1. 9/19/2018
ST3131, Lecture 6

2. Chapter 3 Multiple Linear Regression
Lecture 6
Review of Lecture 5
Chapter 3 Multiple Linear Regression
Motivation Example
MLR Model
Estimation of the MLR Model
Methods for Assessment of Linearity
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ST3131, Lecture 6

3. Review of Lecture 5: Special SLR Models
No Intercept Model :
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ST3131, Lecture 6

4. Review of Lecture 5: Special SLR Models
No Slope Model :
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ST3131, Lecture 6

5. Review of Lecture 5: Special SLR Models
Trivial Regression Model
One-sample t-test:
Trivial Model against No-slope Model
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ST3131, Lecture 6

6. Review of Lecture 5: Special SLR Models
Paired two-sample t-test
Transformation:
The two-sample t-test becomes one-sample t-test:
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ST3131, Lecture 6

7. Chapter 3 Multiple Linear Regression
Motivation Example: Supervisor Performance Data
In a large financial organization, there are 30 departments, each having a supervisor and 35 employees. To study the supervisor performance, the employees in each department are given a questionnaire with following items:
(1 response variable)
Y: Overall rating of job being done
(6 predictor variables)
X1: handles employee complaints
X2:Does not allow special privileges
X4: Raises based on performance
X3:Opportunity to learn new things
X5: Too critical of poor performance
X6:Rate of advancing to better jobs
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8. Motivation Example (cont.)
For each item, the employees are required to choose a number from
very satisfactory very unsatisfactory
To evaluate the supervisor on a item,
The proportion of “favorable” responses among 35 answers in
a department is regarded as observation of the associated item.
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ST3131, Lecture 6

9. Motivation Example (cont.)
Part of the Supervisor Performance Data
Y X1 X2 X3 X4 X5 X6
===================================================
……..
Each Row :
Observations for a department for all items
Each Column:
Observations for all departments for an item
Examples:
X3=(39,54,69,47,…)’
Y=(43,63,71,61,81,…)’
called a design matrix
X=(X1,X2,X3,X4,X5,X6)
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10. MLR Models Multiple Linear Regression Model: 1 Response variable Y
SLR is not good enough for handling many practical cases where more predictor variables are involved for predicting the response variable. We need to use Multiple Linear Regression Model.
Multiple Linear Regression Model:
1 Response variable Y
MLR generalizes SLR
Remark:
SLR is a special case of MLR
When p=1, MLR reduces to SLR
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11. Estimation of the MLR Model
Least Squares Method:
Solution:
where
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12. Fitted Values and Squares Decomposition
Fitted Regression Line:
Fitted values:
Residuals:
Noise variance estimator:
SSE= Sum of Squared Errors,
n-(p+1)=degrees of freedom of SSE
Observation Decomposition
Squares Decomposition
SST = SSR SSE
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ST3131, Lecture 6

13. Methods of Assessment of Linearity /Quality of Linear Fit
A. Coefficient of Determination
Proportion of total variability
Explained by Linear Regression
B. Correlation between responses and fitted values
R is called Multiple Correlation Coefficient between Y
and multiple predictor variables X1, X2, …, Xp, measuring
the linear relationship between Y and X1, X2, …,Xp.
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14. Methods of Assessment of Linearity /Quality of Linear Fit
That is
Remark:
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15. Analysis of the Supervisor Performance Data
Example
Analysis of the Supervisor Performance Data
Results for: P054.txt
Correlations: Y, X1, X2, X3, X4, X5, X6
Y X X X X X5 X
0.000 X X X X X
Cell Contents: Pearson correlation
P-Value
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ST3131, Lecture 6

16. Regression Analysis: Y versus X1, X2, X3, X4, X5, X6
The regression equation is
Y = X X X X X X6
Predictor Coef SE Coef T P
Constant X X X X X X
S = R-Sq = 73.3% R-Sq(adj) = 66.3%
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17. 9/19/2018 ST3131, Lecture 6 Analysis of Variance Source DF SS MS F P
Regression
Residual Error
Total
Source DF Seq SS X X X X X X
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ST3131, Lecture 6

18. After-class Questions: When is a MLR needed?
Can we fit several SLR models instead fitting a single MLR model?
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ST3131, Lecture 6