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Redundancy and Suppression. Trivariate Regression. Predictors Independent of Each Other. X 1. X 2. a. c. Y. b = error. Redundancy. - PowerPoint PPT Presentation
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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
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