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# Linear Regression Essay

1304 words - 6 pages

Chapter 4
Multiple Linear Regression

Section 4.1
The Model and Assumptions

Objectives
Participants will:  understand the elements of the model  understand the major assumptions of doing a regression analysis  learn how to verify the assumptions  understand a median split

3

The Model
y   o  1x1  ...   p x p   or in Matrix Notation
Dependent Variable nx1 Unknown Parameters (p+1) x 1

Y  X e
Independent Variables – n x(p+1)

Error – nx1

4

Questions
How many unknown parameters are there? Can you name them? How many populations will be sampled? What are conceptual populations?

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Major Requirements for Doing a Regression Analysis
The errors ...view middle of the document...

Determine the number of populations sampled. Run the regression analysis and save the residuals. Plot the residuals versus independent variables. Check for normality of residuals. Check for constant variance.

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Exercises
Use the Store24 data set.

Remind yourself of the model and the parameters. How many populations were sampled? Are the residuals normal? Do you have constant variance?

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Section 4.2
Possible Problem Points, Multicollinearity, and Additional Residual Analyses

Objectives
Participants will learn:  About using Cook’s D, leverage values, and residuals to identify possible problem points.  What is multicollinearity? How to detect it? Why you should use the COLLINOINT option.  About partial F tests and why the p values can be misleading.  About other residuals and the importance of residual graphs.

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Cook’s D, Leverage Values, and Residuals
Cook’s D helps identify possible problem points in the y direction and in the x direction. Cook’s D statistic is a measure of the simultaneous change in the parameter estimates when an observation is deleted from the analysis. Flag if > 1. Leverage values(Hat Diag H )help identify possible problem points in the x direction. Flag if > .5. Residuals help identify possible problem points in the y direction. Flag if abs (STUDENT) > 3.

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Possible Problem Points in the Y Direction

Click Here

http://www.stat.tamu.edu/~mspeed/stat608/content/applets/regress0/regress.htm

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Possible Problem Points in the X Direction

Click Here

http://www.stat.tamu.edu/~mspeed/stat608/content/applets/regress1/regress.html

23

Multicollinearity
Occurs when the x’s predictors (independent variables) are highly correlated. Problems if VIF > 10. Some people use the condition index. In order to avoid false positives, use the COLLINOINT option.

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Variance Inflation Factor (VIF) Example

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Collinearity Diagnostics – Not Adjusted

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Collinearity Diagnostics – Adjusted

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Body Fat Example
Variables
              28

Percent body fat from Siri’s (1956) equation – dependent Age (years) Weight (lbs) Height (inches) Neck circumference (cm) Chest circumference (cm) Abdomen 2 circumference (cm) Hip circumference (cm) Thigh circumference (cm Knee circumference (cm) Ankle circumference (cm) Biceps (extended) circumference (cm) Forearm circumference (cm) Wrist circumference (cm)

What Is Being Tested by |t|

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continued...

What Is Being Tested by Pr >|t|

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Partial F-Tests

H o : 3  0 | all other  's are in the model

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Interpretation – The Stable Table

Do I need this leg to have a stable table?

Nope!

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...

Interpretation – The Stable Table

Do I need this leg to have a stable table?

Nope!

34

...

Interpretation – The Stable Table

Do I need this leg to have a stable table?

Nope!

35

...

Graphs
Predicted versus Y Residual versus...

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