**The Meaning of R: It
is the bivariate r between actual Y_{i} and
the Y_{i} predicted by the regression
equation**

- 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.

- We can't simply add
together their separate explanations of variation --
67% + 69% = 136%
- Here's the result of
that multiple regression analysis, first the overall
summary, with the
**R**and**R**:^{2}

- 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

- 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
**R**is exactly equal to the^{2}**R**from the multiple regression analysis.^{2 } - 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.