Business Forecasting

Business Forecasting Chapter 7Forecasting with Simple Regression

Chapter TopicsTypes of Regression ModelsDetermining the Simple Linear Regression EquationMeasures of VariationAssumptions of Regression and CorrelationResidual Analysis Measuring AutocorrelationInferences about the Slope

Chapter TopicsCorrelationMeasuring the Strength of the AssociationEstimation of Mean Values and Prediction of Individual ValuesPitfalls in Regression and Ethical Issues(continued)

Purpose of Regression AnalysisRegression Analysis is Used Primarily to Model Causality and Provide PredictionPredict the values of a dependent (response) variable based on values of at least one independent (explanatory) variable.Explain the effect of the independent variables on the dependent variable.

Types of Regression ModelsPositive Linear RelationshipNegative Linear RelationshipRelationship NOT LinearNo Relationship

Sample regression line provides an estimate of the population regression line as well as a predicted value of Y.Linear Regression EquationSample Y InterceptSample Slope CoefficientResidual Simple Regression Equation (Fitted Regression Line, Predicted Value)

Simple Linear Regression: ExampleYou wish to examine the linear dependency of the annual sales of produce stores on their sizes in square footage. Sample data for seven stores were obtained. Find the equation of the straight line that best fits the data.Annual Store Square Sales Feet($1000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760

Scatter Diagram: ExampleExcel Output

Chart6

3681

3395

6653

9543

3318

5563

3760

Annual Sales ($000)

Square Feet

Annual Sales

Sheet1

StoreSquare FeetAnnual Sales ($000)

11,7263,681

21,5423395

32,8166653

45,5559543

51,2923318

62,2085563

71,3133760

Sheet1

Annual Sales ($000)

Square Feet

Annual Sales

Sheet2

Sheet3

Simple Linear Regression Equation: ExampleFrom Excel Printout:

Sheet4

SUMMARY OUTPUT

Regression Statistics

Multiple R0.9705572038

R Square0.9419812858

Adjusted R Square0.930377543

Standard Error611.7515173314

Observations7

ANOVA

dfSSMSFSignificance F

Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478616374239.918957231

Total632251655.7142857

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Intercept1636.4147260759451.49533084953.62443333130.0151488191475.80903014342797.0204220085475.80903014342797.0204220085

Square Footage1.48663365650.16499921239.00994395940.00028120091.06248967861.91077763451.06248967861.9107776345

RESIDUAL OUTPUTPROBABILITY OUTPUT

ObservationPredicted YResidualsPercentileY

14202.3444172716-521.34441727167.14285714293318

23928.8038244674-533.803824467421.42857142863395

35822.775102905830.22489709535.71428571433681

49894.6646881801-351.6646881801503760

53557.1454103313-239.145410331364.28571428575563

64918.901839726644.09816027478.57142857146653

73588.3647171187171.635282881392.85714285719543

Sheet4

1726

1542

2816

5555

1292

2208

1313

X Variable 1

Residuals

X Variable 1 Residual Plot

Sheet1

Sample Percentile

Y

Normal Probability Plot

Sheet2

Sq. Ft spaceSales

11,7263,681

21,5423,395

32,8166,653

45,5559,543

51,2923,318

62,2085,563

71,3133,760

Sheet3

Graph of the Simple Linear Regression Equation: ExampleYi = 1636.415 +1.487Xi

Chart2

3681

3395

6653

9543

3318

5563

3760

Annual Sales ($000)

Square Feet

Annual Sales ($)

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT

ObservationPredicted YResiduals

14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet1


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet1

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet2

Sheet3

Interpretation of Results: ExampleThe slope of 1.487 means that, for each increase of one unit in X, we predict the average of Y to increase by an estimated 1.487 units.The equation estimates that, for each increase of 1 square foot in the size of the store, the expected annual sales are predicted to increase by $1,487.

Simple Linear Regression in ExcelIn Excel, use | Regression EXCEL Spreadsheet of Regression Sales on Square Footage

Scatter

3681

3395

6653

9543

3318

5563

3760

Sales

X

Y

Scatter Diagram

DataCopy

FootageSales(X-XBar)^2

17263681389732.653061224

15423395653325.795918367

28166653216889.795918367

5555954310270193.6530612

129233181119968.65306122

2208556320245.2244897959

131337601075961.65306122

SLR

Regression Analysis



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%

Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536

Footage1.48663365650.16499921239.00994395940.00028120091.06249037151.9107769416

RESIDUAL OUTPUT

ObservationPredicted SalesResiduals

14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

SLR

1726

1542

2816

5555

1292

2208

1313

Footage

Residuals

Footage Residual Plot

Estimate

Confidence Interval Estimate

X Value2000

Confidence Level95%

Sample Size7

Degrees of Freedom5

t Value2.5705776352

Sample Mean2350.2857142857

Sum of Squared Difference13746317.43

Standard Error of the Estimate611.7515173314

h Statistic0.1517831758

Average Predicted Y (YHat)4609.6820391647

For Average Predicted Y (YHat)

Interval Half Width612.6572787957

Confidence Interval Lower Limit3997.024760369

Confidence Interval Upper Limit5222.3393179604

For Individual Response Y


Prediction Interval Lower Limit2921.9979923185

Prediction Interval Upper Limit6297.3660860109

This sheet is generated by:(1)| Regression | Simple Linear Regression (2) checking the "ANOVA and Coefficients Table", and the "Residual Plot" boxes

&A

Page &P

This sheet is generated by:(1) PHStat | Regression | Simple Linear Regression (2) checking the "Confidence and Prediction Intervals for X=" box

data

StoreFootageSales

11,7263,681

21,5423,395

32,8166,653

45,5559,543

51,2923,318

62,2085,563

71,3133,760

Measures of Variation: The Sum of SquaresSST = SSR + SSETotal Sample Variability=Explained Variability+Unexplained Variability

The ANOVA Table in Excel

ANOVAdfSSMSFSignificance FRegressionkSSRMSR=SSR/kMSR/MSEP-value of the F TestResidualsnk1SSEMSE=SSE/(nk1)Totaln1SST

Measures of VariationThe Sum of Squares: ExampleExcel Output for Produce StoresSSRSSERegression (explained) dfDegrees of freedomError (residual) dfTotal dfSST

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130,380,456.1230,380,456.1281.1790901520.0002812009

Residual51,871,199.59374,239.92

Total632,251,655.71


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet1


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet1

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet2

Sheet3

The Coefficient of Determination

Measures the proportion of variation in Y that is explained by the independent variable X in the regression model.

Measures of Variation: Produce Store ExampleExcel Output for Produce Stores r2 = 0.94 94% of the variation in annual sales can be explained by the variability in the size of the store as measured by square footage.Syx

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet1


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet1

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet2

Sheet3

Linear Regression AssumptionsNormalityY values are normally distributed for each X.Probability distribution of error is normal.2.Homoscedasticity (Constant Variance)3.Independence of Errors

Residual AnalysisPurposesExamine linearity Evaluate violations of assumptionsGraphical Analysis of ResidualsPlot residuals vs. X and time

Residual Analysis for LinearityNot LinearLinearX e eXYXYX

Residual Analysis for Homoscedasticity HeteroscedasticityHomoscedasticitySRXSRXYXXY

Residual Analysis: Excel Output for Produce Stores ExampleExcel Output

Chart3

-521.3444172716

-533.8038244674

830.224897095

-351.6646881801

-239.1454103313

644.098160274

171.6352828813

Square Feet

Residual Plot

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet6

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT

ObservationPredicted YResidualsStandard Residuals

14202.3444172716-521.3444172716-0.9335558294

23928.8038244674-533.8038244674-0.9558665166

35822.775102905830.2248970951.486658851

49894.6646881801-351.6646881801-0.6297154218

53557.1454103313-239.1454103313-0.4282305219

64918.901839726644.0981602741.1533672795

73588.3647171187171.63528288130.3073421591

Sheet6

1726

1542

2816

5555

1292

2208

1313

Square Feet

Residuals

Residual Plot

Sheet1

36811726

33951542

66532816

95435555

33181292

55632208

37601313

Y

Predicted Y

X Variable 1

Y

X Variable 1 Line Fit Plot

Sheet2


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet2

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet3

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14,202.34442-$521.34442

23,928.80382-$533.80382

35,822.77510$830.22490

49,894.66469-$351.66469

53,557.14541-$239.14541

64,918.90184$644.09816

73,588.36472$171.63528

Sheet1


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet1

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet2

Sheet3

Residual Analysis for IndependenceThe DurbinWatson StatisticUsed when data are collected over time to detect autocorrelation (residuals in one time period are related to residuals in another period).Measures the violation of the independence assumptionShould be close to 2. If not, examine the model for autocorrelation.

DurbinWatson Statistic in MinitabMinitab | Regression | Simple Linear Regression

Check the box for DurbinWatson Statistic

Obtaining the Critical Values of DurbinWatson StatisticFinding the Critical Values of DurbinWatson Statistic

k=1

k=2

n

dL

dU

dL

dU

15

1.08

1.36

0.95

1.54

16

1.10

1.37

0.98

1.54

_1013327115.unknown

_1044645607.unknown

_1013326860.unknown

Accept H0 (no autocorrelation)Using the DurbinWatson Statistic : No autocorrelation (error terms are independent). : There is autocorrelation (error terms are not independent).042dL4-dLdU4-dUReject H0 (positive autocorrelation)InconclusiveReject H0 (negative autocorrelation)

Residual Analysis for IndependenceNot IndependentIndependenteeTimeTimeResidual is plotted against time to detect any autocorrelationNo Particular PatternCyclical PatternGraphical Approach

Inference about the Slope: t Testt Test for a Population SlopeIs there a linear dependency of Y on X ?Null and Alternative HypothesesH0: = 0(No Linear Dependency)H1: 0(Linear Dependency)Test Statistic

df = n - 2

Example: Produce StoreData for Seven Stores:Estimated Regression Equation:The slope of this model is 1.487. Does Square Footage Affect Annual Sales?Annual Store Square Sales Feet($000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760

Inferences about the Slope: t Test ExampleH0: = 0H1: 0 0.05df 7 - 2 = 5Critical Value(s):Test Statistic: Decision:

Conclusion:

There is evidence that square footage affects annual sales.t02.57062.57060.025RejectReject0.025From Excel Printout Reject H0

SLR

Regression Analysis



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147451.49533.62440.01515475.81092619832797.0185259536

Footage1.48660.16509.00990.000281.06249037151.9107769416

Sheet1

FootageAnnual Sales

11,7263,681

21,5423,395

32,8166,653

45,5559,543

51,2923,318

62,2085,563

71,3133,760

Sheet2

Sheet3

Inferences about the Slope: F TestF Test for a Population SlopeIs there a linear dependency of Y on X ?Null and Alternative HypothesesH0: = 0(No Linear Dependency)H1: 0(Linear Dependency)Test Statistic

Numerator df=1, denominator df=n-2

Relationship between a t Test and an F TestNull and Alternative HypothesesH0: = 0(No Linear Dependency)H1: 0(Linear Dependency)

Inferences about the Slope: F Test ExampleTest Statistic: Decision:Conclusion:

H0: = 0H1: 0 0.05numerator df = 1denominator df 7 2 = 5

There is evidence that square footage affects annual sales.From Excel Printout Reject H006.61Reject = 0.05

Scatter

3681

3395

6653

9543

3318

5563

3760

Sales

X

Y

Scatter Diagram

DataCopy

FootageSales(X-XBar)^2

17263681389732.653061224

15423395653325.795918367

28166653216889.795918367

5555954310270193.6530612

129233181119968.65306122

2208556320245.2244897959

131337601075961.65306122

SLR

Regression Analysis



R Square0.9419812858



Observations7

ANOVA


Regression130,380,456.1230,380,456.1281.1790901520.000281

Residual51,871,199.59374,239.92

Total632,251,655.71


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536

Footage1.48663365650.16499921239.00994395940.00028120091.06249037151.9107769416

RESIDUAL OUTPUT

ObservationPredicted SalesResiduals

14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

SLR

1726

1542

2816

5555

1292

2208

1313

Footage

Residuals

Footage Residual Plot

Estimate

Confidence Interval Estimate

X Value2000

Confidence Level95%

Sample Size7

Degrees of Freedom5

t Value2.5705818347

Sample Mean2350.2857142857

Sum of Squared Difference13746317.43

Standard Error of the Estimate611.7515173314

h Statistic0.1517831758

Average Predicted Y (YHat)4609.6820391647

For Average Predicted Y (YHat)


Confidence Interval Lower Limit3997.0237594833

Confidence Interval Upper Limit5222.3403188462

For Individual Response Y


Prediction Interval Lower Limit2921.9952351833

Prediction Interval Upper Limit6297.3688431462

&A

Page &P

data

StoreFootageSales

11,7263,681

21,5423,395

32,8166,653

45,5559,543

51,2923,318

62,2085,563

71,3133,760

Purpose of Correlation AnalysisPopulation Correlation Coefficient (rho) is used to measure the strength between the variables.Sample Correlation Coefficient r is an estimate of and is used to measure the strength of the linear relationship in the sample observations.(continued)

r = 0.5r = 1Sample of Observations from Various r ValuesYXYXYXYXYXr = 1r = 0.5r = 0

Features of r and rUnit FreeRange between 1 and 1The Closer to 1, the Stronger the Negative Linear Relationship.The Closer to 1, the Stronger the Positive Linear Relationship.The Closer to 0, the Weaker the Linear Relationship.

Pitfalls of Regression AnalysisLacking an Awareness of the Assumptions Underlining Least-squares Regression.Not Knowing How to Evaluate the Assumptions.Not Knowing What the Alternatives to Least-squares Regression are if a Particular Assumption is Violated.Using a Regression Model Without Knowledge of the Subject Matter.

Strategy for Avoiding the Pitfalls of RegressionStart with a scatter plot of X on Y to observe possible relationship.Perform residual analysis to check the assumptions.Use a normal probability plot of the residuals to uncover possible non-normality.

Strategy for Avoiding the Pitfalls of RegressionIf there is violation of any assumption, use alternative methods (e.g., least absolute deviation regression or least median of squares regression) to least-squares regression or alternative least-squares models (e.g., curvilinear or multiple regression).If there is no evidence of assumption violation, then test for the significance of the regression coefficients and construct confidence intervals and prediction intervals.(continued)

Chapter SummaryIntroduced Types of Regression Models.Discussed Determining the Simple Linear Regression Equation.Described Measures of Variation.Addressed Assumptions of Regression and Correlation.Discussed Residual Analysis.Addressed Measuring Autocorrelation.

Chapter SummaryDescribed Inference about the Slope.Discussed CorrelationMeasuring the Strength of the Association.Discussed Pitfalls in Regression and Ethical Issues.

(continued)

Documents

Business Forecasting