multiple correlation & regression spss. analyze, regression, linear notice that we have added...
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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