Multicollinearity and Two-Predictor Suppressor Effects

For several decades, the concept of two-predictor suppressor effects has been the subject of debate. It seems the confusion, controversy and dismay that surround the definition and explanation of suppression situations (e.g., Holling, 1983; Ludlow & Klein, 2014; McFatter, 1979) has been caused by impossibilities or contradictory results.

Linear regression modelling with two or more predictors such as $x_1, x_2, ..., x_n$ and one response variable like $y$ is subject to multicollinearity effects. When a second predictor say $x_2$ which is correlated with $x_1$ is included in the regression equation, researchers have to carefully consider the effects of three zero-order correlations on the results of the linear regression model: $Cor(y,x_1)$, $Cor(y,x_2)$, and $Cor(x_1,x_2)$, or simply $r_{y1}$, $r_{y2}$, and $r_{12}$, respectively.

Under the condition of $|r_{y1}| > |r_{y2}|$, a two-predictor suppressor variable is a predictor included in the equation as the second predictor ($x_2$) which its absolute correlation with the previous predictor in the equation ($x_1$) exceeds the absolute ratio of $\gamma = r_{y2}/r_{y1}$ (i.e., $|r_{12}| > |r_{y2}/r_{y1}|$) (Negative Suppression). In addition, when $r_{y1}$ and $r_{y2}$ show similar signs the sign of $r_{12}$ is expected to be positive, otherwise $x_1$ and $x_2$ will reciprocally act as suppressor variables to each other. And when $r_{y1}$ and $r_{y2}$ show opposite signs the sign of $r_{12}$ is expected to be negative, otherwise $x_1$ and $x_2$ will reciprocally act as suppressor variables to each other (Reciprocal suppression). Friedman and Wall's graph (Friedman and Wall, 2005) refers to reciprocal suppression situations as region I enhancement and divides negative suppression situations into two regions: region III suppression in which $|\gamma| < |r_{12}| \leq |2\gamma/(1+\gamma^2)|$; and region IV enhancement in which $|r_{12}| > |2\gamma/(1+\gamma^2)|$. Therefore, there are four regions on Friedman and Wall's graph: region I enhancement (= reciprocal suppression), region II redundancy, region III suppression, region IV enhancement (see Friedman and Wall's application).

To help facilitate the research on two-predictor suppressor effects, Morteza Nazifi (Ph.D. in psychology) and Hamid Fadishei (Ph.D. in computer engineering) have developed a computerized algorithm called "Supsim" that enables researchers to easily generate numerous series of random data vectors $x_1$, $x_2$, and $y$ that when one regresses $y$ on both $x_1$ and $x_2$ by using each of these randomly generated datasets, numerous situations with or without suppression is produced. For each randomly generated datasets, Supsim itself automatically regresses $y$ on both $x_1$ and $x_2$ to build random, two-predictor models (RTM's) and then evaluates $R^2$, $\beta_1$, and $\beta_2$ values to classify the resulting situations either as enhancement, suppression, or redundancy. Visit web-based Supsim to produce numerous RTM's falling within suppression and non-suppression situations (also download and read Users Guide for Supsim).

References

Holling, H. (1983). Suppressor Structures in the General Linear Model. Educational and Psychological Measurement, 43(1), 1-9. doi:10.1177/001316448304300101

Ludlow, L., & Klein, K. (2014). Suppressor Variables: The Difference Between ‘is’ Versus ‘Acting As’. Journal of Statistics Education, 22(2), null-null. doi:10.1080/10691898.2014.11889703

McFatter, R. M. (1979). The Use of Structural Equation Models in Interpreting Regression Equations Including Suppressor and Enhancer Variables. Applied Psychological Measurement, 3(1), 123-135. doi:10.1177/014662167900300113