Tbs910 regression models
-
Upload
stephen-ong -
Category
Business
-
view
253 -
download
0
Transcript of Tbs910 regression models
Regression ModelsRegression ModelsRegression ModelsRegression Models
TBS910 BUSINESS ANALYTICSTBS910 BUSINESS ANALYTICS
byProf. Stephen Ong
Visiting Professor, Shenzhen University
Visiting Fellow, Sydney Business School, University of Wollongong
Learning ObjectivesLearning Objectives
1.1. Identify variables and use them in a regression model.Identify variables and use them in a regression model.2.2. Develop simple linear regression equations. from sample data and Develop simple linear regression equations. from sample data and
interpret the slope and intercept.interpret the slope and intercept.3.3. Compute the coefficient of determination and the coefficient of Compute the coefficient of determination and the coefficient of
correlation and interpret their meanings.correlation and interpret their meanings.4.4. Interpret the Interpret the FF-test in a linear regression model.-test in a linear regression model.5.5. List the assumptions used in regression and use residual plots to List the assumptions used in regression and use residual plots to
identify problems.identify problems.6.6. Develop a multiple regression model and use it for prediction Develop a multiple regression model and use it for prediction
purposes.purposes.7.7. Use dummy variables to model categorical data.Use dummy variables to model categorical data.8.8. Determine which variables should be included in a multiple Determine which variables should be included in a multiple
regression model.regression model.9.9. Transform a nonlinear function into a linear one for use in regression.Transform a nonlinear function into a linear one for use in regression.10.10. Understand and avoid common mistakes made in the use of Understand and avoid common mistakes made in the use of
regression analysis.regression analysis.
After completing this lecture, students will be able to:After completing this lecture, students will be able to:
Regression Models : OutlineRegression Models : Outline4.14.1 IntroductionIntroduction4.24.2 Scatter DiagramsScatter Diagrams4.34.3 Simple Linear RegressionSimple Linear Regression4.44.4 Measuring the Fit of the Regression ModelMeasuring the Fit of the Regression Model4.54.5 Using Computer Software for RegressionUsing Computer Software for Regression4.64.6 Assumptions of the Regression ModelAssumptions of the Regression Model4.74.7 Testing the Model for SignificanceTesting the Model for Significance4.84.8 Multiple Regression AnalysisMultiple Regression Analysis4.94.9 Binary or Dummy VariablesBinary or Dummy Variables4.104.10 Model BuildingModel Building4.114.11 Nonlinear Regression Nonlinear Regression 4.124.12 Cautions and Pitfalls in Regression AnalysisCautions and Pitfalls in Regression Analysis
5-5
Regression Regression AnalysisAnalysis
MultipleMultiple
RegressionRegression
MovingAverage
Exponential Smoothing
Trend Projections
Decomposition
Delphi Methods
Jury of Executive Opinion
Sales ForceComposite
Consumer Market Survey
Time-Series Time-Series MethodsMethods
Qualitative Qualitative ModelsModels
Causal Causal MethodsMethods
Forecasting ModelsForecasting ModelsForecasting Forecasting TechniquesTechniques
Figure 5.1
IntroductionIntroduction
Regression analysisRegression analysis is a very valuable tool is a very valuable tool for a manager.for a manager.
Regression can be used to:Regression can be used to: Understand the relationship between variables.Understand the relationship between variables. Predict the value of one variable based on Predict the value of one variable based on
another variable.another variable.
Simple linear regression models have only Simple linear regression models have only two variables.two variables.
Multiple regression models have more Multiple regression models have more variables.variables.
IntroductionIntroduction The variable to be predicted is called The variable to be predicted is called
the the dependent variabledependent variable.. This is sometimes called the This is sometimes called the response response
variable.variable.
The value of this variable depends on The value of this variable depends on the value of the the value of the independent variable.independent variable. This is sometimes called the This is sometimes called the explanatoryexplanatory
or or predictor variable.predictor variable.
Independent Independent variablevariable
Dependent Dependent variablevariable
Independent Independent variablevariable
= +
4-8
Scatter DiagramScatter Diagram A A scatter diagramscatter diagram or or scatter plotscatter plot
is often used to investigate the is often used to investigate the relationship between variables.relationship between variables.
The independent variable is The independent variable is normally plotted on the normally plotted on the XX axis. axis.
The dependent variable is The dependent variable is normally plotted on the normally plotted on the YY axis. axis.
Triple A ConstructionTriple A Construction Triple A Construction renovates old homes.Triple A Construction renovates old homes. Managers have found that the dollar volume Managers have found that the dollar volume
of renovation work is dependent on the area of renovation work is dependent on the area payroll.payroll.
TRIPLE A’S TRIPLE A’S SALESSALES
($100,000s)($100,000s)
LOCAL PAYROLLLOCAL PAYROLL($100,000,000s)($100,000,000s)
66 33
88 44
99 66
55 44
4.54.5 22
9.59.5 55Table 4.1
4-10
Triple A ConstructionTriple A Construction
Figure 4.1
Scatter Diagram of Triple A Construction Company DataScatter Diagram of Triple A Construction Company Data
Simple Linear RegressionSimple Linear Regression
wherewhereYY = dependent variable (response)= dependent variable (response)
XX = independent variable (predictor or explanatory)= independent variable (predictor or explanatory) ββ 00 = intercept (value of = intercept (value of YY when when XX = 0) = 0)
ββ 11 = slope of the regression line = slope of the regression line
εε = random error= random error
Regression models Regression models are used to test if there is are used to test if there is a relationship between variables.a relationship between variables.
There is some There is some random error random error that cannot be that cannot be predicted.predicted.
εββ ++= XY 10
Simple Linear RegressionSimple Linear Regression True values for the slope and intercept True values for the slope and intercept
are not known so they are estimated are not known so they are estimated using sample data.using sample data.
XbbY 10 +=ˆwherewhere
YY = predicted value of = predicted value of YYbb00 = estimate of = estimate of ββ00, based on sample results, based on sample results
bb11 = estimate of = estimate of ββ11, based on sample results, based on sample results
^
Triple A ConstructionTriple A Construction
Triple A Construction is trying to Triple A Construction is trying to predict sales based on area payroll.predict sales based on area payroll.
YY = Sales = SalesXX = Area payroll = Area payroll
The line chosen in Figure 4.1 is the one The line chosen in Figure 4.1 is the one that minimizes the errors.that minimizes the errors.
Error = (Actual value) – (Predicted value)Error = (Actual value) – (Predicted value)
YYe ˆ−=
Triple A ConstructionTriple A ConstructionFor the simple linear regression model, the For the simple linear regression model, the values of the intercept and slope can be values of the intercept and slope can be calculated using the formulas below.calculated using the formulas below.
XbbY 10 +=ˆ
values of (mean) average Xn
XX == ∑
values of (mean) average Yn
YY == ∑
∑∑
−−−
= 21 )(
))((
XX
YYXXb
XbYb 10 −=
Triple A ConstructionTriple A Construction
YY XX ((XX – – XX))22 ((XX – – XX)()(YY – – YY))
66 33 (3 – 4)(3 – 4)22 = 1 = 1 (3 – 4)(6 – 7) = 1(3 – 4)(6 – 7) = 1
88 44 (4 – 4)(4 – 4)22 = 0 = 0 (4 – 4)(8 – 7) = 0(4 – 4)(8 – 7) = 0
99 66 (6 – 4)(6 – 4)22 = 4 = 4 (6 – 4)(9 – 7) = 4(6 – 4)(9 – 7) = 4
55 44 (4 – 4)(4 – 4)22 = 0 = 0 (4 – 4)(5 – 7) = 0(4 – 4)(5 – 7) = 0
4.54.5 22 (2 – 4)(2 – 4)22 = 4 = 4 (2 – 4)(4.5 – 7) = 5(2 – 4)(4.5 – 7) = 5
9.59.5 55 (5 – 4)(5 – 4)22 = 1 = 1 (5 – 4)(9.5 – 7) = 2.5(5 – 4)(9.5 – 7) = 2.5
ΣΣYY = 42= 42YY = 42/6 = 7 = 42/6 = 7
ΣΣXX = 24= 24XX = 24/6 = 4 = 24/6 = 4
ΣΣ((XX – – XX))22 = 10= 10 ΣΣ((XX – – XX)()(YY – – YY) ) = 12.5= 12.5
Regression calculations for Triple A Regression calculations for Triple A ConstructionConstruction
Triple A ConstructionTriple A Construction
46
246
=== ∑ XX
7642
6=== ∑Y
Y
25110
51221 .
.)(
))((==
−−−
= ∑∑
XX
YYXXb
24251710 =−=−= ))(.(XbYb
Regression calculationsRegression calculations
XY 2512 .ˆ +=ThereforeTherefore
Triple A ConstructionTriple A Construction
46
246
=== ∑ XX
7642
6=== ∑Y
Y
25110
51221 .
.)(
))((==
−−−
= ∑∑
XX
YYXXb
24251710 =−=−= ))(.(XbYb
Regression calculationsRegression calculations
XY 2512 .ˆ +=Therefore
sales = 2 + 1.25(payroll)sales = 2 + 1.25(payroll)If the payroll next year is If the payroll next year is $600 million$600 million
000950 $ or 5962512 ,.)(.ˆ =+=Y
Measuring the Fit Measuring the Fit of the Regression Modelof the Regression Model
Regression models can be developed for Regression models can be developed for any variables any variables XX and and Y.Y.
How do we know the model is actually How do we know the model is actually helpful in predicting helpful in predicting YY based on based on XX?? We could just take the average error, but the positive and We could just take the average error, but the positive and
negative errors would cancel each other out.negative errors would cancel each other out.
Three measures of variability are:Three measures of variability are: SSTSST – Total variability about the mean.– Total variability about the mean. SSESSE – Variability about the regression line.– Variability about the regression line. SSRSSR – Total variability that is explained by the model.– Total variability that is explained by the model.
Measuring the Fit Measuring the Fit of the Regression Modelof the Regression Model
Sum of the squares total Sum of the squares total ::2)(∑ −= YYSST
Sum of the squared errorSum of the squared error::∑ ∑ −== 22 )ˆ( YYeSSE
Sum of squares due to regressionSum of squares due to regression::∑ −= 2)ˆ( YYSSR
SSESSRSST +=
Measuring the Fit Measuring the Fit of the Regression Modelof the Regression Model
YY XX ((YY – – YY))22 YY ((YY – – YY))22 ((YY – – YY))22
66 33 (6 – 7)(6 – 7)22 = 1 = 1 2 + 1.25(3) = 5.752 + 1.25(3) = 5.75 0.06250.0625 1.5631.563
88 44 (8 – 7)(8 – 7)22 = 1 = 1 2 + 1.25(4) = 7.002 + 1.25(4) = 7.00 11 00
99 66 (9 – 7)(9 – 7)22 = 4 = 4 2 + 1.25(6) = 9.502 + 1.25(6) = 9.50 0.250.25 6.256.25
55 44 (5 – 7)(5 – 7)22 = 4 = 4 2 + 1.25(4) = 7.002 + 1.25(4) = 7.00 44 00
4.54.5 22 (4.5 – 7)(4.5 – 7)22 = 6.25 = 6.25 2 + 1.25(2) = 4.502 + 1.25(2) = 4.50 00 6.256.25
9.59.5 55 (9.5 – 7)(9.5 – 7)22 = 6.25 = 6.25 2 + 1.25(5) = 8.252 + 1.25(5) = 8.25 1.56251.5625 1.5631.563
∑∑((YY – – YY))22 = 22.5 = 22.5 ∑∑((YY – – YY))22 = 6.875= 6.875 ∑∑((YY – – YY))22 = = 15.62515.625
YY = 7 = 7 SSTSST = 22.5 = 22.5 SSESSE = 6.875= 6.875 SSRSSR = 15.625 = 15.625
^
^^
^^
Table 4.3
Sum of Squares for Triple A ConstructionSum of Squares for Triple A Construction
Sum of the squares total
2)(∑ −= YYSST
Sum of the squared error
∑ ∑ −== 22 )ˆ( YYeSSE
Sum of squares due to regression
∑ −= 2)ˆ( YYSSR
An important relationship
SSESSRSST +=
Measuring the Fit Measuring the Fit of the Regression Modelof the Regression Model
For Triple A ConstructionFor Triple A Construction
SSTSST = 22.5 = 22.5SSESSE = 6.875 = 6.875SSRSSR = 15.625 = 15.625
Measuring the Fit Measuring the Fit of the Regression Modelof the Regression Model
Figure 4.2
Deviations from the Regression Line and from the MeanDeviations from the Regression Line and from the Mean
Coefficient of DeterminationCoefficient of Determination
The proportion of the variability in The proportion of the variability in YY explained by explained by the regression equation is called the the regression equation is called the coefficient coefficient of determination.of determination.
The coefficient of determination is The coefficient of determination is rr22..
SSTSSE
SSTSSR
r −== 12
69440522
625152 ... ==r
About 69% of the variability in About 69% of the variability in YY is explained by is explained by the equation based on payroll (the equation based on payroll (XX).).
4-24
Correlation CoefficientCorrelation Coefficient The correlation coefficient is an expression of the
strength of the linear relationship. It will always be between +1 and –1. The correlation coefficient is r.
2rr = For Triple A Construction:For Triple A Construction:
8333069440 .. ==r
Four Values of the Four Values of the Correlation CoefficientCorrelation Coefficient
**
*
*(a)(a) Perfect PositivePerfect Positive
Correlation: Correlation: rr = +1 = +1
X
Y
*
* *
*
(c)(c) No No Correlation: Correlation: rr = 0 = 0
X
Y
* **
** *
* ***
(d)(d)Perfect Perfect Negative Negative Correlation: Correlation: rr = = ––11
X
Y
**
**
* ***
*(b)(b)PositivePositive
Correlation: Correlation: 0 < 0 < rr < 1 < 1
X
Y
****
*
**
Figure 4.3
4-26
Using Computer Software for Using Computer Software for RegressionRegression
Program 4.1A
Accessing the Regression Option in Excel 2010Accessing the Regression Option in Excel 2010
Using Computer Software for Using Computer Software for RegressionRegression
Program 4.1B
Data Input for Regression in ExcelData Input for Regression in Excel
4-28
Using Computer Software for Using Computer Software for RegressionRegression
Program 4.1C
Excel Output for the Triple A Construction ExampleExcel Output for the Triple A Construction Example
4-29
Assumptions of the Assumptions of the Regression ModelRegression Model
1.1. Errors are independent.Errors are independent.
2.2. Errors are normally distributed.Errors are normally distributed.
3.3. Errors have a mean of zero.Errors have a mean of zero.
4.4. Errors have a constant variance.Errors have a constant variance.
If we make certain assumptions about the errors in a If we make certain assumptions about the errors in a regression model, we can perform statistical tests to regression model, we can perform statistical tests to determine if the model is useful.determine if the model is useful.
A plot of the residuals (errors) will A plot of the residuals (errors) will often highlight any glaring violations often highlight any glaring violations of the assumption.of the assumption.
Residual Plots Residual Plots Pattern of Errors Indicating RandomnessPattern of Errors Indicating Randomness
Figure 4.4A
Err
or
X
4-31
Residual Plots Residual Plots
Nonconstant error varianceNonconstant error variance
Figure 4.4B
Err
or
X
4-32
Residual Plots Residual Plots Errors Indicate Relationship is not LinearErrors Indicate Relationship is not Linear
Figure 4.4C
Err
or
X
Estimating the VarianceEstimating the Variance
Errors are assumed to have a constant Errors are assumed to have a constant variance (variance (σσ 22), but we usually don’t know ), but we usually don’t know this.this.
It can be estimated using the It can be estimated using the mean mean squared errorsquared error ( (MSEMSE), ), ss2.2.
12
−−==
knSSE
MSEs
wherewherenn = number of observations in the sample = number of observations in the samplekk = number of independent variables = number of independent variables
4-34
Estimating the VarianceEstimating the Variance
For Triple A Construction:For Triple A Construction:
718814
87506116
875061
2 ... ==
−−=
−−==
knSSE
MSEs
We can estimate the standard deviation, We can estimate the standard deviation, s.s. This is also called the This is also called the standard error of the standard error of the
estimateestimate or the or the standard deviation of the standard deviation of the regression.regression.
31171881 .. === MSEs
4-35
Testing the Model for Testing the Model for SignificanceSignificance
When the sample size is too small, you When the sample size is too small, you can get good values for can get good values for MSEMSE and and rr22 even if there is no relationship between even if there is no relationship between the variables.the variables.
Testing the model for significance Testing the model for significance helps determine if the values are helps determine if the values are meaningful.meaningful.
We do this by performing a statistical We do this by performing a statistical hypothesis test.hypothesis test.
Testing the Model for SignificanceTesting the Model for Significance
We start with the general linear We start with the general linear modelmodel
εββ ++= XY 10
If If ββ 11 = 0, the null hypothesis is that there is = 0, the null hypothesis is that there is nono relationship between relationship between XX and and Y.Y.
The alternate hypothesis is that there The alternate hypothesis is that there isis a a linear relationship (linear relationship (ββ 11 ≠ 0).≠ 0).
If the null hypothesis can be rejected, we If the null hypothesis can be rejected, we have proven there is a relationship.have proven there is a relationship.
We use the We use the FF statistic for this test. statistic for this test.
4-37
Testing the Model for Testing the Model for SignificanceSignificance
The The FF statistic is based on the statistic is based on the MSEMSE and and MSR:MSR:
kSSR
MSR =
wherewherekk = = number of independent variables in the modelnumber of independent variables in the model
The The FF statistic is: statistic is:
MSEMSR
F = This describes an This describes an FF distribution with: distribution with:
degrees of freedom for the numerator = degrees of freedom for the numerator = dfdf11 = = kk
degrees of freedom for the denominator = degrees of freedom for the denominator = dfdf22 = = nn – – kk – 1 – 1
4-38
Testing the Model for SignificanceTesting the Model for Significance
If there is very little error, the If there is very little error, the MSEMSE would be would be small and the small and the FF--statistic would be large statistic would be large indicating the model is useful.indicating the model is useful.
If the If the FF-statistic is large-statistic is large, the significance , the significance level (level (pp-value) will be low, indicating it is -value) will be low, indicating it is unlikely this would have occurred by unlikely this would have occurred by chance.chance.
So when the So when the FF--value is large, value is large, we can reject we can reject the null hypothesisthe null hypothesis and accept that there is a and accept that there is a linear relationship between linear relationship between XX and and YY and the and the values of the values of the MSEMSE and and rr22 are meaningful. are meaningful.
Steps in a Hypothesis TestSteps in a Hypothesis Test
1.1. Specify null and alternative Specify null and alternative hypotheses:hypotheses: 010 =β:H
011 ≠β:H2.2. Select the level of significance (Select the level of significance (αα ). ).
Common values are 0.01 and 0.05.Common values are 0.01 and 0.05.3.3. Calculate the value of the test statistic Calculate the value of the test statistic
using the formula:using the formula:
MSEMSR
F =
Steps in a Hypothesis TestSteps in a Hypothesis Test4.4. Make a decision using one of the Make a decision using one of the
following methods:following methods:a)a) Reject the null hypothesis if the test statistic is greater than Reject the null hypothesis if the test statistic is greater than
the the FF-value from the table in Appendix D. Otherwise, do not -value from the table in Appendix D. Otherwise, do not reject the null hypothesis:reject the null hypothesis:
21 ifReject dfdfcalculated FF ,,α>
kdf =1
12 −−= kndf
b)b) Reject the null hypothesis if the observed significance Reject the null hypothesis if the observed significance level, or level, or pp-value, is less than the level of significance -value, is less than the level of significance ((αα ). Otherwise, do not reject the null hypothesis:). Otherwise, do not reject the null hypothesis:
)( statistictest calculatedvalue- >= FPp
α<value- ifReject p
Triple A ConstructionTriple A Construction
Step 1.Step 1.HH00: : ββ 11 = 0 = 0 (no linear relationship (no linear relationship between between XX and and YY))HH11: : ββ 11 ≠ 0 ≠ 0 (linear relationship exists (linear relationship exists between between XX and and YY))Step 2.Step 2.
Select Select αα = 0.05 = 0.05
6250151625015
.. ===
kSSR
MSR
09971881625015
... ===
MSEMSR
F
Step 3.Step 3.Calculate the value of the test statistic.
Triple A ConstructionTriple A Construction
Step 4.Step 4.Reject the null hypothesis if the test statistic is greater than the F-value in Appendix D.
dfdf11 = = kk = 1 = 1
dfdf22 = = nn – – kk – 1 = 6 – 1 – 1 = 4 – 1 = 6 – 1 – 1 = 4
The value of The value of FF associated with a 5% level of associated with a 5% level of significance and with degrees of freedom 1 and 4 is significance and with degrees of freedom 1 and 4 is found in Appendix D.found in Appendix D.
FF0.05,1,40.05,1,4 = 7.71 = 7.71
FFcalculatedcalculated = 9.09 = 9.09
Reject Reject HH00 because 9.09 > 7.71 because 9.09 > 7.71
F = 7.71
0.05
9.09
Triple A ConstructionTriple A Construction
Figure 4.5
We can conclude there is a We can conclude there is a statistically significant statistically significant relationship relationship between between XX and and Y.Y.
The The rr22 value of 0.69 means about value of 0.69 means about 69% of the variability in sales (69% of the variability in sales (YY) ) is explained by local payroll (is explained by local payroll (XX).).
4-44
Analysis of Variance Analysis of Variance (ANOVA) Table(ANOVA) Table
When software is used to develop a regression When software is used to develop a regression model, an model, an ANOVA table ANOVA table is typically created that is typically created that shows the observed significance level (shows the observed significance level (pp-value) for -value) for the calculated the calculated FF value. value.
This can be compared to the level of significance This can be compared to the level of significance ((αα ) to make a decision.) to make a decision.
DFDF SSSS MSMS FF SIGNIFICANCESIGNIFICANCE
RegressionRegression kk SSRSSR MSRMSR = = SSRSSR//kk MSRMSR//MSEMSE PP((FF > > MSRMSR//MSEMSE))
ResidualResidual nn - - kk - 1 - 1 SSESSE MSEMSE = = SSESSE//((nn - - kk - 1) - 1)
TotalTotal nn - 1 - 1 SSTSST
Table 4.4
4-45
ANOVA for Triple A ConstructionANOVA for Triple A Construction
Because this probability is less than 0.05, we reject Because this probability is less than 0.05, we reject the null hypothesis of no linear relationship and the null hypothesis of no linear relationship and conclude there is a linear relationship between conclude there is a linear relationship between XX and and Y.Y.
Program 4.1C (partial)
PP((FF > 9.0909) = 0.0394 > 9.0909) = 0.0394
4-46
Multiple Regression AnalysisMultiple Regression Analysis Multiple regression models are extensions
to the simple linear model and allow the creation of models with more than one independent variable.
YY = = ββ 00 + + ββ 11XX11 + + ββ 22XX22 + … + + … + ββ kkXXkk + + εεwherewhere
YY = = dependent variable (response variable)dependent variable (response variable)XXii = = iithth independent variable (predictor or explanatory independent variable (predictor or explanatory variable)variable)ββ 00 = = intercept (value of intercept (value of YY when all when all XXii = 0)= 0)
ββ ii = = coefficient of the coefficient of the iithth independent variable independent variable
kk = = number of independent variablesnumber of independent variablesεε == random errorrandom error
Multiple Regression AnalysisMultiple Regression Analysis
To estimate these values, a sample is To estimate these values, a sample is taken the following equation developedtaken the following equation developed
kk XbXbXbbY ++++= ...ˆ22110
wherewhere == predicted value of predicted value of YYbb00 = = sample intercept (and is an estimate of sample intercept (and is an estimate of ββ 00))
bbii == sample coefficient of the sample coefficient of the iithth variable (and variable (and is an estimate of is an estimate of ββ ii))
Y
4-48
Jenny Wilson RealtyJenny Wilson RealtyJenny Wilson wants to develop a model to Jenny Wilson wants to develop a model to determine the suggested listing price for determine the suggested listing price for houses based on the size and age of the houses based on the size and age of the house.house.
22110ˆ XbXbbY ++=
wherewhere == predicted value of dependent variable predicted value of dependent variable (selling price)(selling price)bb00 = = YY intercept intercept
XX11 and and XX22 = = value of the two independent value of the two independent variables (square footage and age) respectivelyvariables (square footage and age) respectivelybb1 1 andand bb22 = =slopes for slopes for XX11 and and XX22 respectively respectively
Y
She selects a sample of houses that have sold She selects a sample of houses that have sold recently and records the data shown in Table 4.5recently and records the data shown in Table 4.5
4-49
Jenny Wilson Real Estate DataJenny Wilson Real Estate DataSELLING SELLING PRICE ($)PRICE ($)
SQUARE SQUARE FOOTAGEFOOTAGE AGEAGE CONDITIONCONDITION
95,00095,000 1,9261,926 3030 GoodGood
119,000119,000 2,0692,069 4040 ExcellentExcellent
124,800124,800 1,7201,720 3030 ExcellentExcellent
135,000135,000 1,3961,396 1515 GoodGood
142,000142,000 1,7061,706 3232 MintMint
145,000145,000 1,8471,847 3838 MintMint
159,000159,000 1,9501,950 2727 MintMint
165,000165,000 2,3232,323 3030 ExcellentExcellent
182,000182,000 2,2852,285 2626 MintMint
183,000183,000 3,7523,752 3535 GoodGood
200,000200,000 2,3002,300 1818 GoodGood
211,000211,000 2,5252,525 1717 GoodGood
215,000215,000 3,8003,800 4040 ExcellentExcellent
219,000219,000 1,7401,740 1212 MintMintTable 4.5
4-50
Jenny Wilson RealtyJenny Wilson Realty
Program 4.2A
Input Screen for the Jenny Wilson Input Screen for the Jenny Wilson Realty Multiple Regression ExampleRealty Multiple Regression Example
4-51
Jenny Wilson RealtyJenny Wilson Realty
Program 4.2B
Output for the Jenny Wilson Realty Multiple Output for the Jenny Wilson Realty Multiple Regression ExampleRegression Example
Evaluating Multiple Evaluating Multiple Regression ModelsRegression Models
Evaluation is similar to simple linear Evaluation is similar to simple linear regression models.regression models. The The pp-value for the -value for the FF-test and -test and rr22 are are
interpreted the same.interpreted the same. The hypothesis is different because there is The hypothesis is different because there is
more than one independent variable.more than one independent variable. The The FF-test is investigating whether all -test is investigating whether all
the coefficients are equal to 0 at the same the coefficients are equal to 0 at the same time.time.
Evaluating Multiple Evaluating Multiple Regression ModelsRegression Models
To determine which independent To determine which independent variables are significant, tests are variables are significant, tests are performed for each variable.performed for each variable.
010 =β:H011 ≠β:H
The test statistic is calculated and if the The test statistic is calculated and if the pp-value is lower than the level of -value is lower than the level of significance (significance (αα ), the null hypothesis is ), the null hypothesis is rejected.rejected.
4-54
Jenny Wilson RealtyJenny Wilson Realty The model is statistically significantThe model is statistically significant
The The pp-value for the -value for the FF-test is 0.002.-test is 0.002. rr22 = 0.6719 so the model explains about 67% of the = 0.6719 so the model explains about 67% of the
variation in selling price (variation in selling price (YY).). But the But the FF-test is for the entire model and we can’t tell if -test is for the entire model and we can’t tell if
one or both of the independent variables are significant.one or both of the independent variables are significant. By calculating the By calculating the pp-value of each variable, we can -value of each variable, we can
assess the significance of the individual variables.assess the significance of the individual variables. Since the p-value for Since the p-value for XX11 (square footage) and (square footage) and X X22 (age) (age)
are both less than the significance level of 0.05, both are both less than the significance level of 0.05, both null hypotheses can be rejected.null hypotheses can be rejected.
Binary or Dummy VariablesBinary or Dummy Variables BinaryBinary (or (or dummydummy or or indicatorindicator) )
variables are special variables variables are special variables created for qualitative data.created for qualitative data.
A dummy variable is assigned a A dummy variable is assigned a value of 1 if a particular condition value of 1 if a particular condition is met and a value of 0 otherwise.is met and a value of 0 otherwise.
The number of dummy variables The number of dummy variables must equal one less than the must equal one less than the number of categories of the number of categories of the qualitative variable.qualitative variable.
4-56
Jenny Wilson RealtyJenny Wilson Realty Jenny believes a better model can be Jenny believes a better model can be
developed if she includes information developed if she includes information about the condition of the property.about the condition of the property.
XX33 = 1 if house is in excellent condition= 1 if house is in excellent condition
= 0 otherwise= 0 otherwiseXX44 = 1 if house is in mint condition= 1 if house is in mint condition
= 0 otherwise= 0 otherwise Two dummy variables are used to describe the Two dummy variables are used to describe the
three categories of condition.three categories of condition. No variable is needed for “good” condition No variable is needed for “good” condition
since if both since if both XX33 and and XX44 = 0, the house must be in = 0, the house must be in good condition.good condition.
4-57
Jenny Wilson RealtyJenny Wilson Realty
Program 4.3A
Input Screen for the Jenny Wilson Realty Input Screen for the Jenny Wilson Realty Example with Dummy VariablesExample with Dummy Variables
4-58
Jenny Wilson RealtyJenny Wilson Realty
Program 4.3B
Output for the Jenny Wilson Realty Example Output for the Jenny Wilson Realty Example with Dummy Variableswith Dummy Variables
Model BuildingModel Building The best model is a statistically The best model is a statistically
significant model with a high significant model with a high rr22 and and few variables.few variables.
As more variables are added to the As more variables are added to the model, the model, the rr22-value usually increases.-value usually increases.
For this reason, the For this reason, the adjusted adjusted rr22 value value is often used to determine the is often used to determine the usefulness of an additional variable.usefulness of an additional variable.
The adjusted The adjusted rr22 takes into account the takes into account the number of independent variables in number of independent variables in the model.the model.
Model BuildingModel Building
SSTSSE
SSTSSR −== 12r
The formula for The formula for rr22
The formula for adjusted The formula for adjusted rr22
)/(SST)/(SSE
11
1 Adjusted 2
−−−−=
nkn
r
As the number of variables increases, the As the number of variables increases, the adjusted adjusted rr22 gets smaller unless the increase gets smaller unless the increase due to the new variable is large enough to due to the new variable is large enough to offset the change in offset the change in k.k.
Model BuildingModel Building In general, if a new variable increases the In general, if a new variable increases the
adjusted adjusted rr22, it should probably be included in the , it should probably be included in the model.model.
In some cases, variables contain duplicate In some cases, variables contain duplicate information.information.
When two independent variables are correlated, When two independent variables are correlated, they are said to be they are said to be collinear.collinear.
When more than two independent variables are When more than two independent variables are correlated, correlated, multicollinearitymulticollinearity exists. exists.
When multicollinearity is present, When multicollinearity is present, hypothesis hypothesis tests for the individual coefficients are not valid tests for the individual coefficients are not valid but the model may still be useful.but the model may still be useful.
4-62
Nonlinear RegressionNonlinear Regression In some situations, variables are not In some situations, variables are not
linear.linear. Transformations may be used to turn Transformations may be used to turn
a nonlinear model into a linear model.a nonlinear model into a linear model.
** **
** ** *
Linear relationshipLinear relationship Nonlinear relationshipNonlinear relationship
**** **
****
*
4-63
Colonel MotorsColonel Motors Engineers at Colonel Motors want to use
regression analysis to improve fuel efficiency. They have been asked to study the impact of
weight on miles per gallon (MPG).
MPGMPG
WEIGHT WEIGHT (1,000 (1,000 LBS.)LBS.) MPGMPG
WEIGHT WEIGHT (1,000 (1,000 LBS.)LBS.)
1212 4.584.58 2020 3.183.18
1313 4.664.66 2323 2.682.68
1515 4.024.02 2424 2.652.65
1818 2.532.53 3333 1.701.70
1919 3.093.09 3636 1.951.95
1919 3.113.11 4242 1.921.92
Table 4.6
4-65
Colonel MotorsColonel Motors
Program 4.4
This is a useful model with a small This is a useful model with a small FF-test -test for significance and a good for significance and a good rr22 value. value.
Excel Output for Linear Regression Excel Output for Linear Regression Model with MPG DataModel with MPG Data
4-66
Colonel MotorsColonel Motors
Figure 4.6B
Nonlinear Model for MPG DataNonlinear Model for MPG Data
4-67
Colonel MotorsColonel Motors
The nonlinear model is a quadratic model.The nonlinear model is a quadratic model. The easiest way to work with this model is The easiest way to work with this model is
to develop a new variable.to develop a new variable.2
2 weight)(=X This gives us a model that can be This gives us a model that can be
solved with linear regression software:solved with linear regression software:
22110 XbXbbY ++=ˆ
4-68
Colonel MotorsColonel Motors
Program 4.5
A better model with a smaller A better model with a smaller FF-test for -test for significance and a larger adjusted significance and a larger adjusted rr22 value value
21 43230879 XXY ...ˆ +−=
Cautions and PitfallsCautions and Pitfalls If the assumptions are not met, the If the assumptions are not met, the
statistical test may not be valid.statistical test may not be valid. Correlation does not necessarily mean Correlation does not necessarily mean
causation.causation. Multicollinearity makes interpreting Multicollinearity makes interpreting
coefficients problematic, but the model may coefficients problematic, but the model may still be good.still be good.
Using a regression model beyond the range Using a regression model beyond the range of of XX is questionable, as the relationship may is questionable, as the relationship may not hold outside the sample data.not hold outside the sample data.
Cautions and PitfallsCautions and Pitfalls AA t t-test for the intercept (-test for the intercept (bb00) may be ignored ) may be ignored
as this point is often outside the range of as this point is often outside the range of the model.the model.
A linear relationship may not be the best A linear relationship may not be the best relationship, even if the relationship, even if the FF-test returns an -test returns an acceptable value.acceptable value.
A nonlinear relationship can exist even if a A nonlinear relationship can exist even if a linear relationship does not.linear relationship does not.
Even though a relationship is statistically Even though a relationship is statistically significant it may not have any practical significant it may not have any practical value.value.
Further ReadingFurther Reading
Render, B., Stair Jr.,R.M. & Hanna, M.E. (2013) Quantitative Analysis for Management, Pearson, 11th Edition
Waters, Donald (2007) Quantitative Methods for Business, Prentice Hall, 4 th Edition.
Anderson D, Sweeney D, & Williams T. (2006) Quantitative Methods For Business Thompson Higher Education, 10th Ed.