1. Week 14
Chapter 16 – Partial Correlation and Multiple Regression and Correlation

2. Partial Correlation and Multiple Regression and Correlation
Chapter 16
Partial Correlation and Multiple Regression and Correlation

3. In This Presentation Partial correlations Multiple regression
Using the multiple regression line to predict Y
Multiple correlation coefficient (R2)
Limitations of multiple regression and correlation

4. Introduction Multiple Regression and Correlation allow us to:
Disentangle and examine the separate effects of the independent variables.
Use all of the independent variables to predict Y.
Assess the combined effects of the independent variables on Y.

5. Partial Correlation
Partial Correlation measures the correlation between X and Y controlling for Z
Comparing the bivariate (“zero-order”) correlation to the partial (“first-order”) correlation allows us to determine if the relationship between X and Y is direct, spurious, or intervening
Interaction cannot be determined with partial correlations

6. Partial Correlation
Note the subscripts in the symbol for a partial correlation coefficient:
rxy●z
which indicates that the correlation coefficient is for X and Y controlling for Z

7. Partial Correlation Example
The table below lists husbands’ hours of housework per week (Y), number of children (X), and husbands’ years of education (Z) for a sample of 12 dual-career households

8. Partial Correlation Example A correlation matrix appears below
The bivariate (zero-order) correlation between husbands’ housework and number of children is +0.50
This indicates a positive relationship

9. Partial Correlation Example
Calculating the partial (first-order) correlation between husbands’ housework and number of children controlling for husbands’ years of education yields +0.43

10. Partial Correlation Example
Comparing the bivariate correlation (+0.50) to the partial correlation (+0.43) finds little change
The relationship between number of children and husbands’ housework controlling for husbands’ education has not changed
Therefore, we have evidence of a direct relationship

11. Multiple Regression Previously, the bivariate regression equation was:
In the multivariate case, the regression equation becomes:

12. Multiple Regression Y = a + b1X1 + b2X2 Notation
a is the Y intercept, where the regression line crosses the Y axis
b1 is the partial slope for X1 on Y
b1 indicates the change in Y for one unit change in X1, controlling for X2
b2 is the partial slope for X2 on Y
b2 indicates the change in Y for one unit change in X2, controlling for X1

13. Multiple Regression using SPSS
Suppose we are interested in the link between Daily Calorie Intake and Female Life Expectancy in a third world country
Suppose further that we wish to look at other variables that might predict Female life expectancy
One way to do this is to add additional variables to the equation and conduct a multiple regression analysis.
E.g. literacy rates with the assumption that those who read can access health and medical information

14. Multiple Regression using SPSS: Steps to Set Up the Analysis
In Data Editor go to Analyze/ Regression/ Linear and click Reset
Put Average Female Life Expectancy into the Dependent box
Put Daily Calorie Intake and People who Read % into the Independents box
Under Statistics, select Estimates, Confidence Intervals, Model Fit, Descriptives, Part and Partial Correlation, R Square Change, Collinearity Diagnostics, and click Continue
Under Options, check Include Constant in the Equation, click Continue and then OK
Compare your output to the next several slides

15. Interpreting Your SPSS Multiple Regression Output
First let’s look at the zero-order (pairwise) correlations between Average Female Life Expectancy (Y), Daily Calorie Intake (X1) and People who Read (X2). Note that these are .776 for Y with X1, .869 for Y with X2, and .682 for X1 with X2
r YX1
r X1X2
r YX2

16. Examining the Regression Weights
Above are the raw (unstandardized) and standardized regression weights for the regression of female life expectancy on daily calorie intake and percentage of people who read.
The standardized regression coefficient (beta weight) for daily caloric intake is .342.
The beta weight for percentage of people who read is much larger, .636.
What this weight means is that for every unit change in percentage of people who read (that is, for every increase by a factor of one standard deviation on the people who read variable), Y (female life expectancy) will increase by a multiple of .636 standard deviations.
Note that both the beta coefficients are significant at p < .001

17. R, R Square, and the SEE
Above is the model summary, which has some important statistics. It gives us R and R square for the regression of Y (female life expectancy) on the two predictors. R is .905, which is a very high correlation. R square tells us what proportion of the variation in female life expectancy is explained by the two predictors, a very high It gives us the standard error of estimate, which we can use to put confidence intervals around the unstandardized regression coefficients

18. F Test for the Significance of the Regression Equation
Next we look at the F test of the significance of the
Regression equation, Y = .342 X X2. Is this so much better a predictor of female literacy (Y) than simply using the mean of Y that the difference is statistically significant? The F test is a ratio of the mean square for the regression equation to the mean square for the “residual” (the departures of the actual scores on Y from what the regression equation predicted). In this case we have a very large value of F, which is significant at p < Thus it is reasonable to conclude that our regression equation is a significantly better predictor than the mean of Y.

19. Confidence Intervals around the Regression Weights
Finally, your output provides confidence intervals around the unstandardized regression coefficients. Thus we can say with 95% confidence that the unstandardized weight to apply to daily calorie intake to predict female life expectancy ranges between .004 and .010, and that the undstandardized weight to apply to percentage of people who read ranges between .247 and .383

20. Limitations
Multiple regression and correlation are among the most powerful techniques available to researchers. But powerful techniques have high demands.
These techniques require:
Every variable is measured at the interval-ratio level
Each independent variable has a linear relationship with the dependent variable
Independent variables do not interact with each other
Independent variables are uncorrelated with each other
When these requirements are violated (as they often are), these techniques will produce biased and/or inefficient estimates. There are more advanced techniques available to researchers that can correct for violations of these requirements. Such techniques are beyond the scope of this text.