Pearson’s Correlation and Bivariate Regression

Lab Exercise:Chapter 9

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Example Questions:

• Do opposites really attract? Is there a negative correlation between the educational levels of spouses?

• One more year in school typically results in how much more annual income?

• Schooling accounts for how much of the differences in persons’ incomes?

• What annual income would we predict for someone with 16 years of schooling?

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Interval/Ratio Measures of Association

• Pearson’s r– ranges from −1.00 to 1.00– symmetric

• Analyze | Correlate | Bivariate– pairwise and listwise deletion of missing data

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Bivariate Correlation

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Scatterplot: Do opposites attract?*Check linearity, strength, direction, and homoscedasticity

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Bivariate Linear Regression: Income on Schooling

• Equation for a straight line

• “Best-fitting” straight line

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Y = a + b(X)

Bivariate Linear Regression (cont.)• Analyze | Regression | Linear

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Answering Questions with Statistics Chapter 9 8

Regression Output of INCOME86 on

EDUC for 1980 GSS Young Adults

Bivariate Linear Regression (cont.)• Unstandardized coefficients

• Regression equation

• Predicted value Ŷ: substitute value for X (16 yrs?)= $21,604.089

• Regression residual: Y - Ŷ9

Y = -3089.255 + 1543.334(X) INCOME86 = -3089.255 + 1543.334(EDUC)

Bivariate Linear Regression (cont.)

• Multiple correlation coefficient (R)– indicates strength but not direction

• Coefficient of determination (R2)

• Coefficient of alienation (residual or unexplained)

coefficient of determination = R2

coefficient of alienation = 1 - R2

Bivariate Linear Regression (cont.)

• Some limitations to remember– regression does not prove causality– for interval-ratio level variables• Can be used with caution (requires special

interpretation) for grouped interval ratio or ordinal variables with >5 categories

– linear means only linear

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