that higher-stage-reasoning subjects more adequately integrated information in a real-world problem and demonstrated metasytematic equivalence relations.

Data Coding

Subjects rated how similar 10 doctor pairs were on a 7-point scale. The next section describes how subjects' ratings were coded. Assessment of the ratings helps to answer the first research question: Can the Doctor-Patient Task detect postformal (Stages 5a, systematic and 5b, metasystematic) use of equivalence classes?

Classifying Subjects' Ratings

Subjects' ratings of the 10 doctor pairs were coded and scored using techniques from signal-detection theory (Green & Swets, 1966; Richards & Commons, 1990; Swets, 1964), choice theory (Luce, 1959), and the decision theory upon which they are built. The signal-detection scheme derives the probability that the subject has detected differences, in the vignettes, that require postformal perspective-taking.

In the present adaptation of signal detection to stage theory, the underlying assumption is that a subject detects signals of a specific order of "hierarchical complexity" (signals that represent a given stage). The hierarchical order of the signals that subjects identify represents the stage of their performance.

Computing the Maximal Non-normal d[prime]

The subjects scaled from 0 to 7 how similar each of 10 doctor pairs were. From rating data, we obtained their sensitivity to the metasystematic and systematic stimuli embedded in the 10 doctor-patient vignette pairs. By using rating data, we obtained their decision rules. We used a process of "sweeping" of subject ratings from 7 downward to score performance independent of what was the highest rating. For instance, one subject may give a highest rating of similarity of 7, whereas another subject may have a highest rating of 4.

The following describes how metasystematic-stage sensitivity was calculated. Calculating the systematic stage sensitivities differs from calculating metasystematic sensitivity only in which doctor-patient pairs were designated as most similar. There were three steps in calculating sensitivity. First, the doctor pair that was similar at the metasystematic stage was noted (Pair 10--Casey and Greys). This was the metasystematic stimulus--metasystematic signal present.

Second, responses were divided into positive assertions and negative assertions in the following manner. The highest rating given to any doctor pair set the first criterion rating level. For example, if the subject's highest rating of any of the 10 pairs was 7, then that became the first criterion value for determining what was a positive and what was a negative assertion. All 10 doctor pairs were then assessed. Each pair rated at or above the criterion rating (7 in this example) was scored a positive assertion. Each pair rated lower than the criterion rating (7) was scored a negative assertion. In traditional applications of choice theory (Lee, 1971; Luce, 1959; Swets, 1964), Hits and False Alarms both describe responses that are positive assertions, statements that claim certain events are present. Misses and Correct Rejections, in contrast, signify negative assertions, statements that claim certain events are not present. As shown in Table 2, positive assertions to the metasystematic pair were hits, negative assertions were misses. Positive assertions to other pairs were false alarms, negative assertions were correct rejections.

Next, a rating of 6 was selected as the criterion. Pairs rated 6 or above were scored as positive assertions. Pairs rated lower than 6 were scored as negative assertions. A pair that was metasystematic at a rating of 7 remains a metasystematic pair. Nontargets rated 6 are scored false alarms and nontargets rated lower than 6 are scored correct rejection. Pairs rated 6 were scored as positive assertion hits. Pairs rated lower than 6 were scored misses. Nontargets rated 6 are scored false alarms and nontargets rated lower than 6 are scored correct rejections. Note that the "sweep" of the ratings works for any degree of difference subjects detect. For example, subjects that choose 1 as their highest rating will score hits for targets that are rated 1, misses for targets that are rated lower than 1, and so forth.

Doctors Casey and Greys are similar with respect to informed-consent. When subjects rate that Casey and Greys are similar at or above a criterion rating, that is coded a positive assertion and a hit. If subjects rate as more dissimilar, that is, at a rating below the criterion rating, that is coded a negative assertion and a miss. However, Adams and Flynn are dissimilar with respect to informed-consent. If subjects assert that they are similar, that is coded a positive assertion and a false alarm. If subjects assert that they are dissimilar, it is coded a negative assertion and a correct rejection.

Sensitivity. Sensitivity is represented by the maximum value of non-normal d[prime], non-normal d[prime] = p(Hits) - p(False Alarms), the difference between the probability of hits and the probability of false alarms (Munsinger, 1970) at a given order of hierarchical complexity (Commons & Richards, 1984b; Kantrowitz, Buhlman, & Commons, 1985). For brevity of reference, the maximum value of non-normal d[prime] will also be referred to as d: Traditionally d[prime] is derived from z scores. Non-normal d[prime] is derived from raw scores. The following example shows how sensitivity is computed. Subject 9 gave a rating of 5 to two doctor pairs, #7 and #10. Because 5 is the subject's highest rating, it is a positive assertion. The other eight pairs receive ratings lower than 5 and therefore they are negative assertions. Pair #10 (Greys and Casey) is metasystematically equivalent because they both successfully inform but fail to obtain consent. Therefore, the rating of 5 is not only a positive assertion but is scored a hit. Pair #7 (Dr. Flynn and Dr. Adams) also was rated a 5, but is not metasystematically similar, however. Therefore, it is scored a false alarm. The other eight pairs are scored correct rejections. At this criterion rating level,

p(hits) = # hits (#hits + #misses) = 1/(1+0) = 1, and p(false alarm = # false alarms/(#false alarms + #correct rejections), = 1/(1 +8) = .11, therefore Sensitivity d[prime] = p(hits) - p(false alarms) = 1 -.11 = .89.

At lower rating levels, d[prime] decreases because the number of false alarms must increase while the numbers of hits stays constant.


Metasystematic Performance Found

Subject performance shows that some subjects were able to detect the metasystematic doctor-pair from the 10 pairs presented in Question 1. The performances of all the subjects are presented in Table 5. From left to right, Table 2 shows the subjects' ID number, their d[prime] scores and statistical significance for sensitivity to Stage 5b, and Stage 5a equivalence classification.

Table 4 shows that 7 out of 18 subjects perfectly detected the identical "no informed-consent" doctor pair on Question 1. Each of these d[prime] erformances was statistically significant, d[prime]= 1.0, and p [is less than] .0152. Such a detection is the essential part of the metasystematic task. These subjects gave the highest rating to the doctor pair that was most similar with respect to informed-consent. Also, their performance correctly differentiates the dissimilar doctors. To meet these two demands, subjects had to carry out a multiplicity of transformations, indicating how many values of a variable (informed, consent, and stage of the doctor) would have to be switched for the systems of social perspectives to be identical. Accordingly, they are classified as fully metasystematic performers (Commons, Richards, & Kuhn, 1982; Richards & Commons, 1984).

Further Classification of Subject Performance

Using their d[prime] scores, subjects were placed into four groups as shown in Table 4. Classifying performances of the subjects on Question 1 was used for the purpose of further analysis. Table 4 shows that the metasystematic Group 1 contains the seven subjects who obtained a d[prime] = 1, with a p [is less than] .0152. The transitional Group 2 contains six subjects who were marginally sensitive with a d[prime] = 0.889, p [is less than] .0455. Their alpha of p [is less than] .0455 indicates barely significant d[prime] values. One must be very cautious about classifying these subjects. They would be slightly transitional to metasystematic performance at best. Their slightly lower d[prime] scores indicate a tendency to confuse systematic similarities with metasystematic similarities. Because some of these transitional subjects did so poorly in detecting systematic stage similarities, it could be possible that their performances are only transitional to systematic stage. Such subjects might be comparing a list of the formal-operational statements rather than systems. The systematic Group 3 contains four subjects who failed to obtain statistically significant d[prime] values, but did detect some degree of difference in informed-consent when prompted under a "supported condition." These four subjects clearly had lots of difficulty differentiating the target pair from other pairs, as their high number of false alarms and nonsignificant p-values indicate. The presystematic Group 4 contains one subject who completely failed the metasystematic task, obtaining d[prime] = 0, p = 1.000.

Differentiating Stage 5a Vignettes

Table 4 also shows subjects' sensitivity scores of the Stage 5a vignettes. After scoring the ratings for detection of Stage 5b perspectives, the researcher analyzed the ratings again to determine how well subjects detected Stage 5a perspectives. The reader is reminded that the Stage 5a criterion requires detection of vignettes that are similar only in how the doctors informed or attempted to gain consent, not both. Note that many subjects said they had difficulty with rating scales, and particularly knowing just how to rate all of the vignettes. Table 4 shows subjects' sensitivity to Stage 5a perspectives.

The "sweep" of the ratings shows subject detection of the most similar Stage 5a vignettes. The sweep also reveals whether subjects correctly differentiated Stage 5a equivalence relationships from lower order equivalence classes. A subject's sensitivity is challenged by vignettes that on the surface appear similar but represent adjacent stages. Subjects in transition should exhibit greater difficulty coordinating those differences. Kohlberg (1984) has also found that subjects in transition evidence drops in performance.

Of all groups, Group 1 did the best at differentiating the non-metasystematic vignettes. All of the subjects in Group 1 differentiated the systematic pairs, as their d[prime] scores and significant p-values indicate.

ln contrast to Group 1, most of the subjects in Group 2 (transitional performance) were not able to differentiate the lower stage perspectives. This means they were not able to perform all the necessary combinations of the variable informed-consent. The hypothesis that subjects in Group 2 are not truly metasystematic performers is thus supported. Subjects 9 and 14 obtained perfect sensitivity scores (d[prime] = 1 ) with respect to the Stage 5a criterion, thus showing a stronger sensitivity to Stage 5a task demands than Stage 5b demands. However, the fact that subjects in Group 2 were able to differentiate the metasystematic vignettes to a significant degree suggests that they are more sensitive to the higher stage perspectives than subjects in the systematic group.

The scores for all four subjects in the systematic Group 3 improved given the lower stage criterion. Three of the four subjects, who had failed to detect the metasystematic vignettes, were able to detect the systematic vignettes. Subject 12 performed best, d[prime] = .875, p [is less than] .0182, and Subjects 2 and 5 slightly lower, d[prime] = .75, p [is less than] .0455. Unlike the subjects in the transitional group, these subjects were not able to detect the metasystematic perspectives, but exhibited better sensitivity to systematic perspectives. This supports the hypothesis that they are systematic performers. Subject 10 in the presystematic group appeared to show improvement; however, her d[prime] = .333 was not statistically significant, p [is less than] 1.0.

In summary, the "sweep" of the ratings shows that subjects who performed fully metasystematic equivalence class also differentiated the systematic vignettes from lower stage vignettes. This supports their perfect detection scores of d[prime] = 1. The performance of subjects. in the transitional group fluttered, some responding perfectly to the systematic perspectives whereas others' performance deteriorated. This supports their lower d[prime] scores, which indicated they were confusing higher order equivalence sets with lower order equivalence sets. In contrast, the improved scores for three of the four subjects in the systematic group support their nonsignificant d[prime] scores for detection of metasystematic perspectives.

How the Information Embedded in the Vignettes Explains the Ratings

A sensitivity analysis in conjunction with a correlation analysis between the information embedded in the vignettes and the subjects' ratings of the doctor pairs will help explain subjects' ratings. The correlation analysis describes the decision rules the experimenter infers that the subjects are using when responding to the information embedded in the vignettes. The results of the correlation analysis complement the signal detection analysis of subject sensitivity to the metasystematic equivalence class.

The hypothesis addressed here is: As stage of performance decreases, control by the critical variables in the doctor-patient vignettes decreases. Do subjects who obtained d[prime] = 1 scores in their detection of the metasystematic equivalence class use more of the information in the vignettes than subjects who perform transitionally, subjects who perform systematically, and so forth? To the extent to which the higher-stage performing subjects use all the information, the results generally support the d[prime] assessment. The correlation analysis suggests how strongly the difference in informed-consent and the difference in the stages of the doctors influenced subjects' ratings.

On the right side of Table 4 are the correlations between the subjects' ratings and the information in the vignettes that comprise the equivalence classes. There are two factors contributing to the equivalence classes, DIFIC and DIFDS. The variable DIFIC is the degree of difference in informed-consent between the two doctors; DIFDS is the degree of difference in the stages of the doctors' methods as shown in Table 5. Larger differences should produce lower ratings of similarity and a negative correlation coefficient. First, "informed" is presented and the four abstract propositions that it is comprised of: telling about a treatment, understanding, side effects, and treatment options. The value presented under each abstract proposition indicates whether the doctors performed or failed to perform that action. There are two values under each proposition, one for each doctor in that pair. A value of 1 means the doctor performed the action. A value of 0 means the doctor failed to perform the action. A value of .5 means that one cannot tell. The difference between the two doctors on "informed" (DIFI) is shown in Table 5. That value is derived by adding up all the propositions each doctor performs, then subtracting one doctor's total from the other's. Likewise, the value of the difference between two doctors on consent (DIFC) is derived and shown. The sum of these differences is the total difference in informed-consent (DIFIC).

The second is the stages the vignettes represent. Note that the stages in the vignettes do not always correspond to informed-consent. For example, two vignettes may represent systematic perspective-taking (Stage 5a) but each may also represent a different subset of informed-consent, either "informed" or "consent." This is also true for the lower stages represented in the vignettes. Some vignettes may represent subsets of either "informed" or "consent." Lastly, when considering ordering the differences between pairs of vignettes, the difference may be 0 stages (both represent the systematic stage) but differences in informed-consent could be eight abstract propositions apart.

Note that the size of the negative correlations between DIFIC and DIFDS with the ratings tend to drop as the subjects' stages of performance drop. The "average" correlation estimates for each group highlight this tendency. This means that the lower the perspective-taking stage of the subjects, the less information embedded in the vignettes they tend to use. For Group 1, the average correlation between the subjects' performance and their use of the information embedded in the vignettes is r = -.62 for how the doctors differed with respect to informed-consent, and r = -.42 for the different stages the doctors' actions represented. Group 2's average estimates indicate less use of the information embedded in the vignettes, r = -.47 for difference in informed consent and r = -.28 for difference in the stages of the doctors. Group 3's average estimates are even lower with respect to informed consent r = -.24, but a little higher with respect to the stages of the doctors, r = -.37. And lastly, the one subject who performed presystematic perspective-taking obtained the lowest correlation estimates, r= .09, and r = -.17.

The correlation analysis also reflects how well subjects differentiated all 10 doctor-pair combinations--large correlation being associated with seeing differences among the vignettes. The correlations increased with stage of subjects' performance. This finding supports the notion that increases in stage are associated with properly ordering the degree of nonequivalence. In contrast, the d' analysis shows subject detection of specific hierarchical pairs. Together, the d[prime] analysis of Stages 5b and 5a and the correlation analysis produce an informative picture of subject performance.

In summary, the correlation analysis supports the finding from the d' analysis that subjects in Group 1 detected more complex relationships between the vignettes than subjects in the other three groups. The correlation scores are averaged to illustrate that as stage decreases so does detection of the higher order equivalence relationships in the vignettes. Although subjects in transition to metasystematic stage detected the metasystematic vignettes, they did not use much more information embedded in the vignettes than subjects performing systematic equivalence class use. Lastly, the one subject who performed presystematic equivalence class use used the least amount of information embedded in the vignettes.


The hierarchical complexity of the equivalence relations seems to vary with the development of the organism, the situation, and materials used to test it and some species characteristics. The equivalence relations themselves are probably sensory-motor tasks, Stage-1a, as shown in Table 1. At this stage, discrimination occurs between conceptual classes, such as Vaughan and Green's study of pigeons classifying trees and nontrees and especially arbitrarily defined large sets (Vaughan, 1983). Zentall, Edwards, and Hogan (1983) among others have found minimally complex equivalence relations in pigeons (see Zentall & Urcuioli, 1993: this issue). For example, if an organism responds correctly to a new picture of a tree, the organism is said to possess the concept of tree. With arbitrarily defined sets, a great deal of training was necessary with the set of items used.

Children by around 18 months of age develop such equivalence classes. Watson and Ramey (1972) showed that young infants can discriminate between causal and noncausal contingencies. Killeen (1981 ) showed that pigeons can do likewise.

After the sensory-motor Stage 1a, some organisms can be trained to name the equivalence classes and move into the nominal Stage 1b, albeit with support. Pepperberg's (1993-a, 1993-b) grey parrots seem to perform at this stage on these tasks. In contrast, we show that without training or other forms of support, a large proportion of a purposeful sample of adults with graduate school training identify equivalence relations at the 12th (metasystematic stage 5b) out of 14 orders of hierarchical complexity.


ALEXANDER, C. N., DRUKER, S. M., & LANGER, E. J. (1990). Introduction: Major issues in the exploration of adult growth. In C. N. Alexander & E. J. Langer (Eds.), Higher stages of human development: Perspectives on adult growth (pp. 3-32). New York: Oxford University Press.

ARMON, C. (1984a). Ideals of the good life and moral judgment: Ethical reasoning across the life span. In M. L. Commons, F. A. Richards, & C. Armon (Eds.), Beyond formal operations: Vol. 1. Late adolescent and adult cognitive development (pp. 357-380). New York: Praeger.

ARMON C. (1984b). Ideals of the good life: A cross-sectional/longitudinal study of evaluative reasoning in children and adults. Unpublished doctoral dissertation, Harvard Graduate School of Education, Cambridge, MA.

CASE, R. (1985). Intellectual development: Birth to adulthood. Orlando, FL: Academic Press.
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Title Annotation:Special Issue: Stimulus Equivalence
Author:Commons, Michael L.; Rodriguez, Joseph Anthony
Publication:The Psychological Record
Date:Sep 22, 1993
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