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Using Microsoft® Excel4th Edition

Chapter 13

Introduction to Multiple Regression

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

The Multiple Regression Model

Idea: Examine the linear relationship between

1 dependent (Y) & 2 or more independent variables (Xi)

Multiple Regression Model with k Independent Variables:

Population slopes

Random Error

Y-intercept

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Multiple Regression Equation

The coefficients of the multiple regression model are estimated using sample data

Multiple regression equation with k independent variables:

Estimated

(or predicted)

value of Y

Estimated

intercept

Estimated slope coefficients

In this chapter we will always use Excel to obtain the regression slope coefficients and other regression summary measures.

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Multiple Regression Equation

(continued)

Two variable model

Y

Slope for variable X1

X2

Slope for variable X2

X1

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Example: 2 Independent Variables

- A distributor of frozen desert pies wants to evaluate factors thought to influence demand
- Dependent variable: Pie sales (units per week)
- Independent variables: Price (in $)

Advertising ($100’s)

- Data are collected for 15 weeks

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Pie Sales Example

Multiple regression equation:

Sales = b0 + b1 (Price)

+ b2 (Advertising)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Estimating a Multiple Linear Regression Equation

- Excel will be used to generate the coefficients and measures of goodness of fit for multiple regression
- Excel:
- Tools / Data Analysis... / Regression

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Multiple Regression Output

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

The Multiple Regression Equation

where

Sales is in number of pies per week

Price is in $

Advertising is in $100’s.

b1 = -24.975: sales will decrease, on average, by 24.975 pies per week for each $1 increase in selling price, net of the effects of changes due to advertising

b2 = 74.131: sales will increase, on average, by 74.131 pies per week for each $100 increase in advertising, net of the effects of changes due to price

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Using The Equation to Make Predictions

Predict sales for a week in which the selling price is $5.50 and advertising is $350:

Note that Advertising is in $100’s, so $350 means that X2 = 3.5

Predicted sales is 428.62 pies

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Coefficient of Multiple Determination

- Reports the proportion of total variation in Y explained by all X variables taken together

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Multiple Coefficient of Determination

(continued)

52.1% of the variation in pie sales is explained by the variation in price and advertising

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Adjusted r2

- r2 never decreases when a new X variable is added to the model
- This can be a disadvantage when comparing models
- What is the net effect of adding a new variable?
- We lose a degree of freedom when a new X variable is added
- Did the new X variable add enough explanatory power to offset the loss of one degree of freedom?

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Adjusted r2

(continued)

- Shows the proportion of variation in Y explained by all X variables adjusted for the number of Xvariables used

(where n = sample size, k = number of independent variables)

- Penalize excessive use of unimportant independent variables
- Smaller than r2
- Useful in comparing among models

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Adjusted r2

(continued)

44.2% of the variation in pie sales is explained by the variation in price and advertising, taking into account the sample size and number of independent variables

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Simple and Multiple Regression Compared

- Coefficients in a simple regression pick up the impact of that variable plus the impacts of other variables that are correlated with it and the dependent variable.
- Coefficients in a multiple regression net out the impacts of other variables in the equation.

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Simple and Multiple Regression Compared:Example

- Two simple regressions:
- ABSENCES=a+b1AUTONOMY
- ABSENCES= a +b2SKILLVARIETY
- Multiple Regression:
- ABSENCES= a +b1AUTONOMY+

b2SKILLVARIETY

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Venn Diagrams and Regression

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Venn Diagrams and Regression

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Overlap in Explanation

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Is the Model Significant?

- F-Test for Overall Significance of the Model
- Shows if there is a linear relationship between all of the X variables considered together and Y
- Use F test statistic
- Hypotheses:

H0: β1 = β2 = … = βk = 0 (no linear relationship)

H1: at least one βi≠ 0 (at least one independent

variable affects Y)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

F-Test for Overall Significance

- Test statistic:

where F has (numerator) = k and

(denominator) = (n – k - 1)

degrees of freedom

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

F-Test for Overall Significance

(continued)

With 2 and 12 degrees of freedom

P-value for the F-Test

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

H0: β1 = β2 = 0

H1: β1 and β2 not both zero

= .05

df1= 2 df2 = 12

F-Test for Overall Significance(continued)

Test Statistic:

Decision:

Conclusion:

Critical Value:

F = 3.885

Since F test statistic is in the rejection region (p-value < .05), reject H0

= .05

0

F

There is evidence that at least one independent variable affects Y

Do not

reject H0

Reject H0

F.05 = 3.885

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Are Individual Variables Significant?

- Use t-tests of individual variable slopes
- Shows if there is a linear relationship between the variable Xi and Y
- Hypotheses:
- H0: βi = 0 (no linear relationship)
- H1: βi≠ 0 (linear relationship does exist

between Xi and Y)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Are Individual Variables Significant?

(continued)

H0: βi = 0 (no linear relationship)

H1: βi≠ 0 (linear relationship does exist

between xi and y)

Test Statistic:

(df = n – k – 1)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Are Individual Variables Significant?

(continued)

t-value for Price is t = -2.306, with p-value .0398

t-value for Advertising is t = 2.855, with p-value .0145

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

H0: βi = 0

H1: βi 0

Inferences about the Slope: tTest ExampleFrom Excel output:

- d.f. = 15-2-1 = 12
- = .05

t/2 = 2.1788

The test statistic for each variable falls in the rejection region (p-values < .05)

Decision:

Conclusion:

Reject H0 for each variable

a/2=.025

a/2=.025

There is evidence that both Price and Advertising affect pie sales at = .05

Reject H0

Do not reject H0

Reject H0

-tα/2

tα/2

0

-2.1788

2.1788

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Job Earnings Example

- http://www.ilir.uiuc.edu/courses/lir593/jearnoutput.htm
- ERTEN: 27.4/2.5= 10.9 t1171
- UNEM: -229.1/45.9= -5.0 t1171
- EDU: 885.13/76.5= 11.6 t1171

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Confidence Interval Estimate for the Slope

Confidence interval for the population slope βi

where t has

(n – k – 1) d.f.

Here, t has

(15 – 2 – 1) = 12 d.f.

Example: Form a 95% confidence interval for the effect of changes in price (X1) on pie sales:

-24.975 ± (2.1788)(10.832)

So the interval is (-48.576 , -1.374)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Confidence Interval Estimate for the Slope

(continued)

Confidence interval for the population slope βi

Example: Excel output also reports these interval endpoints:

Weekly sales are estimated to be reduced by between 1.37 to 48.58 pies for each increase of $1 in the selling price

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Qualitative Independent Variables

- Can we handle variable such as gender or race or state or schooling categories in a regression?
- YES. In fact, we can exactly duplicate ANOVA (differences of means across categories) and t-tests for differences of means with regression.
- Therefore, regression can handle almost all the tests that we have previously done!

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Using Dummy Variables

- A dummy variable is a categorical explanatory variable with two levels:
- yes or no, on or off, male or female
- coded as 0 or 1
- Regression intercepts are different if the variable is significant
- If more than two levels, the number of dummy variables needed is (number of levels - 1)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Regression with only dummy variables

- Two sample t-test: Suppose that we have two groups: M and F and we want to know if their job satisfaction is the same.
- Code variable female=1 if the person is female and female=0 if male
- JS = α + β1FEMALE
- Then for males the equation predicts:
- JS = α + β1(0) = α therefore α is the avg of JS for males
- For females the equation predicts:
- JS = α + β1(1) = α + β1 therefore α + β1 is avg of JS for females

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Regression with only dummy variables

- Thus, β1 is the difference of means in JS for males and females. The t-test is identical to the test that we did previously!
- Now suppose that we have 3 groups: Blacks, Whites and Asians. We form TWO dummies variables (# groups – 1). For example, BLACK=1 iff race is Black and ASIAN=1 iff race is Asian and zero otherwise

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Regression with only dummy variables

- The regression is JS= α + β1ASIAN + β2BLACK
- For Whites JS= α + β1(0) + β2(0) = α
- For Blacks JS= α + β1(0) + β2(1) = α + β2
- For Asians JS= α + β1(1) + β2(0) = α + β1
- Therefore, the coefficients on ASIAN and BLACK represent the difference of means of that group and the EXCLUDED group (Whites)
- What is the difference of means between Asians and Blacks?

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Regression with only dummy variables

- To test the hypothesis that there is a difference of means across the 3 groups, we can just use the F-test for the entire regression.
- This yields an IDENTICAL test to One-way ANOVA!
- These tests can be generalized to include continuous variables and interactions

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Dummy-Variable Example (with 2 Levels) and 1 continuous variable

Let:

Y = pie sales

X1 = price

X2 = holiday (X2 = 1 if a holiday occurred during the week) (X2 = 0 if there was no holiday that week)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Dummy-Variable Example (with 2 Levels)

(continued)

Holiday

No Holiday

Different intercept

Same slope

Y (sales)

If H0: β2 = 0 is rejected, then

“Holiday” has a significant effect on pie sales

b0 + b2

Holiday (X2 = 1)

b0

No Holiday (X2 = 0)

X1 (Price)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Interpreting the Dummy Variable Coefficient (with 2 Levels)

Example:

Sales: number of pies sold per week

Price: pie price in $

Holiday:

1 If a holiday occurred during the week

0 If no holiday occurred

b2 = 15: on average, sales were 15 pies greater in weeks with a holiday than in weeks without a holiday, given the same price

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Dummy-Variable Models (more than 2 Levels)

- The number of dummy variables is one less than the number of levels
- Example:

Y = house price ; X1 = square feet

- If style of the house is also thought to matter:

Style = ranch, split level, condo

Three levels, so two dummy variables are needed

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Dummy-Variable Models (more than 2 Levels)

(continued)

- Example: Let “condo” be the default category, and let X2 and X3 be used for the other two categories:

Y = house price

X1 = square feet

X2 = 1 if ranch, 0 otherwise

X3 = 1 if split level, 0 otherwise

The multiple regression equation is:

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Interpreting the Dummy Variable Coefficients (with 3 Levels)

Consider the regression equation:

For a condo: X2 = X3 = 0

With the same square feet, a ranch will have an estimated average price of 23.53 thousand dollars more than a condo

For a ranch: X2 = 1; X3 = 0

With the same square feet, a split-level will have an estimated average price of 18.84 thousand dollars more than a condo.

For a split level: X2 = 0; X3 = 1

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Testing Portions of the Multiple Regression Model

- Contribution of a Single Independent Variable Xj

SSR(Xj | all variables except Xj)

= SSR (all variables) – SSR(all variables except Xj)

- Measures the contribution of Xj in explaining the total variation in Y (SST)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Testing Portions of the Multiple Regression Model

(continued)

- Contribution of a Single Independent Variable Xj, assuming all other variables are already included
- (consider here a 3-variable model):
- SSR(X1 | X2 and X3)
- = SSR (all variables) – SSR(X2 and X3)

From ANOVA section of regression for

From ANOVA section of regression for

Measures the contribution of X1 in explaining SST

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

The Partial F-Test Statistic

- Consider the hypothesis test:

H0: variable Xj does not significantly improve the model after all

other variables are included

H1: variable Xj significantly improves the model after all other

variables are included

- Test using the F-test statistic:

(with 1 and n-k-1 d.f.)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Testing Portions of Model: Example

Example: Frozen desert pies

Test at the = .05 level to determine whether the price variable significantly improves the model given that advertising is included

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Testing Portions of Model: Example

(continued)

H0: X1 (price) does not improve the model

with X2 (advertising) included

H1: X1 does improve model

= .05, df = 1 and 12

F critical Value = 4.75

(For X1 and X2)

(For X2 only)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Testing Portions of Model: Example

(continued)

(For X1 and X2)

(For X2 only)

Conclusion: Reject H0; adding X1 does improve model

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Do I need to do this for one variable?

- The F test for the inclusion of a single variable after all other variables are included in the model isIDENTICAL to the t test of the slope for that variable
- The only reason to do an F test is to test several variables together.

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Simultaneous Contribution of Explanatory Variables

- Use partial F-test for the simultaneous contribution of multiple variables to the model
- Let m variables be an additional set of variables added simultaneously
- To test the hypothesis that the set of m variables improves the model:

(where F has m and n-k-1 d.f.)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Example: Collinear Variables

20,000 Execs in 439 Corps: Dependent Variable=base pay+bonus

Individual Simple Regression Multiple Regression

R2 Contribution to R2

Company Dummies .33 .08

Occupational Dummies .52 .022

Position in hierarchy .69 .104

Human Capital Vars .28 .032

Shared .632

TOTAL .87

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Testing hypotheses about included categories and subsets of variables

- http://www.ilir.uiuc.edu/courses/lir593/spss_example.htm

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Regression Example

- http://www.ilir.uiuc.edu/courses/lir593/spss_example.htm

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Interaction Between Explanatory Variables

- Hypothesizes interaction between pairs of X variables
- Response to one X variable may vary at different levels of another X variable
- Contains two-way cross product terms

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Effect of Interaction

- Given:
- Without interaction term, effect of X1 on Y is measured by β1
- With interaction term, effect of X1 on Y is measured by β1 + β3 X2
- Effect changes as X2 changes

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Interaction Example

Suppose X2 is a dummy variable and the estimated regression equation is

= 1 + 2X1 + 3X2 + 4X1X2

Y

12

X2 = 1:

Y = 1 + 2X1 + 3(1) + 4X1(1) = 4 + 6X1

8

4

X2 = 0:

Y = 1 + 2X1 + 3(0) + 4X1(0) = 1 + 2X1

0

X1

0

0.5

1

1.5

Slopes are different if the effect of X1 on Y depends on X2 value

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Significance of Interaction Term

- Can perform a partial F-test for the contribution of a variable to see if the addition of an interaction term improves the model
- Multiple interaction terms can be included
- Use a partial F-test for the simultaneous contribution of multiple variables to the model

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

Regression and Two-way ANOVA

- With interaction terms among the dummies, we can exactly duplicate Two-way or N-way ANOVA as well.
- Regression automatically controls for different cell sizes: Can be ANOVA with or without duplication
- Be sure that EACH cell has enough observations to allow application of CLT

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.

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