Linear Regression Models. Week 7

Август 24, 2022

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Linear Regression Models. Week 7

Содержание

2. Learning Outcomes Distinguish between deterministic and probabilistic relations. Understand the concepts of correlation and regression. Be
3. Deterministic (Functional) Relationship A functional relation between two variables is expressed by a mathematical formula. If
4. Probabilistic (Statistical) Relationship A probabilistic relationship, unlike a deterministic one, is not a perfect one. In
5. Correlation Coefficient
6. Correlation is not causation
7. Historical Origins of Regression Regression analysis was first developed by Sir Francis Galton in the latter
8. Simple Linear Regression Simple linear regression is a statistical method that allows us to summarize and
9. Simple Linear Regression Basic simple linear model: Yi = β0 + β1Xi + εI Yi is
10. Method of Least Squares
11. Computing the least-squares regression line.
12. Example
13. Results from R R Syntax: lm.model summary(lm.model)
14. Interpretation of b0 and b1
15. Multiple Linear Regression Multiple linear regression is an extension of simple linear regression used to predict
16. Example
17. Interpretations For a given predictor variable, the coefficient (b) can be interpreted as the average effect
18. Qualitative (Categorical) Variables Categorical independent variables can be incorporated into a regression model by converting them
19. Example
20. Output and Interpretations If the quality of shelving is Good, ceteris paribus*, average amount of sales
21. Fitted Models
22. Literature Lind et al. Basic Statistics for Business and Economics. Chapter 13. Holmes et al. Introductory
23. Practice Exercises Refer to Carseats data. Compute the correlation coefficient between competitor’s price and the company’s
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Learning Outcomes
Distinguish between deterministic and probabilistic relations.
Understand the concepts of correlation

and regression.
Be able to fit linear models.
Understand the method of least squares.
Interpret regression coefficients.

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Deterministic (Functional) Relationship
A functional relation between two variables is expressed by

a mathematical formula. If X denotes the independent variable and Y the dependent variable, a functional relation is of the form: Y = f(X)

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Probabilistic (Statistical) Relationship
A probabilistic relationship, unlike a deterministic one, is

not a perfect one. In general, the observations for a probabilistic relation do not fall directly on-the curve of relationship.

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Correlation Coefficient

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Correlation is not causation

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Historical Origins of Regression
Regression analysis was first developed by Sir Francis

Galton in the latter part of the 19th century. Galton had studied the relation between heights of parents and children and noted that the heights of children of both tall and short parents appeared to "revert" or "regress" to the mean of the group. He considered this tendency to be a regression to "mediocrity." Galton developed a mathematical description of this regression tendency, the precursor of today's regression models.
The term regression persists to this day to describe statistical relations between variables.

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Simple Linear Regression
Simple linear regression is a statistical method that allows us

to summarize and study relationships between two quantitative variables:
One variable, denoted x, is regarded as the predictor, explanatory, or independent variable.
The other variable, denoted y, is regarded as the response, outcome, or dependent variable.
It is called “simple”, because there is only one predictor variable.
It is called “linear”, because no parameter appears as an exponent or is multiplied or divided by another parameter.

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Simple Linear Regression
Basic simple linear model:
Yi = β0 + β1Xi +

εI
Yi is the value of the response variable in the ith trial.
β0 (intercept) and β1 (slope) are parameters of the regression.
Xi is a known constant, namely, the value of the predictor variable in the ith trial.
εi is a random error term with mean E{εi} = 0 and variance σ2{εi} = σ2; Error terms are uncorrelated with each other.
Expected value of Y: E(Y) = β0 + β1*X
When scope of the model includes X=0, β0 gives mean of Y at X=0. When the scope of the model does not cover X=0, β0 does not have any particular meaning as a separate term in the model.

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Method of Least Squares

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Computing the least-squares regression line.

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Example

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Results from R

R Syntax:
lm.model <- lm(price ~ size, data = housing)
summary(lm.model)

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Interpretation of b0 and b1

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Multiple Linear Regression
Multiple linear regression is an extension of simple linear regression used to

predict an outcome variable (y) on the basis of multiple distinct predictor variables (x).

y – dependent (response, outcome) variable.
x1, x2, …, xk - independent (predictor, explanatory) variables.
β0 – intercept.
β1, β2, …, βk – partial slopes or partial regression coefficients.
ε – error terms.

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Example

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Interpretations
For a given predictor variable, the coefficient (b) can be interpreted

as the average effect on y of a one unit increase in predictor, holding all other predictors fixed.
For example, for a fixed amount of youtube and newspaper advertising budget, spending an additional 1,000 dollars on facebook advertising leads to an increase in sales by approximately 0.1885*1,000 = 189 sale units, on average.
The youtube coefficient suggests that for every 1,000 dollars increase in youtube advertising budget, holding all other predictors constant, we can expect an increase of 0.046*1,000 = 46 sales units, on average.

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Qualitative (Categorical) Variables
Categorical independent variables can be incorporated into a regression

model by converting them into 0/1 (“dummy”) variables
Use k – 1 = 3 – 1 = 2 dummy variables to code this information like this:

In R, categorical variables need to be converted into factor before running the lm function. R will automatically keep the first factorial level as reference group.

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Example

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Output and Interpretations
If the quality of shelving is Good, ceteris paribus*,

average amount of sales increases by 4.751*1000 = 4751 compared to the Bad shelving.

If the quality of shelving is Medium, while holding other variables constant, average amount of sales increases by 1.862*1000 = 1862 compared to the Bad shelving.

*ceteris paribus = all other things being equal

If the competitor increases its price by $1, while holding other variables constant, average carseat sales increases by 0.010*1000 = 10

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Fitted Models

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Literature
Lind et al. Basic Statistics for Business and Economics. Chapter

13.
Holmes et al. Introductory Business Statistics. Chapter 13.
McClave, Sincich. Statistics. Chapter 11 & 12.
http://www.sthda.com/english/articles/40-regression-analysis/168-multiple-linear-regression-in-r/

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Practice Exercises
Refer to Carseats data. Compute the correlation coefficient between competitor’s

price and the company’s price. Comment on their relationships.
Fit a simple linear model to predict the carseat price using the competitor’s price as an independent variable.
What is the expected price if the competitor’s price is $120?
3. Fit a multiple linear model to predict the sales using Income, Advertising and Urban as predictor variables.
Use the stores in urban locations as reference group.
What is the predicted sales if Community Income level is $75,000, Local advertising budget is $10,000 and the store is located in rural area?
4. Interpret your estimated regression coefficients in task 3.
5. (Homework).
Fit a multiple linear model to predict the sales using Income, Age, Education level and US as predictor variables. Provide the fitted model for US and non-US locations.

Linear Regression Models. Week 7

Содержание

Learning OutcomesDistinguish between deterministic and probabilistic relations.Understand the concepts of correlation

Deterministic (Functional) RelationshipA functional relation between two variables is expressed by

Probabilistic (Statistical) Relationship A probabilistic relationship, unlike a deterministic one, is

Correlation Coefficient

Correlation is not causation

Historical Origins of RegressionRegression analysis was first developed by Sir Francis

Simple Linear RegressionSimple linear regression is a statistical method that allows us

Simple Linear RegressionBasic simple linear model:Yi = β0 + β1Xi +

Method of Least Squares

Computing the least-squares regression line.

Example

Results from R R Syntax:lm.model <- lm(price ~ size, data = housing)summary(lm.model)

Interpretation of b0 and b1

Multiple Linear RegressionMultiple linear regression is an extension of simple linear regression used to

Example

InterpretationsFor a given predictor variable, the coefficient (b) can be interpreted

Qualitative (Categorical) VariablesCategorical independent variables can be incorporated into a regression

Example

Output and InterpretationsIf the quality of shelving is Good, ceteris paribus*,

Fitted Models

Literature Lind et al. Basic Statistics for Business and Economics. Chapter

Practice ExercisesRefer to Carseats data. Compute the correlation coefficient between competitor’s

Похожие презентации

Learning Outcomes
Distinguish between deterministic and probabilistic relations.
Understand the concepts of correlation

Deterministic (Functional) Relationship
A functional relation between two variables is expressed by

Probabilistic (Statistical) Relationship
A probabilistic relationship, unlike a deterministic one, is

Historical Origins of Regression
Regression analysis was first developed by Sir Francis

Simple Linear Regression
Simple linear regression is a statistical method that allows us

Simple Linear Regression
Basic simple linear model:
Yi = β0 + β1Xi +

Results from R

R Syntax:
lm.model <- lm(price ~ size, data = housing)
summary(lm.model)

Multiple Linear Regression
Multiple linear regression is an extension of simple linear regression used to

Interpretations
For a given predictor variable, the coefficient (b) can be interpreted

Qualitative (Categorical) Variables
Categorical independent variables can be incorporated into a regression

Output and Interpretations
If the quality of shelving is Good, ceteris paribus*,

Literature
Lind et al. Basic Statistics for Business and Economics. Chapter

Practice Exercises
Refer to Carseats data. Compute the correlation coefficient between competitor’s