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Chapter 11. Introduction to Linear Regression and Correlation Analysis. Chapter 11 - Chapter Outcomes. After studying the material in this chapter, you should be able to: Calculate and interpret the simple correlation between two variables. Determine whether the correlation is significant.

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chapter 11

Chapter 11

Introduction to Linear Regression and Correlation Analysis

chapter 11 chapter outcomes

Chapter 11 - Chapter Outcomes

After studying the material in this chapter, you should be able to:

Calculate and interpret the simple correlation between two variables.

Determine whether the correlation is significant.

Calculate and interpret the simple linear regression coefficients for a set of data.

Understand the basic assumptions behind regression analysis.

Determine whether a regression model is significant.

chapter 11 chapter outcomes continued

Chapter 11 - Chapter Outcomes(continued)

After studying the material in this chapter, you should be able to:

Calculate and interpret confidence intervals for the regression coefficients.

Recognize regression analysis applications for purposes of prediction and description.

Recognize some potential problems if regression analysis is used incorrectly.

Recognize several nonlinear relationships between two variables.

scatter diagrams

Scatter Diagrams

A scatter plot is a graph that may be used to represent the relationship between two variables. Also referred to as a scatter diagram.

dependent and independent variables

Dependent and Independent Variables

A dependent variable is the variable to be predicted or explained in a regression model. This variable is assumed to be functionally related to the independent variable.

dependent and independent variables1

Dependent and Independent Variables

An independent variable is the variable related to the dependent variable in a regression equation. The independent variable is used in a regression model to estimate the value of the dependent variable.

correlation

Correlation

The correlation coefficient is a quantitative measure of the strength of the linear relationship between two variables. The correlation ranges from + 1.0 to - 1.0. A correlation of  1.0 indicates a perfect linear relationship, whereas a correlation of 0 indicates no linear relationship.

correlation1

Correlation

SAMPLE CORRELATION COEFFICIENT

where:

r = Sample correlation coefficient

n = Sample size

x = Value of the independent variable

y = Value of the dependent variable

correlation2

Correlation

SAMPLE CORRELATION COEFFICIENT

or the algebraic equivalent:

correlation example 11 12
Correlation(Example 11-1)

Correlation between Years and Sales

Excel Correlation Output

(Figure 11-5)

correlation3

Correlation

TEST STATISTIC FOR CORRELATION

where:

t = Number of standard deviations r is from 0

r = Simple correlation coefficient

n = Sample size

slide19

Correlation Significance Test(Example 11-1)

Rejection Region  = 0.025

Since t=4.752 > 2.228, reject H0, there is a significant linear relationship

correlation4

Correlation

Spurious correlation occurs when there is a correlation between two otherwise unrelated variables.

simple linear regression analysis

Simple Linear Regression Analysis

Simple linear regression analysis analyzes the linear relationship that exists between a dependent variable and a single independent variable.

simple linear regression analysis1

Simple Linear Regression Analysis

SIMPLE LINEAR REGRESSION MODEL (POPULATION MODEL)

where:

y = Value of the dependent variable

x = Value of the independent variable

= Population’s y-intercept

= Slope of the population regression line

= Error term, or residual

simple linear regression analysis2
Simple Linear Regression Analysis

The simple linear regression model has four assumptions:

  • Individual values if the error terms, i, are statistically independent of one another.
  • The distribution of all possible values of  is normal.
  • The distributions of possible i values have equal variances for all value of x.
  • The means of the dependent variable, for all specified values of the independent variable, y, can be connected by a straight line called the population regression model.
simple linear regression analysis3

Simple Linear Regression Analysis

REGRESSION COEFFICIENTS

In the simple regression model, there are two coefficients: the intercept and the slope.

simple linear regression analysis4

Simple Linear Regression Analysis

The interpretation of the regression slope coefficient is the average change in the dependent variable for a unit increase in the independent variable. The slope coefficient may be positive or negative, depending on the relationship between the two variables.

simple linear regression analysis5

Simple Linear Regression Analysis

The least squares criterion is used for determining a regression line that minimizes the sum of squared residuals.

simple linear regression analysis6

Simple Linear Regression Analysis

A residual is the difference between the actual value of the dependent variable and the value predicted by the regression model.

simple linear regression analysis7
Simple Linear Regression Analysis

y

390

400

Sales in Thousands

300

312

200

Residual = 312 - 390 = -78

100

x

4

Years with Company

simple linear regression analysis8

Simple Linear Regression Analysis

ESTIMATED REGRESSION MODEL

(SAMPLE MODEL)

where:

= Estimated, or predicted, y value

b0 = Unbiased estimate of the regression intercept

b1 = Unbiased estimate of the regression slope

x = Value of the independent variable

simple linear regression analysis9

Simple Linear Regression Analysis

LEAST SQUARES EQUATIONS

algebraic equivalent:

and

simple linear regression analysis table 11 3
Simple Linear Regression Analysis(Table 11-3)

The least squares regression line is:

simple linear regression analysis figure 11 11
Simple Linear Regression Analysis(Figure 11-11)

Excel Midwest Distribution Results

least squares regression properties
Least Squares Regression Properties
  • The sum of the residuals from the least squares regression line is 0.
  • The sum of the squared residuals is a minimum.
  • The simple regression line always passes through the mean of the y variable and the mean of the x variable.
  • The least squares coefficients are unbiased estimates of 0 and 1.
simple linear regression analysis11

Simple Linear Regression Analysis

SUM OF RESIDUALS

SUM OF SQUARED RESIDUALS

simple linear regression analysis12

Simple Linear Regression Analysis

TOTAL SUM OF SQUARES

where:

TSS = Total sum of squares

n = Sample size

y = Values of the dependent variable

= Average value of the dependent variable

simple linear regression analysis13

Simple Linear Regression Analysis

SUM OF SQUARES ERROR (RESIDUALS)

where:

SSE = Sum of squares error

n = Sample size

y = Values of the dependent variable

= Estimated value for the average of y for the given x value

simple linear regression analysis14

Simple Linear Regression Analysis

SUM OF SQUARES REGRESSION

where:

SSR = Sum of squares regression

= Average value of the dependent variable

y = Values of the dependent variable

= Estimated value for the average of y for the given x value

simple linear regression analysis16

Simple Linear Regression Analysis

The coefficient of determination is the portion of the total variation in the dependent variable that is explained by its relationship with the independent variable. The coefficient of determination is also called R-squared and is denoted as R2.

simple linear regression analysis17

Simple Linear Regression Analysis

COEFFICIENT OF DETERMINATION (R2)

simple linear regression analysis midwest example1

Simple Linear Regression Analysis(Midwest Example)

COEFFICIENT OF DETERMINATION (R2)

69.31% of the variation in the sales data for this sample can be explained by the linear relationship between sales and years of experience.

simple linear regression analysis18

Simple Linear Regression Analysis

COEFFICIENT OF DETERMINATION SINGLE INDEPENDENT VARIABLE CASE

where:

R2 = Coefficient of determination

r = Simple correlation coefficient

simple linear regression analysis19

Simple Linear Regression Analysis

STANDARD DEVIATION OF THE REGRESSION SLOPE COEFFICIENT (POPULATION)

where:

= Standard deviation of the regression slope (Called the standard error of the slope)

= Population standard error of the estimate

simple linear regression analysis20

Simple Linear Regression Analysis

ESTIMATOR FOR THE STANDARD ERROR OF THE ESTIMATE

where:

SSE= Sum of squares error

n = Sample size

k = number of independent variables in the model

simple linear regression analysis21

Simple Linear Regression Analysis

ESTIMATOR FOR THE STANDARD DEVIATION OF THE REGRESSION SLOPE

where:

= Estimate of the standard error of the least squares slope

= Sample standard error of the estimate

simple linear regression analysis22

Simple Linear Regression Analysis

TEST STATISTIC FOR TEST OF SIGNIFICANCE OF THE REGRESSION SLOPE

where:

b1 = Sample regression slope coefficient

1 = Hypothesized slope

sb1 = Estimator of the standard error of the slope

slide49

Significance Test of Regression Slope(Example 11-5)

Rejection Region  /2 = 0.025

Rejection Region  /2 = 0.025

Since t=4.753 > 2.228, reject H0: conclude that the true slope is not zero

simple linear regression analysis23

Simple Linear Regression Analysis

MEAN SQUARE REGRESSION

where:

SSR = Sum of squares regression

k = Number of independent variables in the model

simple linear regression analysis24

Simple Linear Regression Analysis

MEAN SQUARE ERROR

where:

SSE = Sum of squares error

n = Sample size

k = Number of independent variables in the model

significance test example 11 6
Significance Test(Example 11-6)

Rejection Region  = 0.05

Since F= 22.59 > 4.96, reject H0: conclude that the regression model explains a significant amount of the variation in the dependent variable

simple regression steps
Simple Regression Steps
  • Develop a scatter plot of y and x. You are looking for a linear relationship between the two variables.
  • Calculate the least squares regression line for the sample data.
  • Calculate the correlation coefficient and the simple coefficient of determination, R2.
  • Conduct one of the significance tests.
simple linear regression analysis25

Simple Linear Regression Analysis

CONFIDENCE INTERVAL ESTIMATE FOR THE REGRESSION SLOPE

or equivalently:

where:

sb1 = Standard error of the regression slope coefficient

s = Standard error of the estimate

simple linear regression analysis26

Simple Linear Regression Analysis

CONFIDENCE INTERVAL FOR

where:

= Point estimate of the dependent variable

t = Critical value with n - 2 d.f.

s = Standard error of the estimate

n = Sample size

xp = Specific value of the independent variable

= Mean of independent variable observations

simple linear regression analysis27

Simple Linear Regression Analysis

PREDICTION INTERVAL FOR

residual analysis

Residual Analysis

Before using a regression model for description or prediction, you should do a check to see if the assumptions concerning the normal distribution and constant variance of the error terms have been satisfied. One way to do this is through the use of residual plots.

key terms
Coefficient of Determination

Correlation Coefficient

Dependent Variable

Independent Variable

Least Squares Criterion

Regression Coefficients

Key Terms
  • Regression Slope Coefficient
  • Residual
  • Scatter Plot
  • Simple Linear Regression Analysis
  • Spurious Correlation