Multiple Correlation & Regression
SPSS
Analyze, Regression, Linear
Notice that we have added “ideal” to the
model we tested earlier.
Statistics, Part and Partial Correlations
Plots: Zresid Zpredict, Histogram
ANOVA
F (2, 151)  4.468, p  .013
R2
In our previous model, without idealism, r2
= .049. Adding idealism has increased r2
by .056 - .049 = .007, not much of a
change.
Intercept and Slopes
Aˆ r  1.637  .185  Misanth .086  Ideal
zˆAr  .233  zMisanth  .086  zIdeal
Aˆ r  1.637  .185  Misanth .086  Ideal
• When Misanth and Ideal are both zero,
predicted Ar is 1.637.
• Holding Ideal constant, predicted Ar increases
by .185 point for each one point increase in
Misanth.
• Holding Misanth constant, predicted Ar
increases by .086 for each one point increase
in Ideal.
zˆAr  .233  zMisanth  .086  zIdeal
• Holding Ideal constant, predicted Ar increases
by .233 standard deviations for each one
standard deviation increase in Misanth.
• Holding Misanth constant, predicted Ar
increases by .086 standard deviation for each
one standard deviation increase in Ideal.
Tests of Partial (Unique) Effects
•Removing misanthropy from the model would
significantly reduce the R2.
•Removing idealism from the model would not
significantly reduce the R2.
sri2
• The squared semipartial correlation
coefficient is the amount of variance in Y that
is explained by Xi, above and beyond the
variance that has already been explained by
other predictors in the model.
• In other words, it is the amount by which R2
would drop if Xi were removed from the
model.
a+b+c+d=1
a + b = r2 for Ar_Mis
c + b = r2 for A_Ideal
R2
=a+b+c
b = redundancy between Mis and
Ideal with respect to predicting Ar
a = sr2 for Mis – the unique contribution of Mis
c = sr2 for Ideal – the unique contribution of Ideal
“Part” is the square root of sr2
The sr2 for Misanth is .232 = .0529
The sr2 for Ideal is .0852 = .007
We previously calculated the sr2 for Ideal as the
reduction in R2 when we removed it from the
model.
pr2
• The squared partial correlation coefficient is
the proportional reduction in error variance
caused by adding a new predictor to the
current model.
• Of the variance in Y that is not already
explained by the other predictors, what
proportion is explained by Xi?
sr2 versus pr2
• sr2 is the proportion of all of Y that is
explained uniquely by Xi.
• pr2 is the proportion of that part of Y not
already explained by the other predictors that
is explained by Xi.
pr2 for Mis is a/(a+d); sr2 is a/(a+b+c+d) = sr2/1.
pr2 for Ideal is c/(c+d); sr2 is c/(a+b+c+d) = sr2/1.
pr2 will be larger than sr2.
The pr2 for Misanth is .2312 = .053.
The pr2 for Ideal is .0872 = .008.
The Marginal Distribution of the
Residuals (error)
We have
assumed that
this is normal.
Standardized Residuals Plot
Standardized Residuals Plot
• As you scan from left to right, is the variance
in the columns of dots constant?
• Are the normally distributed?
Put a CI on R2
• If you want the CI to be consistent with the
test of significance of R2, use a confidence
coefficient of 1-2, not 1-.
The CI extends from .007 to .121.
Effect of Misanth Moderated by Ideal
• I had predicted that the relationship between
Ar and Misanth would be greater among
nonidealists than among idealists.
• Let us see if that is true.
• Although I am going to dichotomize Idealism
here, that is generally not good practice.
• There is a better way, covered in advanced
stats classes.
Split File by Idealism
Predict Ar from Misanth by Ideal
For the NonIdealists
Ar = 1.626 + .30 Misanth
Among Idealists
Ar = 2.405 + .015 Misanth
Confidence Intervals for 
• http://faculty.vassar.edu/lowry/rho.html
• For the NonIdealists,
CI for the Idealists