Class 5

1
Class 5

• Multiple Regression Models

2
Multiple Regression Models

• We can readily imagine that there may be several
  factors that we can include in our model to
  explain test scores.

3
Using EXCEL

• The procedure is the same tools/data
  analysis/regression.
• Note that the independent variables have to be in
  contiguous columns.
• The F-test now tests to see if all of the
  variables are explaining variation in y.
• The problem becomes tricky because the degree to
  which a variable appears to be important in
  explaining the variation in y depends on the
  other variables present!

4
Hypothesis Testing

• The F-test tests to see if all of the
  coefficients of the independent variables are
  zero. For our model

• The t-test tests to see if each coefficient of an
  independent variables is zero.

5
Some Final Comments

• The first step in building a regression model is
  to develop a list of candidate variables.
• Notice that measurement might be a problem.
• Note that the t-test now takes on an important
  role. But all you need are the p-values!
• Examination of residuals may provide clues about
  other factors that you have left out.

6
Adding Qualitative Factors

• Qualitative factors can be added to the model
  through the use of dummy variables.
• Consider the following data

7
Coding the Data

• We can add the gender factor by coding a variable
  in the following way
• If Female ? then x 1,
• If Male ? then x 0.
• What does our model say about salary?

E(y) expected salary ?0 ?1x
8
Doing the Analysis

• After doing the regression analysis, what
  hypothesis should we test?
• Is there another way of doing this test? From
  prior material?

9
Coding Variables with More than Two Levels

• Consider the following data set. How would you
  code the qualitative factor additive for the
  model?

The additives were added to the gasoline and
resulted in the following miles per gallon (MPG).
Is there a difference in the additives? What
model should we build to check this? Be careful
about what the model implies!
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Coding Qualitative Variables--Summary

• The coding of dummy variables depends upon the
  number of levels that the qualitative factor has.
  For k levels, use k-1 dummy variables. The case
  where k5

This adds four variables to the model (four
columns in your spreadsheet).
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More on Dummy Variables

• Of course, these dummy variables just define
  different populations of which we are comparing
  the means.
• If there are only two populations (one dummy
  variable), you can use the pooled t-test.
• In a regression model, we have the luxury of
  including other factors!

Controlling for other factors!
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More on Dummy Variables

• If you have only a set of dummy variables (like
  the fuel additive problem), you can use ANOVA.