Redundancy and Suppression

Trivariate Regression

b a cX1 X2

Y

Predictors Independent of Each Other

21

21 srary 2

222 srcry 02

12 r

22

21

212 yyY rrR

b = error

Redundancy

sr12 = b sr22 = d

Redundancy:Example

• 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)

Extreme Redundancy• X1 and X2 are highly

correlated with each other• Each X is well correlated

with Y (B + C; C + D)• Each X has a unique

contribution (B or C) toosmall to be significant

• But the R2 (B + C + D) is significant.

Formulas Used Here

212

122122

212

12 12r

rrrrrR yyyyy

212

12211 1 r

rrr yy

2)1...(12

2123

212

21

2...12 pypyyypy srsrsrrR

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

X1

X2

,4255.181.45.1

38.2

2

12.

yR

,476.45.1

)45(.038.21

.214.45.1

)45(.38.022

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)

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.

2)2.1(

2)2.1(

22

2 0 yyy rrrR

Classical Suppression Math

• is less than

144.38. 221

b

dcbbr y

212.22

12

212

)2.1( 181.45.1

144.11 y

yy R

rr

db

cbbr

Y

X1

X2

Net Suppression YX1

X2

ry1 = .65, ry2 = .25, and r12 = .70.

. > 93.70.1

)70(.25.65.121 yr

.40.70.1

)70(.65.25.22

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.

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.

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).

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.

Cooperative Suppression

• ry1 = .30, ry2 = .25, and r12 = 0.35.

.405.35.1

)35.(30.25.22

.442.35.1

)35.(25.30.21

.234.)405(.25.)442(.3.212. iyiy rR

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.

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.

Psychologist Investigating Suppressor Effects in a Five

Predictor Model