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The Meaning of R: It is the bivariate r between actual Yi and the Yi predicted by the regression equation

Predicting Female Life Expectancy using two variables in multiple regression

  • Yesterday, we saw the separate effects of two variables on female life expectancy
  • This graph showed that Female Literacy explained 67% of the variance
  • And this graph showed that the wealth of a society, measured by the logarithm of GDP per capita, explained 69% of the variance:
  • Suppose we used both variables, female literacy and society's wealth, to explain female life span?
    • We can't simply add together their separate explanations of variation -- 67% + 69% = 136%
      • It makes no sense to explain more than 100% of the variance.
      • One can't add together their explained variance, for female literacy and social wealth are themselves correlated at .632
    • That means that the two variables are sharing the variation that they explain.
    • We can use muliple regression analysis to separate their explanations.
  • Here's the result of that multiple regression analysis, first the overall summary, with the R and R2:
  • This box shows that the combined effects of the two variables increased the variance explained to 80%.
  • The ANOVA box shows that the multiple correlation, R, is significant far beyond the .05 level, for two variables and 85 cases.
  • The box above reports separate t test for the variables in the equation, which indicate that each is significant far beyond .05.
  • Here is the final regression equation, built from information in the box above:
    Y = 26.229 + 8.738*Log GDP_CAP + .197*FemaleLiteracy
  • To reproduce the multiple R between the actual life span and that predicted by the above equation by computing the estimated value from the equation, using "Compute" under the Transform Menu in SPSS 10.

  • Then we use the new variable Estimate in a simple linear scatterplot against Female Life Expectancy:
  • Note that this R2 is exactly equal to the R2 from the multiple regression analysis.
  • Thus, the R for a multiple regression equation is equal to the simple r computed between the original dependent variable and the estimated variable predicted by the regression equation.