Multiple Correlation & RegressionSPSS

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

013. ,468.4)151 ,2( pF

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

IdealMisanthrA 086.185.637.1ˆ

IdealMisanthAr zzz 086.233.ˆ

• 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.

IdealMisanthrA 086.185.637.1ˆ

• 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.

IdealMisanthAr zzz 086.233.ˆ

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 + cb = 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