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Quantitative Analysis for Management

Thirteenth Edition

Chapter 4

Regression Models

Learning Outcomes

Create models of simple and multiple regression analysis.

Formulate a simple linear regression model and generate by hand or with the assistance of computer software all required numerical terms that comprise the whole package of a standard simple linear regression analysis.

Chapter Outline

4.1 Scatter Diagrams

4.2 Simple Linear Regression

4.3 Measuring the Fit of the Regression Model

4.4 Assumptions of the Regression Model

4.5 Testing the Model for Significance

4.6 Using Computer Software for Regression

4.7 Multiple Regression Analysis

4.8 Binary or Dummy Variables

4.9 Model Building

4.10 Nonlinear Regression

4.11 Cautions and Pitfalls in Regression Analysis

Introduction (1 of 2)

Regression analysis – very valuable tool for a manager

Understand the relationship between variables

Predict the value of one variable based on another variable

Simple linear regression models have only two variables

Multiple regression models have more than one independent variable

Introduction (2 of 2)

Variable to be predicted is called the dependent variable or response variable

Value depends on the value of the independent variable(s)

Explanatory or predictor variable

Scatter Diagram

Scatter diagram or scatter plot often used to investigate the relationship between variables

Independent variable normally plotted on X axis

Dependent variable normally plotted on Y axis

Triple A Construction (1 of 7)

Triple A Construction renovates old homes

The dollar volume of renovation work is dependent on the area payroll

TABLE 4.1 Triple A Construction Company Sales and Local Payroll

TRIPLE A’S SALES ($100,000s)	LOCAL PAYROLL ($100,000,000s)
6	3
8	4
9	6
5	4
4.5	2
9.5	5

Triple A Construction (2 of 7)

FIGURE 4.1 Scatter Diagram of Triple A Construction Company Data

Simple Linear Regression (1 of 2)

Regression models used to test relationships between variables

Random error

where

Y = dependent variable (response)

X = independent variable (predictor or explanatory)

β0 = intercept (value of Y when X = 0)

β1 = slope of the regression line

e = random error

Simple Linear Regression (2 of 2)

True values for the slope and intercept are not known

Estimated using sample data

where

Ŷ = predicted value of Y

b0 = estimate of β0, based on sample results

b1 = estimate of β1, based on sample results

Triple A Construction (3 of 7)

Predict sales based on area payroll

Y = Sales X = Area payroll

The line Figure 4.1 minimizes the errors

Error = (Actual value) − (Predicted value

Regression analysis minimizes the sum of squared errors

Least-squares regression

Triple A Construction (4 of 7)

Formulas for simple linear regression, intercept and slope

Triple A Construction (5 of 7)

TABLE 4.2 Regression Calculations for Triple A Construction

Y	X	(X − X̅)2	(X − X̅)(Y − Y̅)
6	3	(3 − 4)2 = 1	(3 − 4)(6 − 7) = 1
8	4	(4 − 4)2 = 0	(4 − 4)(8 − 7) = 0
9	6	(6 − 4)2 = 4	(6 − 4)(9 − 7) = 4
5	4	(4 − 4)2 = 0	(4 − 4)(5 − 7) = 0
4.5	2	(2 − 4)2 = 4	(2 − 4)(4.5 − 7) = 5
9.5	5	(5 − 4)2 = 1	(5 − 4)(9.5 − 7) = 2.5
ΣY = 42 Y̅ = 42÷6 = 7	ΣX = 24 X̅ = 24÷6 = 4	Σ(X − X̅)2 = 10	Σ(X − X̅)(Y − Y̅) = 12.5

Triple A Construction (7 of 7)

Regression calculations

Therefore

sales = 2 + 1.25(payroll)

If the payroll next year is $600 million

Ŷ = 2 + 1.25(6) = 9.5 or $ 950,000

Measuring the Fit of the Regression Model (1 of 5)

Regression models can be developed for any variables X and Y

How helpful is the model in predicting Y?

With average error positive and negative errors cancel each other out

Three measures of variability

SST – Total variability about the mean

SSE – Variability about the regression line

SSR – Total variability that is explained by the model

Measuring the Fit of the Regression Model (2 of 5)

Sum of squares total

Sum of squares error

Sum of squares regression

An important relationship

Measuring the Fit of the Regression Model (3 of 5)

TABLE 4.3 Sum of Squares for Triple A Construction

Y	X	(Y − Y̅)2	Ŷ	(Y − Ŷ)2	(Ŷ − Y̅)2
6	3	(6 − 7)2 = 1	2 + 1.25(3) = 5.75	0.0625	1.563
8	4	(8 − 7)2 = 1	2 + 1.25(4) = 7.00	1	0
9	6	(9 − 7)2 = 4	2 + 1.25(6) = 9.50	0.25	6.25
5	4	(5 − 7)2 = 4	2 + 1.25(4) = 7.00	4	0
4.5	2	(4.5 − 7)2 = 6.25	2 + 1.25(2) = 4.50	0	6.25
9.5	5	(9.5 − 7)2 = 6.25	2 + 1.25(5) = 8.25	1.5625	1.563
Y̅ = 7	Blank	∑(Y − Y̅)2 = 22.5	Blank	∑(Y − Ŷ)2 = 6.875	∑(Ŷ − Y̅)2 = 15.625
Blank	Blank	SST = 22.5	Blank	SSE = 6.875	SSR = 15.625

Measuring the Fit of the Regression Model (4 of 5)

For Triple A Construction

SST = 22.5

SSE = 6.875

SSR = 15.625

Measuring the Fit of the Regression Model (5 of 5)

FIGURE 4.2 Deviations from the Regression Line and from the Mean

Coefficient of Determination (1 of 2)

The proportion of the variability in Y explained by the regression equation

The coefficient of determination is r2.

For Triple A Construction

About 69% of the variability in Y is explained by the equation based on payroll (X)

Correlation Coefficient

An expression of the strength of the linear relationship

Always between +1 and −1

The correlation coefficient is r

For Triple A Construction

Four Values of the Correlation Coefficient

FIGURE 4.3 Four Values of the Correlation Coefficient

Assumptions of the Regression Model

With certain assumptions about the errors, statistical tests can be performed to determine the model’s usefulness

Errors are independent

Errors are normally distributed

Errors have a mean of zero

Errors have a constant variance

A plot of the residuals (errors) often highlights glaring violations of assumptions

Residual Plots (1 of 3)

FIGURE 4.4A Pattern of Errors Indicating Randomness

WHEN THE PATTERN IS RANDOM, THE ASSUMPTIONS ARE MET AND THE MODEL IS APPROPRIATE

Residual Plots (2 of 3)

FIGURE 4.4B Non-constant Error Variance

THE ERRORS INCREASE AS X INCREASES, AND THEREFORE VIOLATES THE CONSTANT ERROR ASSUMPTION.

Residual Plots (3 of 3)

FIGURE 4.4C Pattern of Errors Indicating Relationship Is Not Linear

THE ERRORS ARE NON-LINEAR WITH RESPECT TO X, AND SO SOME OTHER MODEL MUST BE USED, E.G., A QUADRATIC EQ.

Estimating the Variance (1 of 2)

Although errors are assumed to have a constant variance (σ2), it is usually unknown but can estimated from the sample results:

Estimated using the mean squared error (MSE), s2

where

n = number of observations in the sample

k = number of independent variables

Estimating the Variance (2 of 2)

For Triple A Construction

We can then estimate the standard deviation, s

The standard error of the estimate or the standard deviation of the regression

IN-CLASS EXERCISE

Summary

Created models of simple and multiple regression analysis.

Formulated a simple linear regression model and generated by hand or with the assistance of computer software all required numerical terms that comprise the whole package of a standard simple linear regression analysis.

bbe

=++

YbbX

eYY

YbbX

average (mean) of values

()()

()

XXYY

bYbX

===

()()

12.5

1.25

()

7 – (1.25)(4)2

X–XY–Y

X–X

bYbX

===

=-==

2 + 1.25

SST()

SSE()

eYY

==-

åå

SSR()

SSTSSR + SSE

SSRSSE

1–

SSTSST

15.625

0.6944

22.5

=±

0.69440.8333

SSE

MSE

SSE6.87506.8750

MSE1.7188

16114

=====

----

MSE 1.71881.31

===