1. Redundancy and Suppression
Trivariate Regression

2. Predictors Independent of Each Otherb X1 X2 a c Y
b = error

3. Redundancy
For each X, sri and i will be smaller than ryi, and the sum of the squared semipartial r’s (a + c) will be less than the multiple R2. (a + b + c)

4. Formulas Used Here

5. Classical Suppression
ry1 = .38, ry2 = 0, r12 = .45.
the sign of  and sr for the classical suppressor variable may be opposite that of its zero-order r12. Notice also that for both predictor variables the absolute value of  exceeds that of the predictor’s r with Y. Y X2 X1

6. Classical Suppression WTF
adding a predictor that is uncorrelated with Y (for practical purposes, one whose r with Y is close to zero) increased our ability to predict Y?
X2 suppresses the variance in X1 that is irrelevant to Y (area d)

7. Classical Suppression Math
r2y(1.2), the squared semipartial for predicting Y from X2 (sr22 ), is the r2 between Y and the residual (X1 – X1.2). It is increased (relative to r2y1) by removing from X1 the irrelevant variance due to X2  what variance is left in partialed X1 is better correlated with Y than is unpartialed X1.

8. Classical Suppression Math
is less than Y X2 X1

9. Net Suppression ry1 = .65, ry2 = .25, and r12 = .70.X1 X2
Note that 2 has a sign opposite that of ry2. It is always the X which has the smaller ryi which ends up with a  of opposite sign. Each  falls outside of the range 0  ryi, which is always true with any sort of suppression.

10. Reversal Paradox Aka, Simpson’s Paradox
treating severity of fire as the covariate, when we control for severity of fire, the more fire fighters we send, the less the amount of damage suffered in the fire.
That is, for the conditional distributions (where severity of fire is held constant at some set value), sending more fire fighters reduces the amount of damage.

11. Cooperative Suppression
Two X’s correlate negatively with one another but positively with Y (or positively with one another and negatively with Y)
Each predictor suppresses variance in the other that is irrelevant to Y
both predictor’s , pr, and sr increase in absolute magnitude (and retain the same sign as ryi).

12. Cooperative Suppression
Y = how much the students in an introductory psychology class will learn
Subjects are graduate teaching assistants
X1 is a measure of the graduate student’s level of mastery of general psychology.
X2 is an SOIS rating of how well the teacher presents simple easy to understand explanations.

13. Cooperative Suppression
ry1 = .30, ry2 = .25, and r12 = 0.35.

14. Summary
When i falls outside the range of 0  ryi, suppression is taking place
If one ryi is zero or close to zero, it is classic suppression, and the sign of the  for the X with a nearly zero ryi may be opposite the sign of ryi.

15. Summary
When neither X has ryi close to zero but one has a  opposite in sign from its ryi and the other a  greater in absolute magnitude but of the same sign as its ryi, net suppression is taking place.
If both X’s have absolute i > ryi, but of the same sign as ryi, then cooperative suppression is taking place.

16. Psychologist Investigating Suppressor Effects in a Five Predictor Model