Originally published as Krus, D. J., & Wilkinson, S. M. (1986) Demonstration of properties of a suppressor variable. Behavior Research Methods, Instruments, and Computers, 18, 21-24.

Demonstrating properties of suppressor variables
USING propositional CALCULUS

David J. Krus and Susan M. Wilkinson

Arizona State University

A general logical model of properties of suppressor variables is proposed. Consistent exploration of possible manifestations of suppressor variables within this theoretical framework accounts for extant classifications of suppressor variables into the classical, net, and cooperative categories and suggests existence of new subcategories, not detected previously. The discussed model also leads to consistent identification and classification of suppressor variables.

Since Horst (1941) introduced the concept of the suppressor variable, this "quasiparadoxical curiosity" (to use Cohen & Cohen's term) has received sustained attention (Darlington, 1968; Dayton, 1972; Lubin, 1957). In its classical rendering (Conger, 1974, pp. 36‑37).

Suppressor variable has a zero correlation with the criterion, but nevertheless contributes to the predictive validity of a test battery. The current definition of a suppressor variable is that it is a variable that increases regression weights and, thus, increases the predictive validity of other variables in a regression equation

Classification of suppressor variables into several categories was a logical outcome of the increase in generality of the suppressor variable concept. Conger (1974) identified three kinds of suppressor variables: traditional, negative, and reciprocal. Cohen and Cohen (1975, pp. 8491) named the same categories classical, net, and cooperative. As defined by (Cohen & Cohen, 1975, p. 91)

If the [predictor variable] in question has a zero (in practice very small) correlation with the [criterion variable], the situation is one of classical suppression. If its beta weight is of opposite sign from its [correlation with the criterion], it is serving as a net suppressor. If its beta weight exceeds its correlation with the criterion and is of the same sign, cooperative suppression is indicated.

The same authors also call attention to the fact that a beta coefficient which falls outside the limits defined by the correlation of its corresponding variable with the criterion and zero, signals the presence of a suppressor variable.

McNEMAR’S MODEL OF SUPPRESSOR VARIABLES

To gain insight into the logical relationships underlying the seemingly paradoxical behavior of suppressor variables, one might well consult the McNemar (1969, pp. 210‑211) model. Suppose a predictor variable X₁ consists of three elements, a predictor variable X₂ of one element, and a criterion variable Y of two elements. Furthermore, suppose that X₁ and X₂ have one element in common, X₁ and Y have two elements in common, and X₂ and Y have no overlapping elements, which can be visualized as:

In the above diagram, inclusion of the predictor variable X₂ into the multiple regression removes the element a from the predictor variable X₁, as this element is not necessary for predicting the criterion variable Y. The purpose of this paper is to elaborate and generalize the McNemar model. Also, consistent unfolding of the new model was attempted to demonstrate its ability to account for extant categories of suppressor variables and to indicate whether existence of alternate or additional classification categories is a theoretical possibility.

A GENERAL LOGICAL MODEL OF SUPPRESSOR VARIABLES

Consider a set of attributes  is contained by a set of predictor variables  and by a criterion variable Y. Let us restrict the discussion to a case in which an attribute is either present (1) or absent (0). An exhaustive pattern of all possible configurations of the attributes contained by variables in a regression equation will be . Assume that we could identify all attributes contained by each predictor variable and by the criterion. The response patterns then could serve as a template for construction of a hypothetical data matrix, containing the minimal, but exhaustive, set of response patterns as if all individuals with all possible attributive configurations were included in the analysis. Next, assume that for a minimal regression model of two predictors and a criterion, the predictor variable X₁ contains u attributes, the predictor variable X₂ is composed of v attributes, and the criterion variable Y has w attributes. Thus, if v is a subset of u and w is a subset of both u and v, then the correlation matrix computed from this data and the results of the multiple regression analysis will indicate that among the predictor variables is a suppressor variable.

For instance, consider a case of a hypothetical test, consisting of two predictor variables X_1, and X₂, and a criterion variable Y, used for selection of applicants for a position that requires knowledge of addition and multiplication. It may happen that one of the arithmetic problems used to assess these abilities contains an item requiring the knowledge of division for its correct solution. The attributes present in this test battery thus are the abilities to add, multiply, and divide, which may be symbolized as a₁, a₂, and a₃. For instance, a test item X₁

[(3 + 5) / 7] 8 = ?

may contain all three attributes. A test item X₂

9 / 3 = ?

may require the knowledge of division, and the criterion performance Y could be assessed by observing on‑the-job additions and multiplications. As the context of this hypothetical study contains three attributes, the matrix of all possible response patterns, assuming (unrealistically) the independence of all three attributes, will have  response patterns, such as

As the successful performance on the variable X₁, necessitates the knowledge of addition, multiplication, and division, only an individual with all three abilities will respond correctly. Thus X₁ = [1, 0, 0, 0, 0, 0, 0, 0] for a hypothetical set of 8 individuals unique with respect to configuration of attributes measured. Since the second predictor variable X₂ is a question solvable by division, X₂ will be the same as a₃. Finally, the criterion performance Y, based on abilities to add and multiply, will be correct only if both the a₁ and a₂ abilities are present with Y = [1, 1, 0, 0, 0, 0, 0, 0]. Thus the data matrix antecedent to this hypothetical multiple regression analysis will be

The correlation matrix for the above hypothetical data is

The above matrix shows a configuration of correlations that suggests presence of a suppressor variable within the set of the predictor variables.

CATEGORIES OF SUPPRESSOR VARIABLES

A minimal configuration of the discussed model requires two predictors of which one must contain at least two attributes and a criterion variable having a singular attribute. Suppose X₁ contains two attributes, X₂ shares its singular attribute with the predictor variable X₁ but not with the criterion variable Y, and the criterion variable Y contains a single attribute which is also present in the variable X1, but not in the variable X2. The template for these attributes may be constructed as

and its corresponding data matrix is shown below.

To facilitate the understanding of how this data matrix was constructed from the above template, assume that the variableX1 consisted of a single question, testing for an ability to add and subtract; for instance 2 + 3 - 1 = ? The variable X2 consisted of single-item testing for an ability to subtract; for instance 2 - 3 = ? The criterion variable Y tested for addition; for instance 3 + 7 = ? Using the above template, the first row of the data matrix was written as a string of ones because the first hypothetical subject possessed all skills necessary to solve the problems. The digit in the second row and first column s zero, because two abilities are necessary to solve the first problem, and the second subject possesses only one. The next zero indicates that even though only a single ability is necessary, the subject, knowing how to add but not how to subtract, will not pass the item. The last number, a one, indicates that the second subject knows how to add. The remaining two lines were formed in the same fashion. The matrix of Intercorrelations, adjacent to the above data matrix in the first display in Table 1, together with the vector of predictor-criterion cross-correlations and its corresponding vector of beta weights indicate that the variable X2 is a classical suppressor variable.

In the case of real data matrices, the presence of exactly equal proportions of all four possible response patterns is unlikely. This situation was simulated in the second display in Table 1, where the first theoretical response pattern A was included twice. The resulting matrix of intercorrelations displayed the properties of a net suppression. The other displays in the Table 1 were created by successively doubling the remaining response patterns. Their corresponding correlation matrices and vectors of beta weights demonstrate patterns characteristics of the cooperative and net suppression.

Table 1. Logical Model of Suppressor Variables

Negative Classical Suppression

Negative Net Suppression

Negative Cooperative Suppression

Negative Cooperative Suppression

Negative Net Suppression

Reflection of a variable reverses its orientation, but does not alter the latent structure of the data set. Reflecting the suppressor variable X2. resulted in patterns of intercorrelations and their corresponding beta weights as shown in Table 2.

Table 2. Logical Model of Suppressor Variables with the
 Reflected Suppressor Variable.

Positive Classical Suppression

Positive Net Suppression

Positive Cooperative Suppression

Positive Cooperative Suppression

Positive Net Suppression

The regression solution for the data matrix in the upper display of Table 2 can again be classified as a classical suppression, but of a different type. The remaining data matrices in Table 2 simulate the variations of over- and under representations of the identified response patterns, simulating the categories of net and cooperative suppressions together with their subcategories. Inspection of relevant parameters in the above tables reveals six distinctive patterns, as shown in Table 3.

Table 3. Classification Table of the Suppressor Variables.

Generic categories of the suppressor variables are the  positive classical suppression and the negative classical suppression. Subcategories of positive classical suppression  are the positive cooperative and the positive net suppression. Subcategories of the negative classical suppression  are the negative cooperative and the negative net suppression. The above table can  be thus rearranged as shown in Table 4.

Table 4. Classification Table for the Positive and Negative Suppression.

DISCUSSION

The logical model of properties of suppressor variables offers a method for systematic construction of idealized data matrices displaying configurations of intercorrelations, predictor ‑ criterion correlations, beta weights, and resulting multiple regression solutions relevant for description of categories of suppressor variables. The proposed classification scheme naturally unfolds from the postulates of the extended McNemar (1969) model. It is concise and closed within the framework of the mutually exclusive and exhaustive categories delineated at the inception of our discussion. The explanatory potential of the described theoretical model is substantial, as it provides means for consistent identification and classification of suppressor variables.

REFERENCES

CONGER, A. J. (1974). A revised definition for suppressor variables: A guide to their identification and interpretation. Educational and Psychological Measurement, 34, 35‑46.

COHEN, J., & COHEN, P. (1975). Applied multiple regression / correlation analysts for the behavioral sciences. New York: Wiley.

DARLINGTON, R. B. (1968). Multiple regression in psychological research and practice. Psychological Bulletin, 69, 161‑182.

DAYTON, C. M. (1972). A method for constructing data which illustrate a suppressor variable. The American Statistician, 26(5), 36.

HORST, P. (1941). The role of predictor variables which are independent of the criterion. Social Science Research Bulletin, 48, 431 ‑436.

LUBIN, A. (1957). Some formulae for use with suppressor variables. Educational and Psychological Measurement, 17, 286‑296.

MCNEMAR, Q. (1969). Psychological statistics. New York: Wiley.

(Manuscript received May 8, 1985; revision accepted for publication February 18, 1986.)