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Multiple Correlation & Regression SPSS

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Analyze, Regression, Linear Notice that we have added “ideal” to the model we tested earlier.

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Statistics, Part and Partial Correlations

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Plots: Zresid Zpredict, Histogram

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ANOVA

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R2R2 In our previous model, without idealism, r 2 =.049. Adding idealism has increased r 2 by.056 -.049 =.007, not much of a change.

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Intercept and Slopes

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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.

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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.

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Tests of Partial (Unique) Effects Removing misanthropy from the model would significantly reduce the R 2. Removing idealism from the model would not significantly reduce the R 2.

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sr i 2 The squared semipartial correlation coefficient is the amount of variance in Y that is explained by X i, 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 R 2 would drop if X i were removed from the model.

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a + b + c + d = 1 a + b = r 2 for Ar_Mis c + b = r 2 for A_Ideal R 2 = a + b + c b = redundancy between Mis and Ideal with respect to predicting Ar a = sr 2 for Mis – the unique contribution of Mis c = sr 2 for Ideal – the unique contribution of Ideal

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“Part” is the square root of sr 2 The sr 2 for Misanth is.23 2 =.0529 The sr 2 for Ideal is.085 2 =.007 We previously calculated the sr 2 for Ideal as the reduction in R 2 when we removed it from the model.

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pr 2 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 X i ?

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sr 2 versus pr 2 sr 2 is the proportion of all of Y that is explained uniquely by X i. pr 2 is the proportion of that part of Y not already explained by the other predictors that is explained by X i.

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pr 2 for Mis is a/(a+d); sr 2 is a/(a+b+c+d) = sr 2 /1. pr 2 for Ideal is c/(c+d); sr 2 is c/(a+b+c+d) = sr 2 /1. pr 2 will be larger than sr 2.

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The pr 2 for Misanth is.231 2 =.053. The pr 2 for Ideal is.087 2 =.008.

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The Marginal Distribution of the Residuals (error) We have assumed that this is normal.

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Standardized Residuals Plot

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As you scan from left to right, is the variance in the columns of dots constant? Are the normally distributed?

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Put a CI on R 2 If you want the CI to be consistent with the test of significance of R 2, use a confidence coefficient of 1-2 , not 1- .

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The CI extends from.007 to.121.

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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.

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Split File by Idealism

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Predict Ar from Misanth by Ideal

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For the NonIdealists

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Ar = 1.626 +.30 Misanth

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Among Idealists

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Ar = 2.405 +.015 Misanth

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Confidence Intervals for http://faculty.vassar.edu/lowry/rho.html For the NonIdealists,

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CI for the Idealists

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Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Simple Linear Regression Analysis Chapter 13.

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Simple Linear Regression Analysis Chapter 13.

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