Linear Regression Models (II). Week 8

Август 8, 2022

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Linear Regression Models (II). Week 8

Содержание

2. Lecture outline Assumptions of Linear Regression R Squared and Adjusted R Squared F-test for model significance
3. Normality: Multiple regression assumes that the error terms are normally distributed. Linearity: There must be linear
4. Normality Normality: Multiple regression assumes that the error terms are normally distributed. Plot QQ (Quantile-quantile) plots
5. Linearity Linearity: There must be linear relationship between response variable and independent variables.
6. No Multicollinearity No Multicollinearity: The independent variables are not highly correlated with each other.
7. Homoscedasticity Homoscedasticity: The variance of error terms are similar across the values of the independent variables
8. R-Squared R-squared (R2), also known as a Coefficient of Determination, is a statistical measure that represents
11. Adjusted R-Squared
12. Testing for Significance: F-test The F test is referred to as the test for overall significance.
13. Hypotheses Rejection Rule Test Statistics Testing for Significance: F-test H0: β1 = β2 = . .
14. Example
15. Hypotheses Rejection Rule Test Statistics Testing for Significance: t-test H0: βi = 0 Ha: βi ≠
16. Example
17. Exercise 1 38 random movies were selected to develop a model for predicting their revenues. We
18. Exercise 1
19. THE END Thank you for your attention!
21. Скачать презентацию

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Lecture outline
Assumptions of Linear Regression
R Squared and Adjusted R Squared
F-test for

model significance
t-test for parameter significance

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Normality: Multiple regression assumes that the error terms are normally distributed.
Linearity:

There must be linear relationship between response variable and independent variables (Scatterplots).
No Multicollinearity: the independent variables are not highly correlated with each other (Correlation matrix).
Homoscedasticity: the variance of error terms are similar across the values of the independent variables (Plot of residuals vs predictor variables).

Assumptions of Linear Regression

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Normality
Normality: Multiple regression assumes that the error terms are normally distributed.
Plot

QQ (Quantile-quantile) plots are
used to visually check the
normality of the data.

R Syntax:
plot(model$residuals)

As all the points fall approximately along the
straight line, we can assume normality.

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Linearity
Linearity:
There must be linear
relationship between
response variable and
independent

variables.

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No Multicollinearity
No Multicollinearity:
The independent variables are not highly correlated with

each other.

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Homoscedasticity
Homoscedasticity:
The variance of error terms are similar across the values

of the independent variables (Plot of residuals vs predictor variables).

par(mfrow=c(1,2))
plot(Carseats$Income,model$residuals)
plot(Carseats$Advertising, model$residuals)

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R-Squared
R-squared (R2), also known as a Coefficient of Determination, is a

statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model.

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Adjusted R-Squared

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Testing for Significance: F-test
The F test is referred to as

the test for overall
significance.

The F test is used to determine whether a significant
relationship exists between the dependent variable
and the set of all the independent variables.

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Hypotheses
Rejection Rule
Test Statistics
Testing for Significance: F-test
H0: β1 = β2 =

. . . = βk = 0
Ha: At least one of the parameters (betas) is not equal to zero.

F = MSR/MSE

Reject H0 if p-value < α or if F > Fα
where Fα is based on an F distribution
with k d.f. in the numerator and
n - k - 1 d.f. in the denominator.

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Example

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Hypotheses
Rejection Rule
Test Statistics
Testing for Significance: t-test
H0: βi = 0
Ha:

βi ≠ 0

Reject H0 if p-value < α or
if |t| > tα/2 where tα/2
is based on a t distribution
with n - k - 1 degrees of freedom.

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Example

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Exercise 1
38 random movies were selected to develop a model for

predicting their revenues.
We have the following variables in the dataset:
USRevenue – movie’s revenue in the US (mln$)
Rating – restrictions based on age (PG, PG-13, R)
Budget – budget (expenditure) of the movie (mln$)
Opening – revenue on the opening weekend (mln$)
Theaters – number of theaters the movie was in for the opening weekend
Opinion – IMDb rating (1 to 10, 10 being the best)

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Exercise 1

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Linear Regression Models (II). Week 8

Содержание

Lecture outlineAssumptions of Linear RegressionR Squared and Adjusted R SquaredF-test for

Normality: Multiple regression assumes that the error terms are normally distributed.Linearity:

NormalityNormality: Multiple regression assumes that the error terms are normally distributed.Plot

LinearityLinearity: There must be linear relationship between response variable and independent

No MulticollinearityNo Multicollinearity: The independent variables are not highly correlated with

HomoscedasticityHomoscedasticity: The variance of error terms are similar across the values

R-SquaredR-squared (R2), also known as a Coefficient of Determination, is a

Adjusted R-Squared

Testing for Significance: F-test The F test is referred to as

HypothesesRejection RuleTest StatisticsTesting for Significance: F-test H0: β1 = β2 =

Example

HypothesesRejection RuleTest StatisticsTesting for Significance: t-test H0: βi = 0 Ha:

Example

Exercise 138 random movies were selected to develop a model for

Exercise 1

THE ENDThank you for your attention!

Похожие презентации

Lecture outline
Assumptions of Linear Regression
R Squared and Adjusted R Squared
F-test for

Normality: Multiple regression assumes that the error terms are normally distributed.
Linearity:

Normality
Normality: Multiple regression assumes that the error terms are normally distributed.
Plot

Linearity
Linearity:
There must be linear
relationship between
response variable and
independent

No Multicollinearity
No Multicollinearity:
The independent variables are not highly correlated with

Homoscedasticity
Homoscedasticity:
The variance of error terms are similar across the values

R-Squared
R-squared (R2), also known as a Coefficient of Determination, is a

Testing for Significance: F-test
The F test is referred to as

Hypotheses
Rejection Rule
Test Statistics
Testing for Significance: F-test
H0: β1 = β2 =

Hypotheses
Rejection Rule
Test Statistics
Testing for Significance: t-test
H0: βi = 0
Ha:

Exercise 1
38 random movies were selected to develop a model for

THE END
Thank you for your attention!