Week 14
Chapter 16 – Partial Correlation and
Multiple Regression and Correlation
Chapter 16
Partial Correlation and Multiple
Regression and Correlation
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
Introduction
 Multiple Regression and Correlation allow us
to:
1.
2.
3.
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.
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
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
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
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
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
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
Multiple Regression
Previously, the bivariate regression equation was:
In the multivariate case, the regression equation
becomes:
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
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
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
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
Correlations
Average
female life
expectancy
Pearson Correlation
r YX1
r YX2
Sig . (1-tailed)
N
Average female life
expectancy
Daily calorie intake
People who read (%)
Average female life
expectancy
Daily calorie intake
People who read (%)
Average female life
expectancy
Daily calorie intake
People who read (%)
Daily calorie
intake
People who
read (%)
1.000
.776
.869
.776
.869
1.000
.682
.682
1.000
.
.000
.000
.000
.000
.
.000
.000
.
74
74
74
74
74
74
74
74
74
r X1X2
Examining the Regression Weights
Coefficientsa
Model
1
(Constant)
People who read (%)
Daily calorie intake
Unstandardized
Coefficients
B
Std. Error
25.838
2.882
.315
.034
.007
.001
Standardized
Coefficients
Beta
.636
.342
t
8.964
9.202
4.949
Sig .
.000
.000
.000
95% Confidence Interval for B
Lower Bound
Upper Bound
20.090
31.585
.247
.383
.004
.010
Zero-order
Correlations
Partial
.869
.776
.738
.506
Part
.465
.250
Collinearity Statistics
Tolerance
VIF
.535
.535
1.868
1.868
a. Dependent Variable: Average female life expectancy
• 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
R, R Square, and the SEE
Model Summary
Chang e Statistics
Model
1
R
.905a
R Square
.818
Adjusted
R Square
.813
Std. Error of
the Estimate
4.948
R Square
Chang e
.818
F Change
159.922
df1
df2
2
71
Sig . F Change
.000
a. Predictors: (Constant), People who read (%), Daily calorie intake
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 .818. It
gives us the standard error of estimate, which we can use
to put confidence intervals around the unstandardized
regression coefficients
F Test for the Significance of the
Regression Equation
ANOVAb
Model
1
Reg ression
Residual
Total
Sum of
Squares
7829.451
1738.008
9567.459
df
2
71
73
Mean Square
3914.726
24.479
F
159.922
Sig .
.000a
a. Predictors: (Constant), People who read (%), Daily calorie intake
b. Dependent Variable: Average female life expectancy
Next we look at the F test of the significance of the
Regression equation, Y = .342 X1 + .636 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 <.001. Thus it is reasonable to conclude that our regression equation is
a significantly better predictor than the mean of Y.
Confidence Intervals around the
Regression Weights
Coefficientsa
Model
1
(Constant)
Daily calorie intake
People who read (%)
Unstandardized
Coefficients
B
Std. Error
25.838
2.882
.007
.001
.315
.034
Standardized
Coefficients
Beta
.342
.636
t
8.964
4.949
9.202
Sig .
.000
.000
.000
95% Confidence Interval for B
Lower Bound
Upper Bound
20.090
31.585
.004
.010
.247
.383
Zero-order
Correlations
Partial
.776
.869
.506
.738
Part
a. Dependent Variable: Average female life expectancy
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
.250
.465
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.