1. Interpretation of Regression Coefficients
Lecture 7
Review of Lecture 6
Interpretation of Regression Coefficients
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ST3131, Lecture 7

2. Multiple Linear Regression Model:
Review of Lecture 6
Multiple Linear Regression Model:
1 Response variable Y
LS Estimators
where
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3. Noise variance estimator:
Fitted values:
Residuals:
Noise variance estimator:
Observation Decomposition
Squares Decomposition
SST = SSR SSE
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4. 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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5. That is
Remark:
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6. How to Interpret the Regression Equation?1
Straight line 2
Plane
>= 3
Hyperplane
How to Interpret the Regression Coefficients?
We have two interpretations.
Interpretation 1: from the view of response change
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ST3131, Lecture 7

7. To address this interpretation, let’s use the simplest MLR model:
Conclusion:
Interpretation 2:
To address this interpretation, let’s use the simplest MLR model:
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ST3131, Lecture 7

8. Results for: P054.txt, the Supervisor Performance data
We use X1 and X2 from the supervisor performance data to illustrate the ideas.
Results for: P054.txt, the Supervisor Performance data
Regression Analysis: Y versus X1, X2
The regression equation is
Y = X X2
Predictor Coef SE Coef T P
Constant X X
What can we see from the above regression results?
(1) Each unit of X1 adds .78 to Y when X2 is fixed
(2) Each unit of X2 subtract from Y when X1 is fixed
(3). .78 and -.05 represents the effects of X1 and X2 respectively on Y after (X1, Y)
(res. (X2,Y)) are linearly adjusted for X2 ( res. X1)
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ST3131, Lecture 7

9. Step 1: SLR Y on X1 (linearly adjust Y on X1)
Let’s consider the effect of X2 on Y as an example. We use the following steps.
Step 1: SLR Y on X1 (linearly adjust Y on X1)
Step 2: SLR X2 on X1 (linearly adjust X2 on X1)
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ST3131, Lecture 7

10. Step 3. Consider the effect of X2 after Y and X2 are linearly adjusted for X1.
Here stands for the effect of X2 after both Y and X2 are linearly adjusted for X1, which is also
the coefficient of X2 in the regression of Y on X1 and X2. Actually, we have
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11. For a general 2-predictor variable MLR model:
In general, we have
For a general 2-predictor variable MLR model:
Plug in the linear adjustment of X2 on X1: Step 2
That is: Step 1
And Step 3
Moreover
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12. Byproduct:
The MLR coefficients and the associated p SLR coefficients are not the same unless all predictor
variables are uncorrelated. For example,
Remark:
In Non-experimental /Observational data, predictor variables are rarely uncorrelated.
However, in an experimental setting, the experimental design is often set up to produce
uncorrelated predictor variables for simple computation and interpretation of coefficients.
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13. In general, for a MLR model
Similarly
In general, for a MLR model
That is
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14. After-class Questions:
How does the regression model quantify the effect of a predictor variable?
How can you check if an effect of a predictor variable is significant or not? Can you suggest some methods based on what you have learned Chapter 2?
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ST3131, Lecture 7