Chapter 11: Multiple Regression
Readings: Chapter 11
Introduction
•
Simple Linear Regression
(Chapters 2 and 10): used when there is
a single quantitative
explanatory variable
.
•
Multiple Regression
: used when there are
2 or more quantitative explanatory variables
which will be used to predict the quantitative response variable.
The Model
•
For
simple linear regression
, the statistical model is:
y
=
β
0
+
β
1
x
+
•
For
multiple regression
, the statistical model is:
y
=
β
0
+
β
1
x
1
+
β
2
x
2
+
. . . β
p
x
p
+
,
where
p
is the number of explanatory variables.
Multiple Regression Assumptions
•
Independence: Responses
y
i
’s are independent of each other (examine the way in which sub
jects/units were selected in the study).
•
Normality: For any fixed value of
x
, the response
y
varies according to a normal distribution
(normal probability plot of the residuals).
•
Linearity: The mean response has a linear relationship with
x
(scatter plot of
y
against each
predictor variable).
•
Constant variability: The standard deviation of
y
(
σ
) is the same for all values of
x
(scatter
plots of residuals against predicted values).
What to do when we have multiple
x
variables?
1.
Look at the variables individually
.
–
Means, standard deviations, minimums, and maximums, outliers (if any), stem plots or
histograms are all good ways to show what is happening with your individual variables.
–
In SPSS, Analyze
→
Descriptive Statistics
→
Explore.
2.
Look at the relationships between the variables using the correlation and scatter
plots
.
–
In SPSS, Analyze
→
Correlate
→
Bivariate. Put all your variables (all the
x
’s and
y
) into
the “variables” box, and hit “ok”.
–
The correlations helps us determine which are the stronger relationships between the
y
and
an
x
.
–
Are there strong
x
to
x
relationships?
–
Look at scatter plots between each pair of variables, too.
–
We are only interested in keeping the variables which had strong relationships with the
response variable
y
.
3.
Do a regression using the all potential explanatory variables
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
–
This will include an ANOVA table and coefficients output like in Chapter 10.
These
regression results will indicate/confirm which relationships are strong.
–
The
regression equation
is
ˆ
y
=
b
0
+
b
1
x
1
+
. . .
+
b
p
x
p
–
The
residual
for the
i
th
observation is
e
i
=
y
i

ˆ
y
i
= observed response  predicted response
–
The estimate of
σ
, the constant standard deviation of
y
, is
s
=
v
u
u
u
t
n
∑
i
=1
e
2
i
n

p

1
ANOVA Table for Multiple Regression
Sum of Squares
Degrees
of
Freedom
Mean Squares
F
Sig.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Statistics, Normal Distribution, Regression Analysis, Errors and residuals in statistics, explanatory variables

Click to edit the document details