Multiple regression analysis MRA

1
Multiple regression analysis (MRA)

• We have seen how simple regression analysis can
  be used to model relationships why then would
  we need to use multiple regression analysis
  (MRA)?
• Because complex relationships may involve more
  than one independent variable!

2
MR equations

• If we are modeling Y as a function of two
  independent variables X1 and X2, our MR equation
  is
• YB0B1X1B2X2
• If Y is a function of 4 independent variables (X1
  X4), our equation
• YB0B1X1B2X2 B3X3 B4X4

3
MRA Problem

• Can we predict the mileage of an automobile
  (measured in MPG) based on its engine horsepower
  and weight?
• How would horsepower affect mileage (would it be
  a positive or negative relationship)?
• How would weight affect mileage (would it be a
  positive or negative relationship)?

4
Data
To do the analysis please use the file Auto.xls
that comes with the text book CD (or email me and
I can send you the data file).
5
EXCEL output
Regression Equation MPG 58.1571
-0.1175Horsepower -0.0069Weight
6
Interpreting MR coefficients Horsepower

• The coefficient for Horsepower is -0.1175 thus,
  holding constant Weight, MPG decrease by 0.1175
  for every 1 unit increase in Horsepower
• Stated another way, according to the regression
  equation, if 2 automobiles have the same weight,
  the auto with a higher horsepower will have lower
  MPG

7
Interpreting MR coefficients Weight

• The coefficient for Weight is -0.0069 thus,
  holding constant Horsepower, MPG decreases by
  0.0069 for every additional lb of Weight
• In other words, our regression model says that
  for 2 automobiles with the same horsepower, the
  automobile that weighs more will have lower MPG

8
Prediction using MR equation

• Can we predict MPG for an automobile with a
  horsepower of 100 and weighs 2000 lbs?
• Yes
• Because the value for horsepower (100) is within
  the range of horsepower values used in developing
  the MR equation
• And because the value for weight (2,000) is also
  within the range of weight values used in
  developing the MR equation

9
Prediction using MR equation
MPG58.1571-0. 1175Horsepower-0.0069Weight
MPG58.1571 - (0. 1175100)
(0.00692000) MPG32.66
Predicted MPG for an automobile with an engine
horsepower of 100 and weighs 2000 lbs
10
Adjusted coefficient of determination (Adj.-r2)

• It is meaningful to use adjusted R-squared (as
  opposed to R-squared) for the MR equation since
  this measure accounts for the number of
  independent variables and observations

11
Adjusted R-square from EXCEL
Adj. R-square0.74 tells us that about 74 of the
variation in MPG is explained by Horsepower and
Weight
12
Is the MR model statistically valid?

• To assess validity of MR model, we need to use
  ANOVA (available in the EXCEL output).
• The hypothesis we are testing is
• H0 Slope (HP)Slope (Weight)0
• H1 Not H0

13
Using ANOVA from EXCEL to test if MR model is
valid
F70.28 and P-value7.50524E-15. This P-value is
(much) smaller than 0.01. Our rule is if
P-value is less than a reject null. Thus, at
a0.01 level, we reject null. Our regression
model is statistically valid.
14
Assessing the contribution of Horsepower to the
MR model
t-Stat-3.6003 and P-value0.000763. Rule-- if
P-value is smaller than a then reject null that
Slope of horsepower0. Because p-value is very
small, we conclude that horsepower is a
significant predictor of MPG in our model.
15
Assessing the contribution of Weight to the MR
model
t-Stat-4.9035 and P-value1.16E-05. Rule-- if
P-value is smaller than a then reject null that
Slope of weight0. Because p-value is very
small, we conclude that weight is a significant
predictor of MPG in our model.
16
Is the analysis over?

• As an academic example of multiple regression,
  yes
• As a modeling exercise perhaps not
• The analyst can consider adding more variables if
  they are available, such as
• Transmission type
• Age of the automobile, etc.