Some Comments on Bivariate Regression

Abstract

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Parametric Collapsibility and the Lack of Moderating Effects in Contingency Tables with a Dichotomous Response Variable

“…Let fJ12.3 denote the partial regression coefficient, the coefficient of V 2 in a linear regression of VI on V 2 and V 3, and fJ12, the marginal regression coefficient, the coefficient of V 2 in a linear regression of VI on V 2 alone. Then it is known (Currie and Korabinski, 1984) that the two are related through…”

Suppression and Enhancement in Bivariate Regression

“…The concept of suppression has not only been widely used in psychology (Cohen & Cohen, 1975) but also appears under different names in other fields of research (Conger, 1974). Currie & Korabinski (1984) shifted the emphasis away from partial regression coefficients entirely, to the increase in the multiple correlation coefficient. They defined 'enhancement' to occur when the squared multiple correlation coefficient exceeds the sum of squared simple correlations with Y.…”

“…Clearly, the concepts of enhancement and suppression are related. Currie & Korabinski (1984) noted that enhancement implies suppression (an unfortunate conflict of terminology) and studied the occurrence of these phenomena in the context of a fixed orientation between the observation vector, Y, and the XiS. We propose to study these phenomena from the more natural sampling viewpoint, where the orientation between the XiS is fixed and Y is allowed to vary.…”

The Geometry of Enhancement in Multiple Regression

“…This so-called "surprising inequality" (Cuadras, 1993), "quirk" (Bertrand & Holder, 1988), or "counterintuitive" (Shieh, 2001) property of multiple regression, is considered theoretically commonplace by some authors (e.g., Currie & Korabinski, 1984;Lewis & Escobar, 1986;Shieh, 2001) and empirically rare by others (Dicken, 1963;Wiggins, 1973). Related notions include "synergism" (Hamilton, 1988), "enhance-synergism" (Lipovetsky & Conklin, 2004), "enhancement-synergism" (Shieh, 2001), "suppression" (Horst, 1941;Meehl, 1945;Sharpe & Roberts, 1997) and "masking" (Kendall & Stuart, 1973, p. 331).…”

“…In linear multiple regression, "enhancement" (Cuadras, 1993;Currie & Korabinski, 1984;Freund, 1988;Friedman & Wall, 2005;Hamilton 1987Hamilton , 1988Mitra, 1988;Routledge, 1990;Shieh, 2001) is said to occur when R 2 = b r > r r, where b is a p × 1 vector of standardized regression coefficients and r is a p × 1 vector of correlations between a criterion y and a set of standardized regressors, x. This so-called "surprising inequality" (Cuadras, 1993), "quirk" (Bertrand & Holder, 1988), or "counterintuitive" (Shieh, 2001) property of multiple regression, is considered theoretically commonplace by some authors (e.g., Currie & Korabinski, 1984;Lewis & Escobar, 1986;Shieh, 2001) and empirically rare by others (Dicken, 1963;Wiggins, 1973).…”