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STAC67H3: Regression Analysis
Fall, 2014
Instructor: Jabed Tomal
Department of Computer and Mathematical Sciences
University of Toronto Scarborough
Toronto, ON
Canada
November 11, 2014
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
1 / 52
Models for Quantitative and Qualitative Predictors
Polynomial Regression Models
Uses: Polynomial regression models have two basic types of uses:
1
When the true curvilinear response function is indeed a
polynomial function.
2
When the true curvilinear response function is unknown (or
complex) but a polynomial function is a good approximation to the
true function. [More Common]
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Polynomial Regression Models
Danger of polynomial regression models:
1
Polynomial regression models may provide good fits for the data
at hand, but may turn in unexpected directions when extrapolated
beyond the range of the data.
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Polynomial Regression Models
One Predictor Variable - Second Order:
1
Polynomial regression models may contain one, two, or more than
two predictor variables, and each predictor variable may be
present in various powers.
2
A polynomial regression model with one predictor variable raised
to the first and second powers:
Yi = β0 + β1 xi + β2 xi2 + i
¯.
where, xi = Xi − X
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Polynomial Regression Models
One Predictor Variable - Second Order:
1
This polynomial regression model is called a second-order model
with one predictor variable because the single predictor variable is
expressed in the model to the first and second powers.
2
The predictor variable is centered – expressed as a deviation
¯ – and that the ith centered observation is
around its mean X
denoted by xi .
3
The reason for centering is to reduce high correlation between X
and X 2 .
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Polynomial Regression Models
One Predictor Variable - Second Order:
1
The regression coefficients in polynomial regression are
frequently written in a slightly different fashion, to reflect the
pattern of the exponents:
Yi = β0 + β1 xi + β11 xi2 + i
2
The response function for the regression model is:
E{Y } = β0 + β1 x + β11 x 2
This response function is a parabola and is frequently called
quadratic response function.
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Polynomial Regression Models
One Predictor Variable - Second Order:
1
The regression coefficient β0 represents the mean response of Y
¯.
when x = 0, i.e., when X = X
2
The regression coefficient β1 is often called the linear effect
coefficient, and β11 is called the quadratic effect coefficient.
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Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Polynomial Regression Models
One Predictor Variable - Third Order:
1
The regression model:
Yi = β0 + β1 xi + β11 xi2 + β111 xi3 + i
¯ is a third-order model with one predictor
where xi = Xi − X
variable.
2
The response function for the regression model is:
E{Y } = β0 + β1 x + β11 x 2 + β111 x 3 .
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Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Polynomial Regression Models
One Predictor Variable - Higher Order:
1
Polynomial models with the predictor variable present in higher
powers than the third should be employed with special caution.
2
The interpretation of the coefficients becomes difficult for such
models, and the models may be highly erratic for interpolations
and even small extrapolations.
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Polynomial Regression Models
Two Predictor Variables - Second Order:
1
The regression model:
2
2
Yi = β0 + β1 xi1 + β2 xi2 + β11 xi1
+ β22 xi2
+ β12 xi1 xi2 + i
¯1 , xi2 = Xi2 − X
¯2 is a second-order model with
where xi1 = Xi1 − X
two predictor variables.
2
The response function is:
E{Y } = β0 + β1 x1 + β2 x2 + β11 x12 + β22 x22 + β12 x1 x2
which contains separate linear and quadratic components for
each of the two predictor variables and a cross-product term.
3
The latter term represents the interaction effect between x1 and
x2 , and β12 is often called the interaction effect coefficient.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Polynomial Regression Models
Three Predictor Variables - Second Order:
1
The regression model:
2 + β x2 + β x2
Yi = β0 + β1 xi1 + β2 xi2 + β3 xi3 + β11 xi1
22 i2
33 i3
+β12 xi1 xi2 + β13 xi1 xi3 + β23 xi2 xi3 + i
¯1 , xi2 = Xi2 − X
¯2 and xi3 = Xi3 − X
¯3 is a
where xi1 = Xi1 − X
second-order model with three predictor variables.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Polynomial Regression Models
Three Predictor Variables - Second Order:
1
The response function is:
E{Y } = β0 + β1 x1 + β2 x2 + β3 x3 + β11 x12 + β22 x22 + β33 x32
+β12 x1 x2 + β13 x1 x3 + β23 x2 x3
2
The coefficients β12 , β13 , and β23 are interaction effect coefficients
for interactions between pairs of predictor variables.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
Fitting of Polynomial Models:
1
Fitting of polynomial regression models presents no new problems
as they are special cases of the general linear regression model.
2
Hence, all earlier results on fitting apply, as do the earlier results
on making inferences.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
Hierarchical Approach to Fitting:
1
First fit a second-order or third-order model and then explore
whether a lower-order model is adequate.
2
For instance, with one predictor variable, the model:
Yi = β0 + β1 xi + β11 xi2 + β111 xi3 + i
may be fitted with the hope that the cubic term and perhaps even
the quadratic terms can be dropped.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
Hierarchical Approach to Fitting:
1
One would wish to test whether or not β111 = 0 or whether or not
both β11 = 0 and β111 = 0.
2
To test whether or not β111 = 0, the appropriate extra sum of
squares is
SSR(x 3 |x, x 2 ).
3
To test whether or not β11 = β111 = 0, the appropriate extra sum
of squares is
SSR(x 2 , x 3 |x) = SSR(x 2 |x) + SSR(x 3 |x, x 2 ).
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Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
Hierarchical Approach to Fitting:
1
With the hierarchical approach, if a polynomial term of a given
order is retained, the all related terms of lower order are also
retained in the model.
2
Thus, one would not drop the quadratic term of a predictor
variable but retain the cubic term in the model.
3
Similarly, an interaction term (second power) would not be
retained without retaining the terms for the predictor variables to
the first power.
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Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
Hierarchical Approach to Fitting:
1
With the hierarchical approach, if a polynomial term of a given
order is retained, the all related terms of lower order are also
retained in the model.
2
Thus, one would not drop the quadratic term of a predictor
variable but retain the cubic term in the model.
3
Similarly, an interaction term (second power) would not be
retained without retaining the terms for the predictor variables to
the first power.
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Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
Regression Function in Terms of X :
1
After a polynomial regression model has been developed, we
often wish to express the final model in terms of the original
variables rather than keeping it in terms of the centered variables.
2
The fitted second-order model for one predictor variable that is
¯:
expressed in terms of centered values x = X − X
ˆ = b0 + b1 x + b11 x 2
Y
becomes in terms of the original X variable:
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
Regression Function in Terms of X :
1
ˆ = b0 + b0 X + b0 X 2
Y
0
1
11
where
¯ + b11 X
¯2
b00 = b0 − b1 X
¯
b10 = b1 − 2b11 X
0
= b11
b11
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
Exercise 6.5. Brand Preference. In a small-scale experimental study
of the relation between degree of brand liking (Y ) and moisture
content (X1 ) and sweetness (X2 ) of the product, the following results
were obtained from the experiment based on a completely randomized
design (data are coded):
i:
Xi1 :
Xi2 :
Yi :
Jabed Tomal (U of T)
1
4
2
64
2
4
4
73
3
4
2
61
···
···
···
···
Regression Analysis
14
10
4
95
15
10
2
94
16
10
4
100
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
1
We want to fit the following polynomial regression model
2
Yi = β0 + β1 xi1 + β2 xi2 + β11 xi1
+ β12 xi1 xi2 + i
¯1 and xi2 = Xi2 − X
¯2 .
where xi1 = Xi1 − X
2
For which the response function is
E{Y } = β0 + β1 x1 + β2 x2 + β11 x12 + β12 x1 x2
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
1
The fitted following polynomial regression model is
ˆ = 82.219 + 4.425x1 + 4.375x2 − 0.0938x 2 − 0.500xi1 xi2
Y
1
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
Coefficient of Multiple Determination:
1
The coefficient of determination
R 2 = 0.9634
The fitted model can explain approximately, 96.35% variability of
the response variable.
2
The adjusted R 2 is
Ra2 = 0.9501
The coefficient is smaller than the unadjusted coefficient because
of the relatively large number of parameters in the polynomial
regression function with two predictor variables.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
Partial F Test:
1
We now turn to consider whether a first-order model would be
sufficient. The hypotheses are:
H0 : β11 = β12 = 0
2
versus
HA : not all β in H0 equalzero
The partial F test statistic is
F∗ =
SSR(x1 x2 , x12 |x1 , x2 )/2
(2.25 + 20.00)/2
=
= 1.698473
MSE
6.55
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
Partial F Test:
1
For level of significance α = 0.05, we require
F (0.95; 2, 11) = 3.982298
2
3
Since F ∗ = 1.698473 < 3.982298, we conclude H0 , that no
curvature on x1 and interaction effects between x1 and x2 are
needed.
A first-order regression model might be appropriate.
Jabed Tomal (U of T)
Regression Analysis
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Models for Quantitative and Qualitative Predictors
Implementation of Polynomial Regression Models
First-Order Model:
1
One the basis of our analysis, we decided to consider the
first-order model as following:
Yi = β0 + β1 xi1 + β2 xi2 + i
2
The fitted model is
ˆ = 81.750 + 4.425x1 + 4.375x2
Y
3
Do check the residuals, and make inferences of the regression
parameters if the model is appropriate.
Jabed Tomal (U of T)
Regression Analysis
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Models for Quantitative and Qualitative Predictors
Interaction Regression Models
1
We have previously noted that regression models with
cross-product interaction effects are special cases of general
linear regression model.
2
The following regression function
E{Y } = β0 + β1 X1 + β2 X2 + β3 X1 X2
contains a cross-production term between X1 and X2 , such as
β3 X1 X2 .
3
The cross-product term is called an interaction term.
Jabed Tomal (U of T)
Regression Analysis
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Models for Quantitative and Qualitative Predictors
Interaction Regression Models
1
When there are three predictor variables whose effects on the
response variable are linear, but the effects on Y of X1 and X2 and
of X1 and X3 are interacting, the response function would be
modeled as follows using cross-product terms:
E{Y } = β0 + β1 X1 + β2 X2 + β3 X3 + β4 X1 X2 + β5 X1 X3
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Interaction Regression Models
Interpretation of Regression Coefficients:
1
The regression model for two quantitative predictor variables with
linear effects on Y and interacting effects of X1 and X2 on Y
represented by a cross-product term is as follows:
E{Y } = β0 + β1 X1 + β2 X2 + β3 X1 X2
2
The meaning of the regression coefficients β1 and β2 here is not
the same as that given earlier because of the interaction term
β3 X1 X2 .
3
The regression coefficients β1 and β2 no longer indicate the
change in the mean response with a unit increase of the predictor
variable, with the other predictor variable held constant at any
given level.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Interaction Regression Models
Interpretation of Regression Coefficients:
1
The change in the mean response with a unit increase in X1 when
X2 is held constant is:
β1 + β3 X2
2
Similarly, the change in the mean response with a unit increase in
X2 when X1 is held constant is:
β2 + β3 X1
3
Hence, both the effect of X1 for given level of X2 and the effect of
X2 for given level of X1 depend on the level of the other predictor
variable.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Interaction Regression Models
Implementation of Interaction Regression Models:
1
Two considerations need to be kept in mind when developing
regression models with interaction effects.
2
When interaction terms are added to a regression model, high
multicollinearities may exist between some of the predictor
variables and some of the interaction tersm, as well as among
some of the interaction terms.
A partial remedy to improve computational accuracy is to center the
predictor variables, i.e., to use
¯k .
xik = Xik − X
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Interaction Regression Models
Implementation of Interaction Regression Models:
1
When the number of predictor variables in the regression model is
large, the potential number of interaction terms can become very
large. For example, if eight predictor variables are present in the
regression model in linear terms, there are potentially 28 pairwise
interaction terms that could be added to the regression model.
It is therefore desirable to identify in advance, whenever possible,
those interactions that are most likely to influence the response
variable in important ways.
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Interaction Regression Models
Exercise 6.5. Brand Preference. In a small-scale experimental study
of the relation between degree of brand liking (Y ) and moisture
content (X1 ) and sweetness (X2 ) of the product, the following results
were obtained from the experiment based on a completely randomized
design (data are coded):
i:
Xi1 :
Xi2 :
Yi :
Jabed Tomal (U of T)
1
4
2
64
2
4
4
73
3
4
2
61
···
···
···
···
Regression Analysis
14
10
4
95
15
10
2
94
16
10
4
100
November 11, 2014
33 / 52
Models for Quantitative and Qualitative Predictors
Interaction Regression Models
Example: Brand Preference. We want to fit the following regression
model with interaction term
Yi = β0 + β1 xi1 + β2 xi2 + β12 xi1 xi2 + i
¯1 and xi2 = Xi2 − X
¯2 .
where xi1 = Xi1 − X
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Interaction Regression Models
Example: Brand Preference. The fitted regression model with
interaction term is
ˆ = 81.750 + 4.425x1 + 4.375x2 − 0.500x1 x2
Y
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Interaction Regression Models
Partial F Test:
1
We now turn to test whether the interaction effect is significant or
not. The hypotheses are:
H0 : β12 = 0
2
versus
HA : β12 6= 0
The partial F test statistic is
F∗ =
Jabed Tomal (U of T)
SSR(x1 x2 |x1 , x2 )/1
20.00
=
= 3.231018
MSE
6.19
Regression Analysis
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Models for Quantitative and Qualitative Predictors
Interaction Regression Models
Partial F Test:
1
For level of significance α = 0.05, we require
F (0.95; 1, 12) = 4.747225
2
Since F ∗ = 3.231018 < 4.747225, we conclude H0 , that is no
interaction effect between x1 and x2 is needed.
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Interaction Regression Models
First-Order Model:
1
One the basis of our analysis, we decided to consider the
first-order model as following:
Yi = β0 + β1 xi1 + β2 xi2 + i
2
The fitted model is
ˆ = 81.750 + 4.425x1 + 4.375x2
Y
3
Do check the residuals, and make inferences of the regression
parameters if the model is appropriate.
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
1
Many variables of interest in business, economics, and the social
and biological sciences are qualitative.
2
Examples of qualitative predictor variables are gender (male,
female), purchase status (purchase, no purchase), and disability
status (not disabled, partly disabled, fully disabled).
3
In order that such a qualitative variable can be used in a
regression model, quantitative indicators for the classes of the
qualitative variable must be employed.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
1
In a study of innovation in the insurance industry, an economist
wished to relate:
2
The speed with which a particular innovation is adopted (Y ):
number of months elapsed between the time the first firm adopted
the innovation and the time the given firm adopted the innovation.
3
The size of the insurance firm (X1 ): measured by the amount of
total assets of the firm.
4
Type of firm (X2 ): composed of two classes - stock companies and
mutual companies.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
1
The qualitative variable (X2 ) is an indicator variable defined as
1 if stock company
X2 =
0 if mutual company
2
The model with the qualitative predictor variable is
Yi = β0 + β1 Xi1 + β2 Xi2 + i
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
1
A qualitative variable with c classes will be represented by c − 1
indicator variables, each taking on the values 0 and 1.
2
The qualitative predictor variable disability status can be
represented by two indicator variables:
1 fully disabled
Z1 =
0 Otherwise
1 partly disabled
Z2 =
0 Otherwise
Here, not disabled is the reference category.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
Interpretation of Regression Coefficients:
1
For the insurance innovation example, the model is
Yi = β0 + β1 Xi1 + β2 Xi2 + i
where:
Xi1 = size of firm
1 if stock company
X2 =
0 if mutual company
2
The response function of this regression model is
E(Y ) = β0 + β1 X1 + β2 X2
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
Interpretation of Regression Coefficients:
1
For mutual firm, X2 = 0 and the response function becomes:
E(Y ) = β0 + β1 X1
which is a straight line with intercept β0 and slope β1 .
2
For stock firm, X2 = 1 and the response function becomes:
E(Y ) = (β0 + β1 ) + β1 X1
is also a straight line with the same slope β1 and intercept β0 + β2 .
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
Interpretation of Regression Coefficients:
1
The mean time alapsed before the innovation is adopted, E{Y }, is
a linear function of size of firm (X1 ), with the same slope β1 for
both types of firms.
2
β2 indicates how much higher (lower) the response function for
stock firms is than the one for mutual firms, for any given size of
firm.
3
In general, β2 shows how much higher (lower) the mean response
line is for the class coded 1 than the line for the class coded 0, for
any given level of X1 .
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
Example:
1
In the insurance innovation example, the economist studied 10
mutual firms and 10 stock firms.
2
The fitted regression model is:
ˆ = 33.87407 − 0.10174X1 + 8.05547X2 .
Y
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
Example:
1
The economist was most interested in the effect of type of firm
(X2 ) on the elapsed time for the innovation to be adopted and
wished to obtain a 95% confidence interval for β2 .
2
The estimates of β2 and σ{b2 } are b2 = 8.055469 and
s{b2 } = 1.459106, respectively. We require t(0.975; 17) = 2.110
and obtain the 95% confidence interval of β2 as following
b2 ± t(0.975; 17) × s{b2 } = (4.98, 11.13).
Jabed Tomal (U of T)
Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
Example:
1
With 95% confidence interval, we conclude that stock companies
tend to adopt the innovation somewhere between 5 and 11
months later, on the average, than mutual companies, for any
given size of firm.
2
A formal test of
H0 : β2 = 0
;
HA : β2 6= 0
with level of significance 0.05 would lead to HA , that type of firm
has an effect, since the 95% confidence interval for β2 does not
include zero.
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
Qualitative Predictor with More than Two Classes:
1
Consider the regression of tool wear (Y ) on tool speed (X1 ) and
tool model, where the latter is a qualitative variable with four
classes (M1, M2, M3, M4). We therefore require three indicator
variables:
1 if tool model M1
X2 =
0 Otherwise
1 if tool model M2
X3 =
0 Otherwise
1 if tool model M3
X4 =
0 Otherwise
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
Qualitative Predictor with More than Two Classes:
1
A first-order regression model is
Yi = β0 + β1 Xi1 + β2 Xi2 + β3 Xi3 + β4 Xi4 + i
2
The response function for the regression model is
E{Y } = β0 + β1 X1 + β2 X2 + β3 X3 + β4 X4
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
Qualitative Predictor with More than Two Classes:
1
The response function for tool models M4 for which X2 = 0,
X3 = 0 and X4 = 0:
E{Y } = β0 + β1 X1
2
For tool models M1 for which X2 = 1, X3 = 0 and X4 = 0:
E{Y } = (β0 + β2 ) + β1 X1
3
For tool models M2 for which X2 = 0, X3 = 1 and X4 = 0:
E{Y } = (β0 + β3 ) + β1 X1
4
For tool models M3 for which X2 = 0, X3 = 0 and X4 = 1:
E{Y } = (β0 + β4 ) + β1 X1
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Regression Analysis
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Models for Quantitative and Qualitative Predictors
Qualitative Predictors
Qualitative Predictor with More than Two Classes:
1
Thus, response function implies that the regression model of tool
wear on tool speed is linear, with the same slope for all four tool
models.
2
The coefficients β2 , β3 and β4 indicate, respectively, how much
higher (lower) the response functions for tool models M1, M2, and
M3 are than the one for tool models M4, for any given level of tool
speed.
Jabed Tomal (U of T)
Regression Analysis
November 11, 2014
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