LECTURE 3Introduction to Linear Regression and Correlation Analysis

1 Simple Linear Regression

2 Regression Analysis

3 Regression Model Validity

Goals

After this, you should be able to: Interpret the simple linear regression equation

for a set of dataUse descriptive statistics to describe the

relationship between X and YDetermine whether a regression model is

significant

Goals

After this, you should be able to:

Interpret confidence intervals for the regression coefficients

Interpret confidence intervals for a predicted value of Y

Check whether regression assumptions are satisfied

Check to see if the data contains unusual values

(continued)

Introduction to Regression Analysis

Regression analysis is used 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 to explain the dependent variable

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

Negative Linear Relationship

Relationship NOT Linear

No Relationship

εxββy 10 Linear component

Population Linear Regression

The population regression model:

Population y intercept

Population SlopeCoefficient

Random Error term, or residualDependent

Variable

Independent Variable

Random Error component

Linear Regression Assumptions

The underlying relationship between the x variable and the y variable is linear

The distribution of the errors has constant variability Error values are normally distributed Error values are independent (over time)

Population Linear Regression

Random Error for this x value

y

x

Observed Value of y for xi

Predicted Value of y for xi

εxββy 10

xi

Slope = β1

Intercept = β0

εi

xbby 10i

The sample regression line provides an estimate of the population regression line

Estimated Regression Model

Estimate of the regression

intercept

Estimate of the regression slope

Estimated (or predicted) y value

Independent variable

b0 is the estimated average value of

y when the value of x is zero

b1 is the estimated change in the

average value of y as a result of a one-unit change in x

Interpretation of the Slope and the Intercept

Finding the Least Squares Equation

The coefficients b0 and b1 will be found using computer software, such as Excel’s data analysis add-in or MegaStat

Other regression measures will also be computed as part of computer-based regression analysis

Simple Linear Regression Example

A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet)

A random sample of 10 houses is selected

Dependent variable (y) = house price in $1000

Independent variable (x) = square feet

Sample Data for House Price Model

House Price in $1000s(y)

Square Feet (x)

245 1400

312 1600

279 1700

308 1875

199 1100

219 1550

405 2350

324 2450

319 1425

255 1700

Regression output from Excel – Data – Data Analysis or MegaStat – Correlation/ regression MegaStat – Correlation/ regression

MegaStat OutputThe regression equation is:

feet) (square 0.10977 98.24833 price house Predicted

Regression Analysis

r² 0.581 n 10 r 0.762 k 1

Std. Error 41.330 Dep. Var. Price($000)

ANOVA tableSource SS df MS F p-value

Regression 18,934.9348 1 18,934.9348 11.08 .0104Residual 13,665.5652 8 1,708.1957

Total 32,600.5000 9      

Regression output confidence intervalvariables coefficients std. error t (df=8) p-value 95% lower 95% upperIntercept 98.2483

Square feet 0.1098 0.0330 3.329 .0104 0.0337 0.1858

0

50

100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500 3000

Square Feet

Ho

use

Pri

ce (

$100

0s)

Graphical Presentation

House price model: scatter plot and regression line

feet) (square 0.10977 98.24833 price house

Slope = 0.10977

Intercept = 98.248

Interpretation of the Intercept, b0

b0 is the estimated average value of Y when the value of X is zero

(if x = 0 is in the range of observed x values)

Here, houses with 0 square feet do not occur, so b0 = 98.24833 just

indicates the height of the line.

feet) (square 0.10977 98.24833 price house

Interpretation of the Slope Coefficient, b1

b1 measures the estimated change in Y as a result of a one-unit

increase in X

feet) (square 0.10977 98.24833 price house

Here, b1 = .10977 tells us that the average value of a house increases by .10977($1000) = $109.77, on average, for each additional one square foot of size

Least Squares Regression Properties

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 β0 and

β1

The percentage of variability in Y that can be explained by variability in X.

Coefficient of Determination, R2

Note: In the single independent variable case, the coefficient of determination is

where:R2 = Coefficient of determination

r = Simple correlation coefficient

22 rR

R2 = 1, correlation = +1

Examples ofR2 Values

y

x

y

x

R2 = 1

R2 = 1, correlation = -1

Perfect linear relationship between x and y:

100% of the variation in y is explained by variation in x

Examples of Approximate R2 Values

y

x

y

x

0 < R2 < 1, correlation is negative

Weaker linear relationship between x and y:

Some but not all of the variation in y is explained by variation in x

0 < R2 < 1, correlation is positive

Examples of Approximate R2 Values

R2 = 0

No linear relationship between x and y:

The value of Y does not depend on x. (None of the variation in y is explained by variation in x)

y

xR2 = 0

Excel Output

58.08% of the variation in house prices is explained by

variation in square feet

Regression Analysis

r² 0.581

r 0.762

Std. Error 41.330

The correlation of .762 shows a fairly strong direct

relationship.

The typical error in predicting Price is 41.33($000) = $41,330

Inference about the Slope: t Test t test for a population slope

Is there a linear relationship between x and y? Null and alternative hypotheses

H0: β1 = 0 (no linear relationship) Ha: β1 0 (linear relationship does exist)

Obtain p-value from ANOVA or across from the slope coefficient (they are the same in simple regression)

House Price in $1000s

(y)

Square Feet (x)

245 1400

312 1600

279 1700

308 1875

199 1100

219 1550

405 2350

324 2450

319 1425

255 1700

(sq.ft.) 0.1098 98.25 price house

Estimated Regression Equation:

The slope of this model is 0.1098

Does square footage of the house affect its sales price?

Inference about the Slope: t Test

(continued)

Inferences about the Slope: t Test Example

H0: β1 = 0

Ha: β1 0

We can be 98.96% confident that square feet is related to house price.

From Excel output:

Reject H0

  Coefficients Standard Error t Stat P-value

Intercept 98.24833 58.03348 1.69296 0.12892

Square Feet 0.10977 0.03297 3.32938 0.01039

P-value

Decision:

Conclusion:

Regression Analysis for DescriptionConfidence Interval Estimate of the Slope:

Excel Printout for House Prices:

We can be 95% confident that house prices increase by between $33.74 and $185.80 for a 1 square foot increase.

 Coefficient

sStandard

Error t Stat P-value Lower 95%Upper 95%

Intercept 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386

Square Feet 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580

Estimates of Expected yfor Different Values of x

y

x xp

y = b0 + b1x

x

y

The relationship describes how x impacts your estimate from y

yp

Interval Estimates for Different Values of x

y

x

Prediction Interval for an individual y, given xp The father from x the less accurate the prediction.

xp

y = b0 + b1x

x

y

House Price in $1000s

(y)

Square Feet (x)

245 1400

312 1600

279 1700

308 1875

199 1100

219 1550

405 2350

324 2450

319 1425

255 1700

(sq.ft.) 0.1098 98.25 price house

Estimated Regression Equation:

Example: House Prices

Predict the price for a house with 2000 square feet

317.85

0)0.1098(200 98.25

(sq.ft.) 0.1098 98.25 price house

Example: House PricesPredict the price for a house with 2000 square feet:

The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,850

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Estimation of Individual Values: Example

Find the 95% confidence interval for an individual house with 2,000 square feet

Predicted Price Yi = 317.85 ($1,000s) = $317, 850

MegaStat will give both the predicted value as well as the lower and upper limits

Prediction Interval Estimate for y|xp

The prediction interval endpoints are from $215,503 to $420,065. We can be 95% confident that the price of a 2000 ft2 home will fall within those limits.

Predicted values for: Price($000)    95% Confidence Interval 95% Prediction Interval

Square feet Predicted lower upper lower upper

2,000 317.784 280.664 354.903 215.503 420.065

Residual Analysis Purposes

Check for linearity assumption Check for the constant variability assumption for all levels of predicted Y Check normal residuals assumption Check for independence over time

Graphical Analysis of Residuals Can plot residuals vs. x and predicted Y Can create NPP of residuals to check for normality (or use

Skewness/Kurtosis) Can check D-W statistic to confirm independence

Residual Analysis for Linearity

Not Linear Linear

x

resi

dua

ls

x

y

x

y

x

resi

dua

ls

Residual Analysis for Constant Variance

Non-constant variance Constant variance

Ŷ Ŷ

y

x x

y

resi

dua

ls

resi

dua

ls

Residual Analysis for Normality Can create NPP of residuals to check for normality. If you see an

approximate straight line residuals are acceptably normal. You can also use Skewness/Kurtosis. If both are within + 1 the residuals are acceptably normal

Residual Analysis for Independence– Can check D-W statistic to confirm

independence. If D-W statistic is greater than 1.3 the residuals are acceptably independent. Needed only if the data is collected over time.

Checking Unusual Data Points Check for outliers from the predicted values

(studentized and studentized deleted residuals do this; MegaStat highlights in blue)

Check for outliers on the X-axis; they are indicated by large leverage values; more than twice as large as the average leverage. MegaStat highlights in blue.

Check Cook’s Distance which measures the harmful influence of a data point on the equation by looking at residuals and leverage together. Cook’s D > 1 suggests potentially harmful data points and those points should be checked for data entry error. MegaStat highlights in blue based on F distribution values.

Patterns of Outliers

a). Outlier is extreme in both X and Y but not in pattern. The point is unlikely to alter regression line.

b). Outlier is extreme in both X and Y as well as in the overall pattern. This point will strongly influence regression line

c). Outlier is extreme for X nearly average for Y. The further it is away from the pattern the more it will change the regression.

d). Outlier extreme in Y not in X. The further it is away from the pattern the more it will change the regression.

e). Outlier extreme in pattern, but not in X or Y. Slope may not be changed much but intercept will be higher with this point included.

Summary Introduced simple linear regression analysis Calculated the coefficients for the simple linear regression equation measures of strength (r, R2 and se)

Summary

Described inference about the slope Addressed prediction of individual values Discussed residual analysis to address assumptions of

regression and correlation Discussed checks for unusual data points

(continued)