We present an overview of regression analysis, theoretical construct, then provide a graphic representation before performing multiple regression analysis step by step using SPSS (audio files accompany the tutorial).

1. Regression Analysis
CMGT 587A
UNIVERSITY OF SOUTHERN CALIFORNIA
AL ARIZMENDEZ/CATHRYN LOTTIER

2. What is Regression Analysis?
 The Regression Method, more commonly referred to
as Regression Analysis, is the assessment of the
relationship of a dependent variable and one or more
multiple independent variable(s).
 It involves techniques for measuring or analyzing
multiple variables and their relationship
 This technique is used to analyze variables with at
least one dependent variable (often y) and one or
multiple independent variables (often x) to
understand a phenomena, make predictions, and/or
test hypotheses

3. Assumptions Underlying the Method
 The validity of regression analysis depends on four
assumptions:
 Linearity: where the relationship between dependent and
independent variables are directly proportional to each other
 Independence: an independence of errors with no serial
correlation (a random value of Y is assumed to be independent
of any other value of Y)
 Constant variance: having your data values be scattered to
the same extent
 Normality: the random variable of interest is distributed is a
normal manner

4. When can you use Regression Analysis?
 Regression Analysis is used to make predictions, so it can
virtually be used by anyone
 Some reasons that you may want to use regression
analysis are:
 To model a phenomena to understand it better in order to make
decisions
 To model a phenomena to understand it better to predict values for
that in other places or times (later in these slides, you will see an
example of this as we created an example to forecast album sales)
 To test a hypotheses, but one should note that regression analysis is
an estimate or guess, not an accurate data set (we will show an
example of this later in the slides with our test of life expectancy vs.
literary rates)

5. Diving a Little Deeper…
 Multiple linear regression analysis begins by positing the
general form of the relationship in the following model:
 ϒi = β0 + β1Χi1 + εi
 More simply put: Outcomei = (b0 + b1xi) + errori
 Where Y is the dependent variable, β0 is the intercept,
β1 is the slope and Χi1 is the independent variable
 The ε is the residual term, which expresses the
composite of all the other types of individual differences
that aren’t explicitly identified in the model (a.k.a.
random error term)…a reminder that it will never be
perfect

6. What does that really mean?
 That equation means that the “outcome” can be predicted
from a model and some error associated with that
prediction (εi)
 The outcome variable is represented as yi, which is
predicted using a predictor variable (xi) and a parameter
(bi) associated with the predictor variable
 Bi is the line the direction or strength of the relationship or effect
 B0 tells us what the value of the outcome is when the predictor is 0
(the intercept)
 The betas tell us what the shape of the model is and what it
looks like

7. Explanation of R Squared
 R2 allows one to assess how well the model fits
 If you square all of the differences, the sum of all the squared
differences is known as the total sum of squares (SST )
 If an optimal model is fitted to the data, the differences
between the observed data points and the values predicted by
the regression line can be squared and summed, which is
referred to as the sum of squared residuals (SSR)
 The difference between SST and SSR is the model sum of
squares (SSM)
 R2 is determined by dividing the model sum of squares by the
total sum of squares, which is used to describe how well the
regression line fits
 An R2 near 1 indicates that a regression line fits the data well,
while an R2 closer to 0 indicates a regression line does not fit
the data very well

8. Example of Regression Analysis
 Regression Analysis can be used to forecast the trend of
album sales (shown on the y-axis) in relation to the
advertising budget (shown on the x-axis)

9. Adding Another Variable to the Equation
 Now, taking it one step further
and adding amount of radio
play to the equation
 This turns into multiple
regression analysis with
more predictors creating a
regression plane (or a 3d
model) with the line turning
into a plane
 It looks more complicated, but
the principles remain the
same as linear regression

10. Explanation of Multiple Regression Analysis
 Multiple Regression Analysis
 Often referred to as OLS (Ordinary Least Squares) regression
 “multiple regression can establish whether a set of
independent variables explains a proportion of the variance in
a dependent variable at a significant level (through a
significance test of R2)” (Garson, 2012, p. 10)
 It can also determine the relative predictive importance of the
independent variable (by comparing regression weights, also
known as beta weights)

11. Multiple Regression Analysis
 While the formula for linear regression analysis
looks like this:
ϒi = β0 + β1Χ1i + εi
 Multiple regression analysis looks more like this:
ϒi = (β0 + β1Χ1i+ β2Χ2i…+ βnΧni) + εi
 This shows that the principles are the same as
linear regression, there are just more predictors!

12. Talking About the Betas
 The betas tell the relationship
between a particular predictor
and the outcome
 The betas also define the shape
of the plane
 In this instance:
 the beta 0 is represent where the
plane hits the y-axis (value of the
outcome when both predictors are
zero)
 b1 represents the slope of the side
associated with radio play
 b2 represents the slope of the side
associated with advertising budget
 This can go on for multiple
dimensions with each of the
predictors defining the shape

13. Simple Linear Regression w/ SPSS
 Life Expectancy of Females (dependent variable)
 Literacy of country in percent (independent variable)

14. Simple Regression w/ SPSS: Open the Data Set

15. Simple Regression w/ SPSS: Scatter
 It’s always a good idea to do a scatter plot
Graphs>Legacy dialogs> Scatter/Dot>Define

16. Simple Regression w/SPSS: Scatter
 Add dependent variable (Life expectancy) on y-axis
 Add independent variable (Literacy) on x-axis

17. Simple Regression : Scatter done, we’re not
 Strong uphill pattern, expectancy increases with literacy
rate, but we need to run a regression line

18. Simple Regression w/SPSS: Scatter and Regression Line

19. Simple Regression: Plotted, now run it
Analyze> Regression> Linear

20. Simple Regression w/SPSS: Output

21. Simple Regression w/SPSS: Scatter
 Coefficients

22. Multiple Linear Regression w/SPSS
Analyze>Regression> Linear

23. Multiple Linear Regression w/SPSS
Top half of output; notice the multiple variables entered
and the single dependent variable (female life expectancy)

24. Multiple Linear Regression w/SPSS
Bottom half of Output:

25. Multiple Linear Regression w/SPSS
 Literacy is one variable, but it is that specific combination of the
variables that Multiple Linear Regression tests for makes MLR so
powerful