3134 Words13 Pages

Introduction to Linear Regression and Correlation Analysis

Goals

After this, 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 equation for a set of data Understand the assumptions behind regression analysis Determine whether a regression model is significant

Goals

(continued)

After this, 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*…show more content…*

sed to:

– Predict the value of a dependent variable based on the value of at least one independent variable – Explain the impact of changes in an independent variable on the dependent variable

Dependent variable: the variable we wish to explain Independent variable: the variable used

Simple Linear Regression Model

• Only one independent variable, x

• Relationship between x and y is described by a linear function

• Changes in y are assumed to be caused by changes in x

Types of Regression Models

Positive Linear Relationship Relationship NOT Linear

Negative Linear Relationship

No Relationship

Population Linear Regression

The population regression model:

Population y intercept Dependent Variable

Population Slope Coefficient

Independent Variable

y β0 β1x ε

Linear component

Random Error term, or residual

Random Error component

Linear Regression Assumptions

• Error values (ε) are statistically independent • Error values are normally distributed for any given value of x

• The probability distribution of the errors is normal

• The probability distribution of the errors has constant variance • The underlying relationship between the x

Population Linear Regression y Observed Value of y for xi

y β0 β1x ε εi (continued)

Slope = β1 Random Error for this x value

Predicted Value of y for xi Intercept = β0

xi

x

Estimated Regression Model

The sample regression line provides an estimate of the

Goals

After this, 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 equation for a set of data Understand the assumptions behind regression analysis Determine whether a regression model is significant

Goals

(continued)

After this, 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

sed to:

– Predict the value of a dependent variable based on the value of at least one independent variable – Explain the impact of changes in an independent variable on the dependent variable

Dependent variable: the variable we wish to explain Independent variable: the variable used

Simple Linear Regression Model

• Only one independent variable, x

• Relationship between x and y is described by a linear function

• Changes in y are assumed to be caused by changes in x

Types of Regression Models

Positive Linear Relationship Relationship NOT Linear

Negative Linear Relationship

No Relationship

Population Linear Regression

The population regression model:

Population y intercept Dependent Variable

Population Slope Coefficient

Independent Variable

y β0 β1x ε

Linear component

Random Error term, or residual

Random Error component

Linear Regression Assumptions

• Error values (ε) are statistically independent • Error values are normally distributed for any given value of x

• The probability distribution of the errors is normal

• The probability distribution of the errors has constant variance • The underlying relationship between the x

Population Linear Regression y Observed Value of y for xi

y β0 β1x ε εi (continued)

Slope = β1 Random Error for this x value

Predicted Value of y for xi Intercept = β0

xi

x

Estimated Regression Model

The sample regression line provides an estimate of the

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