multiple regression and other refgression analysis

Regression Analysis
Multiple Regression
[ Cross-Sectional Data ]

Learning Objectives
 Explain the linear multiple regression
model [for cross-sectional data]
 Interpret linear multiple regression
computer output
 Explain multicollinearity
 Describe the types of multiple regression
models

Regression Modeling Steps
 Define problem or question
 Specify model
 Collect data
 Do descriptive data analysis
 Estimate unknown parameters
 Evaluate model
 Use model for prediction

Simple vs. Multiple
 
 represents the
unit change in Y
per unit change in
X .
 Does not take into
account any other
variable besides
single independent
variable.
 
i represents the unit
change in Y per unit
change in Xi.
 Takes into account
the effect of other

i s.
 “Net regression
coefficient.”

Assumptions
 Linearity - the Y variable is linearly related
to the value of the X variable.
 Independence of Error - the error
(residual) is independent for each value of X.
 Homoscedasticity - the variation around
the line of regression be constant for all values
of X.
 Normality - the values of Y be normally
distributed at each value of X.

Goal
Develop a statistical model that
can predict the values of a
dependent (response
response) variable
based upon the values of the
independent (explanatory
explanatory)
variables.

Simple Regression
A statistical model that utilizes
one quantitative
quantitative independent
variable “X” to predict the
quantitative
quantitative dependent
variable “Y.”

Multiple Regression
A statistical model that utilizes two
or more quantitative and
qualitative explanatory variables
(x1,..., xp) to predict a quantitative
quantitative
dependent variable Y.
Caution: have at least two or more quantitative
explanatory variables (rule of thumb)

Multiple Regression Model
X2
X1
Y
e

Hypotheses
 H0: 
1 = 
2 = 
3 = ... = 
P = 0
 H1: At least one regression
coefficient is not equal to
zero

Hypotheses (alternate format)
H0: 
i
i = 0
H1: 
i
i  0

Types of Models
 Positive linear relationship
 Negative linear relationship
 No relationship between X and Y
 Positive curvilinear relationship
 U-shaped curvilinear
 Negative curvilinear relationship

Multiple Regression Models
Multiple
Regression
Models
Linear
Dummy
Variable
Linear
Non-
Linear
Inter-
action
Poly-
Nomial
Square
Root
Log Reciprocal Exponential

Multiple Regression Equations
This is too
complicated! You’ve got to
be kiddin’!

Linear Model
Relationship between one dependent & two
or more independent variables is a linear
function




 




 P
P X
X
X
Y 
2
2
1
1
0
Dependent
Dependent
(response)
(response)
variable
variable
Independent
Independent
(explanatory)
(explanatory)
variables
variables
Population
Population
slopes
slopes
Population
Population
Y-intercept
Y-intercept
Random
Random
error
error

Method of Least Squares
 The straight line that best fits the data.
 Determine the straight line for which the
differences between the actual values (Y)
and the values that would be predicted
from the fitted line of regression (Y-hat)
are as small as possible.

Measures of Variation
 Explained variation (sum of
squares due to regression)
 Unexplained variation (error sum
of squares)
 Total sum of squares

Coefficient of Multiple Determination
When null hypothesis
is rejected, a
relationship between
Y and the X variables
exists.
Strength measured by
R2
[ several types ]

Coefficient of Multiple
Determination
R2
y.123- - -P
The proportion of Y that is
explained by the set of
explanatory variables selected

Standard Error of the Estimate
s
sy.x
y.x
the measure of
variability
around the line
of regression

Confidence interval estimates
»True mean
Y.X
»Individual
Y-hati

Interval Bands [from simple regression]
X
Y
X
Y i
= b 0
+ b 1
X
^
Xgiven
_

Multiple Regression Equation
Y-hat = 
0 + 
1x1 + 
2x2 + ... + 
PxP + 

where:

0 = y-intercept {a constant value}

1
1 = slope of Y with variable x1 holding the
variables x2, x3, ..., xP effects constant

P = slope of Y with variable xP holding all
other variables’ effects constant

Mini-Case
Predict the consumption of home
heating oil during January for
homes located around Screne Lakes.
Two explanatory variables are
selected - - average daily
atmospheric temperature (o
F) and
the amount of attic insulation (“).

Oil (Gal) Temp Insulation
275.30 40 3
363.80 27 3
164.30 40 10
40.80 73 6
94.30 64 6
230.90 34 6
366.70 9 6
300.60 8 10
237.80 23 10
121.40 63 3
31.40 65 10
203.50 41 6
441.10 21 3
323.00 38 3
52.50 58 10
Mini-Case
(0
F)
Develop a model for
estimating heating oil
used for a single family
home in the month of
January based on average
temperature and amount
of insulation in inches.

Mini-Case
 What preliminary conclusions can home
owners draw from the data?
 What could a home owner expect heating
oil consumption (in gallons) to be if the
outside temperature is 15 o
F when the
attic insulation is 10 inches thick?

[mini-case]
Dependent variable: Gallons Consumed
-------------------------------------------------------------------------------------
Standard T
Parameter Estimate Error Statistic P-Value
--------------------------------------------------------------------------------------
CONSTANT 562.151 21.0931 26.6509 0.0000
Insulation -20.0123 2.34251 -8.54313 0.0000
Temperature -5.43658 0.336216 -16.1699 0.0000
--------------------------------------------------------------------------------------
R-squared = 96.561 percent
R-squared (adjusted for d.f.) = 95.9879 percent
Standard Error of Est. = 26.0138
+

[mini-case]
Y-hat = 562.15 - 5.44x
Y-hat = 562.15 - 5.44x1
1 - 20.01x
- 20.01x2
2
where: x
x1
1 = temperature [degrees F]
x
x2
2 = attic
attic insulation [inches]

[mini-case]
Y-hat = 562.15 - 5.44x
Y-hat = 562.15 - 5.44x1
1 - 20.01x
- 20.01x2
2
thus:
thus:
 For a home with zero inches of attic
insulation and an outside temperature of 0 o
F,
562.15 gallons of heating oil would be consumed.
[ caution .. data boundaries .. extrapolation ]
[ caution .. data boundaries .. extrapolation ]
+

Extrapolation
Y
Interpolation
X
Extrapolation Extrapolation
Relevant Range

[mini-case]
Y-hat = 562.15 - 5.44x
Y-hat = 562.15 - 5.44x1
1 - 20.01x
- 20.01x2
2
 For a home with zero attic insulation and an outside temperature of zero,
562.15 gallons of heating oil would be consumed. [ caution .. data boundaries
[ caution .. data boundaries
.. extrapolation ]
.. extrapolation ]
 For each incremental increase in degree F of
temperature, for a given amount of attic insulation,
for a given amount of attic insulation,
heating oil consumption drops 5.44 gallons.
+

[mini-case]
Y-hat = 562.15 - 5.44x
Y-hat = 562.15 - 5.44x1
1 - 20.01x
- 20.01x2
2
 For a home with zero attic insulation and an outside temperature of zero, 562
gallons of heating oil would be consumed. [ caution … ]
[ caution … ]
 For each incremental increase in degree F of temperature, for a given amount
of attic insulation, heating oil consumption drops 5.44 gallons.
 For each incremental increase in inches of
attic insulation, at a given temperature,
at a given temperature,
heating oil consumption drops 20.01
gallons.

Multiple Regression Prediction
[mini-case]
Y-hat = 562.15 - 5.44x
Y-hat = 562.15 - 5.44x1
1 - 20.01x
- 20.01x2
2
with x1 = 15o
F and x2 = 10 inches
Y-hat = 562.15 - 5.44(15) - 20.01(10)
= 280.45 gallons consumed

[mini-case]
R2
y.12 = .9656
96.56 percent of the variation in
heating oil can be explained by
the variation in temperature and
and
insulation.

 Proportion of variation in Y ‘explained’ by all
X variables taken together
 R2
Y.12 = Explained variation = SSR
Total variation SST
 Never decreases when new X variable is
added to model
– Only Y values determine SST
– Disadvantage when comparing models

 Proportion of variation in Y ‘explained’ by all
X variables taken together
 Reflects
– Sample size
– Number of independent variables
 Smaller [more conservative] than R2
Y.12
 Used to compare models
Adjusted

(adjusted)
R2
(adj) y.123- - -P
The proportion of Y that is explained by the
set of independent [explanatory] variables
selected, adjusted for the number of
independent variables and the sample size.

(adjusted) [Mini-Case]
R2
adj = 0.9599
heating oil consumption can be
explained by the model - adjusted
for number of independent variables
and the sample size

Coefficient of Partial Determination
 Proportion of variation in Y ‘explained’ by
variable XP holding all others constant
 Must estimate separate models
 Denoted R2
Y1.2 in two X variables case
– Coefficient of partial determination of X1 with Y
holding X2 constant
 Useful in selecting X variables

Coefficient of Partial
Determination [p. 878]
R2
y1.234 --- P
The coefficient of partial variation of
variable Y with x1 holding constant
the effects of variables x2, x3, x4, ... xP.

[Mini-Case]
R2
y1.2 = 0.9561
For a fixed (constant) amount of
insulation, 95.61 percent of the variation
in heating oil can be explained by the
variation in average atmospheric
temperature. [p. 879]

[Mini-Case]
R2
y2.1 = 0.8588
For a fixed (constant) temperature,
heating oil can be explained by the
variation in amount of insulation.

Testing Overall Significance
 Shows if there is a linear relationship between
all X variables together & Y
 Uses p-value
 Hypotheses
– H0: 1 = 2 = ... = P = 0
»No linear relationship
– H1: At least one coefficient is not 0
»At least one X variable affects Y

 Examines the contribution of a set of X
variables to the relationship with Y
 Null hypothesis:
– Variables in set do not improve significantly
the model when all other variables are included
 Must estimate separate models
 Used in selecting X variables
Testing Model Portions

Diagnostic Checking
 H0 retain or reject
If reject - {p-value  0.05}
 R2
adj
 Correlation matrix
 Partial correlation matrix

Multicollinearity
 High correlation between X variables
 Coefficients measure combined effect
 Leads to unstable coefficients depending on X
variables in model
 Always exists; matter of degree
 Example: Using both total number of rooms
and number of bedrooms as explanatory
variables in same model

Detecting Multicollinearity
 Examine correlation matrix
– Correlations between pairs of X variables are
more than with Y variable
 Few remedies
– Obtain new sample data
– Eliminate one correlated X variable

Evaluating Multiple Regression Model Steps
 Examine variation measures
 Do residual analysis
 Test parameter significance
– Overall model
– Portions of model
– Individual coefficients
 Test for multicollinearity

Dummy-Variable Regression Model
 Involves categorical X variable with
two levels
– e.g., female-male, employed-not employed, etc.

two levels
 Variable levels coded 0 & 1

two levels
 Variable levels coded 0 & 1
 Assumes only intercept is different
– Slopes are constant across categories

Dummy-Variable Model Relationships
Y
Y
X
X1
1
0
0
0
0
Same slopes b1
b
b0
0
b
b0
0 + b
+ b2
2
Females
Males

Dummy Variables
 Permits use of
qualitative data
(e.g.: seasonal, class
standing, location,
gender).
 0, 1 coding
(nominative data)
 As part of Diagnostic
Checking;
incorporate outliers
(i.e.: large residuals)
and influence
measures.

Interaction Regression Model
 Hypothesizes interaction between pairs of X
variables
– Response to one X variable varies at different
levels of another X variable
 Contains two-way cross product terms
Y = 0 + 1x1 + 2x2 + 3x1x2 + 
 Can be combined with other models
e.g. dummy variable models

Effect of Interaction
 Given:
 Without interaction term, effect of X1 on Y is
measured by 1
 With interaction term, effect of X1 on
Y is measured by 1 + 3X2
– Effect increases as X2i increases
Y X X X X
i i i i i i
    
    
0 1 1 2 2 3 1 2

Interaction Example
X
X1
1
4
4
8
8
12
12
0
0
0
0 1
1
0.5
0.5 1.5
1.5
Y
Y Y
Y = 1 + 2
= 1 + 2X
X1
1 + 3
+ 3X
X2
2 + 4
+ 4X
X1
1X
X2
2

Interaction Example
X
X1
1
4
4
8
8
12
12
0
0
0
0 1
1
0.5
0.5 1.5
1.5
Y
Y Y
Y = 1 + 2
= 1 + 2X
X1
1 + 3
+ 3X
X2
2 + 4
+ 4X
X1
1X
X2
2
Y
Y = 1 + 2
= 1 + 2X
X1
1 + 3(
+ 3(0
0) + 4
) + 4X
X1
1(
(0
0) = 1 + 2
) = 1 + 2X
X1
1

Interaction Example
Y
Y
X
X1
1
4
4
8
8
12
12
0
0
0
0 1
1
0.5
0.5 1.5
1.5
Y
Y = 1 + 2
= 1 + 2X
X1
1 + 3
+ 3X
X2
2 + 4
+ 4X
X1
1X
X2
2
Y
Y = 1 + 2
= 1 + 2X
X1
1 + 3(
+ 3(1
1) + 4
) + 4X
X1
1(
(1
1) = 4 + 6
) = 4 + 6X
X1
1
Y
Y = 1 + 2
= 1 + 2X
X1
1 + 3(
+ 3(0
0) + 4
) + 4X
X1
1(
(0
0) = 1 + 2
) = 1 + 2X
X1
1

Interaction Example
Effect (slope) of
Effect (slope) of X
X1
1 on
on Y
Y does depend on
does depend on X
X2
2 value
value
X
X1
1
4
4
8
8
12
12
0
0
0
0 1
1
0.5
0.5 1.5
1.5
Y
Y Y
Y = 1 + 2
= 1 + 2X
X1
1 + 3
+ 3X
X2
2 + 4
+ 4X
X1
1X
X2
2
Y
Y = 1 + 2
= 1 + 2X
X1
1 + 3(
+ 3(1
1) + 4
) + 4X
X1
1(
(1
1) = 4 +
) = 4 + 6
6X
X1
1
Y
Y = 1 + 2
= 1 + 2X
X1
1 + 3(
+ 3(0
0) + 4
) + 4X
X1
1(
(0
0) = 1 +
) = 1 + 2
2X
X1
1

Inherently Linear Models
 Non-linear models that can be expressed in
linear form
– Can be estimated by least square in linear form
 Require data transformation

Y
X1
Curvilinear Model Relationships
Y
X1
Y
X1
Y
X1

Logarithmic Transformation
Y
X1

1
1 > 0
> 0

1
1 < 0
< 0
Y =  + 1 lnx1 + 2 lnx2 + 

Square-Root Transformation
Y
X1
Y X X
i i i i
   
   
0 1 1 2 2

1
1 > 0
> 0

1
1 < 0
< 0

Reciprocal Transformation
Y
X1

1
1 > 0
> 0

1
1 < 0
< 0
i
i
i
i
X
X
Y 


 



2
2
1
1
0
1
1
Asymptote
Asymptote

Exponential Transformation
Y
X1

1
1 > 0
> 0

1
1 < 0
< 0
Y e
i
X X
i
i i
  
  

0 1 1 2 2

Overview
 Explained the linear multiple regression
model
 Interpreted linear multiple regression
computer output
 Explained multicollinearity
 Described the types of multiple regression
models

Source of Elaborate Slides
Prentice Hall, Inc
Levine, et. all, First Edition

Regression Analysis
[Multiple Regression]
*** End of Presentation ***
Questions?

multiple regression and other refgression analysis

multiple regression and other refgression analysis

More Related Content

Similar to multiple regression and other refgression analysis (20)

More from ssuserd23711 (10)

Recently uploaded (20)

multiple regression and other refgression analysis

Editor's Notes