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StatisticsStatistics
Multiple Regression and Model Multiple Regression and Model BuildingBuilding
Chapter 12 part IChapter 12 part I
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Learning ObjectivesLearning Objectives
1.1. Explain the Linear Multiple Regression Explain the Linear Multiple Regression ModelModel
2.2. Explain Residual AnalysisExplain Residual Analysis
3.3. Test Overall SignificanceTest Overall Significance
4.4. Explain MulticollinearityExplain Multicollinearity
5.5. Interpret Linear Multiple Regression Interpret Linear Multiple Regression Computer OutputComputer Output
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Types of Types of Regression ModelsRegression Models
RegressionModels
LinearNon-
Linear
2+ ExplanatoryVariables
Simple
Non-Linear
Multiple
Linear
1 ExplanatoryVariable
RegressionModels
LinearNon-
Linear
2+ ExplanatoryVariables
Simple
Non-Linear
Multiple
Linear
1 ExplanatoryVariable
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Regression Modeling Regression Modeling Steps Steps
1.1. Hypothesize Deterministic ComponentHypothesize Deterministic Component
2.2. Estimate Unknown Model ParametersEstimate Unknown Model Parameters
3.3. Specify Probability Distribution of Specify Probability Distribution of Random Error TermRandom Error Term Estimate Standard Deviation of ErrorEstimate Standard Deviation of Error
4.4. Evaluate ModelEvaluate Model
5.5. Use Model for Prediction & Estimation Use Model for Prediction & Estimation
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Regression Modeling Regression Modeling Steps Steps
1.1. Hypothesize Deterministic ComponentHypothesize Deterministic Component
2.2. Estimate Unknown Model ParametersEstimate Unknown Model Parameters
3.3. Specify Probability Distribution of Specify Probability Distribution of Random Error TermRandom Error Term Estimate Standard Deviation of ErrorEstimate Standard Deviation of Error
4.4. Evaluate ModelEvaluate Model
5.5. Use Model for Prediction & Estimation Use Model for Prediction & Estimation
Expanded in
Multiple
RegressionExpanded in
Multiple
Regression
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Linear Multiple Linear Multiple Regression ModelRegression Model
Hypothesizing the Hypothesizing the Deterministic ComponentDeterministic Component
Expanded in
Multiple
RegressionExpanded in
Multiple
Regression
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Regression Modeling Regression Modeling Steps Steps
1.1. Hypothesize Deterministic ComponentHypothesize Deterministic Component
2.2. Estimate Unknown Model ParametersEstimate Unknown Model Parameters
3.3. Specify Probability Distribution of Random Specify Probability Distribution of Random Error TermError Term Estimate Standard Deviation of ErrorEstimate Standard Deviation of Error
4.4. Evaluate ModelEvaluate Model
5.5. Use Model for Prediction & Estimation Use Model for Prediction & Estimation
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Linear Multiple Linear Multiple Regression ModelRegression Model
1.1. Relationship between 1 dependent & Relationship between 1 dependent & 2 or more independent variables is a 2 or more independent variables is a linear functionlinear function
Y X X Xi i i k ki i 0 1 1 2 2 Y X X Xi i i k ki i 0 1 1 2 2
Dependent Dependent (response) (response) variablevariable
Independent Independent (explanatory) (explanatory) variablesvariables
Population Population slopesslopes
Population Population Y-interceptY-intercept
Random Random errorerror
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X2
Y
X1E(Y) = 0 + 1X 1i + 2X 2i
0
Y i = 0 + 1X 1i + 2X 2i + i
ResponsePlane
(X 1i,X 2i)
(Observed Y )
i
X2
Y
X1E(Y) = 0 + 1X 1i + 2X 2i
0
Y i = 0 + 1X 1i + 2X 2i + i
ResponsePlane
(X 1i,X 2i)
(Observed Y )
i
PopulationPopulation Multiple Multiple Regression ModelRegression Model
Bivariate modelBivariate model
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Sample Multiple Sample Multiple Regression ModelRegression Model
X2
Y
X1
0
Y i = 0 + 1X 1i + 2X 2i + i
ResponsePlane
(X 1i,X 2i)
(Observed Y)
^
i
Y i = 0 + 1X 1i + 2X 2i
^^^
^^
^^^
^
X2
Y
X1
0
Y i = 0 + 1X 1i + 2X 2i + i
ResponsePlane
(X 1i,X 2i)
(Observed Y)
^
i
Y i = 0 + 1X 1i + 2X 2i
^^^
^^
^^^
^Bivariate modelBivariate model
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Parameter EstimationParameter Estimation
Expanded in
Multiple
RegressionExpanded in
Multiple
Regression
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Regression Modeling Regression Modeling Steps Steps
1.1. Hypothesize Deterministic ComponentHypothesize Deterministic Component
2.2. Estimate Unknown Model ParametersEstimate Unknown Model Parameters
3.3. Specify Probability Distribution of Specify Probability Distribution of Random Error TermRandom Error Term Estimate Standard Deviation of ErrorEstimate Standard Deviation of Error
4.4. Evaluate ModelEvaluate Model
5.5. Use Model for Prediction & Estimation Use Model for Prediction & Estimation
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Multiple Linear Multiple Linear Regression Regression EquationsEquations
Too complicated
by hand! Ouch!
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Interpretation of Interpretation of Estimated Estimated
CoefficientsCoefficients
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Interpretation of Interpretation of Estimated Estimated
CoefficientsCoefficients1.1. Slope (Slope (kk))
Estimated Estimated YY Changes by Changes by kk for Each 1 for Each 1
Unit Increase in Unit Increase in XXkk Holding All Other Holding All Other
Variables ConstantVariables Constant Example: If Example: If 11 = 2, then Sales ( = 2, then Sales (YY) Is Expected ) Is Expected
to Increase by 2 for Each 1 Unit Increase in to Increase by 2 for Each 1 Unit Increase in Advertising (Advertising (XX11) Given the Number of Sales ) Given the Number of Sales
Rep’s (Rep’s (XX22) )
^̂
^̂
^̂
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Interpretation of Interpretation of Estimated Estimated
CoefficientsCoefficients1.1. Slope (Slope (kk))
Estimated Estimated YY Changes by Changes by kk for Each 1 Unit for Each 1 Unit
Increase in Increase in XXkk Holding All Other Variables Holding All Other Variables
ConstantConstant Example: If Example: If 11 = 2, then Sales ( = 2, then Sales (YY) Is Expected to ) Is Expected to
Increase by 2 for Each 1 Unit Increase in Increase by 2 for Each 1 Unit Increase in Advertising (Advertising (XX11) Given the Number of Sales Rep’s ) Given the Number of Sales Rep’s
((XX22) )
2.2. Y-Intercept (Y-Intercept (00)) Average Value of Average Value of YY When When XXkk = 0 = 0
^̂
^̂
^̂
^̂
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Parameter Parameter Estimation ExampleEstimation Example
You work in advertising for You work in advertising for the New York Times. You the New York Times. You want to find the effect of want to find the effect of ad sizead size (sq. in.) & (sq. in.) & newspaper newspaper circulationcirculation (000) on the number of (000) on the number of ad ad responsesresponses (00). (00).
You’ve collected the You’ve collected the following data:following data:
RespResp SizeSize CircCirc
11 11 2244 88 8811 33 1133 55 7722 66 4444 1010 66
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ParameterParameter Estimation Estimation
Computer OutputComputer Output Parameter EstimatesParameter Estimates
ParameterParameter Standard T for H0: Standard T for H0:Variable Variable DF DF Estimate Estimate Error Param=0 Prob>|T|Error Param=0 Prob>|T|
INTERCEP INTERCEP 1 1 0.0640 0.0640 0.2599 0.246 0.82140.2599 0.246 0.8214
ADSIZE ADSIZE 1 1 0.2049 0.2049 0.0588 3.656 0.03990.0588 3.656 0.0399
CIRC CIRC 1 1 0.2805 0.2805 0.0686 4.089 0.02640.0686 4.089 0.0264
P
2
0
1^̂
^̂
^̂
^̂
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Interpretation of Interpretation of Coefficients SolutionCoefficients Solution
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Interpretation of Interpretation of Coefficients SolutionCoefficients Solution
1.1. Slope (Slope (11)) # Responses to Ad Is Expected to Increase # Responses to Ad Is Expected to Increase
by .2049 (20.49) for Each 1 Sq. In. Increase by .2049 (20.49) for Each 1 Sq. In. Increase in Ad Size in Ad Size Holding Circulation ConstantHolding Circulation Constant
^̂
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Interpretation of Interpretation of Coefficients SolutionCoefficients Solution
1.1. Slope (Slope (11)) # Responses to Ad Is Expected to Increase # Responses to Ad Is Expected to Increase
by .2049 (20.49) for Each 1 Sq. In. Increase in by .2049 (20.49) for Each 1 Sq. In. Increase in Ad Size Ad Size Holding Circulation ConstantHolding Circulation Constant
2.2. Slope (Slope (22)) # Responses to Ad Is Expected to Increase # Responses to Ad Is Expected to Increase
by .2805 (28.05) for Each 1 Unit (1,000) by .2805 (28.05) for Each 1 Unit (1,000) Increase in CirculationIncrease in Circulation Holding Ad Holding Ad Size Size ConstantConstant
^̂
^̂
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Evaluating the ModelEvaluating the Model
Expanded in
Multiple
RegressionExpanded in
Multiple
Regression
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Regression Modeling Regression Modeling Steps Steps
1.1. Hypothesize Deterministic ComponentHypothesize Deterministic Component
2.2. Estimate Unknown Model ParametersEstimate Unknown Model Parameters
3.3. Specify Probability Distribution of Specify Probability Distribution of Random Error TermRandom Error Term Estimate Standard Deviation of ErrorEstimate Standard Deviation of Error
4.4. Evaluate ModelEvaluate Model
5.5. Use Model for Prediction & Estimation Use Model for Prediction & Estimation
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Evaluating Multiple Evaluating Multiple Regression Model Regression Model
StepsSteps
1.1. Examine Variation MeasuresExamine Variation Measures
2.2. Do Residual AnalysisDo Residual Analysis
3.3. Test Parameter SignificanceTest Parameter Significance Overall ModelOverall Model Individual CoefficientsIndividual Coefficients
4.4. Test for MulticollinearityTest for Multicollinearity
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Evaluating Multiple Evaluating Multiple Regression Model Regression Model
StepsSteps
1.1. Examine Variation MeasuresExamine Variation Measures
2.2. Do Residual AnalysisDo Residual Analysis
3.3. Test Parameter SignificanceTest Parameter Significance Overall ModelOverall Model Individual CoefficientsIndividual Coefficients
4.4. Test for MulticollinearityTest for Multicollinearity
NewNew
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Variation MeasuresVariation Measures
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Evaluating Multiple Evaluating Multiple Regression Model Regression Model
StepsSteps
1.1. Examine Variation MeasuresExamine Variation Measures
2.2. Do Residual AnalysisDo Residual Analysis
3.3. Test Parameter SignificanceTest Parameter Significance Overall ModelOverall Model Individual CoefficientsIndividual Coefficients
4.4. Test for MulticollinearityTest for Multicollinearity
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Coefficient of Coefficient of MultipleMultiple
DeterminationDetermination1.1. Proportion of Variation in Proportion of Variation in YY ‘Explained’ by ‘Explained’ by
All All XX Variables Variables Taken TogetherTaken Together
RR22 = = Explained VariationExplained Variation = = SSRSSR
Total VariationTotal Variation SS SSyyyy
2.2. Never Decreases When New Never Decreases When New XX Variable Is Variable Is Added to ModelAdded to Model Only Only YY Values Determine SS Values Determine SSyyyy
Disadvantage When Comparing ModelsDisadvantage When Comparing Models
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Residual AnalysisResidual Analysis
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Evaluating Multiple Evaluating Multiple Regression Model Regression Model
StepsSteps
1.1. Examine Variation MeasuresExamine Variation Measures
2.2. Do Residual AnalysisDo Residual Analysis
3.3. Test Parameter SignificanceTest Parameter Significance Overall ModelOverall Model Individual CoefficientsIndividual Coefficients
4.4. Test for MulticollinearityTest for Multicollinearity
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Residual AnalysisResidual Analysis
1.1. Graphical Analysis of ResidualsGraphical Analysis of Residuals Plot Estimated Errors vs. Plot Estimated Errors vs. XXii Values Values
Difference Between Actual Difference Between Actual YYii & Predicted & Predicted YYii
Estimated Errors Are Called ResidualsEstimated Errors Are Called Residuals Plot Histogram or Stem-&-Leaf of ResidualsPlot Histogram or Stem-&-Leaf of Residuals
2.2. PurposesPurposes Examine Functional Form (Linear vs. Examine Functional Form (Linear vs.
Non-Linear Model)Non-Linear Model) Evaluate Violations of AssumptionsEvaluate Violations of Assumptions
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Linear Regression Linear Regression Assumptions Assumptions
1.1. Mean of Probability Distribution of Error Mean of Probability Distribution of Error Is 0Is 0
2.2. Probability Distribution of Error Has Probability Distribution of Error Has Constant VarianceConstant Variance
3.3. Probability Distribution of Error is Probability Distribution of Error is NormalNormal
4.4. Errors Are IndependentErrors Are Independent
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X
e
Residual Plot Residual Plot for Functional Formfor Functional Form
X
e
Add XAdd X22 Term Term Correct SpecificationCorrect Specification
^ ^
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Residual Plot Residual Plot for Equal Variancefor Equal Variance
X
SR
Unequal VarianceUnequal Variance
X
SR
Correct SpecificationCorrect Specification
Fan-shaped.Fan-shaped.Standardized residuals used typically. Standardized residuals used typically.
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Residual Plot Residual Plot for Independencefor Independence
X
SR
Not IndependentNot Independent
X
SR
Correct SpecificationCorrect Specification
Plots reflect sequence data were collected. Plots reflect sequence data were collected.
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Residual Analysis Residual Analysis Computer OutputComputer Output
Dep Var Predict Dep Var Predict StudentStudent
Obs SALES Value Obs SALES Value Residual Residual -2-1-0 1 2 Residual Residual -2-1-0 1 2
1 1.0000 0.6000 1 1.0000 0.6000 0.4000 1.044 | |** | 0.4000 1.044 | |** |
2 1.0000 1.3000 2 1.0000 1.3000 -0.3000 -0.592 | *| | -0.3000 -0.592 | *| |
3 2.0000 2.00003 2.0000 2.0000 0 0.000 | | | 0 0.000 | | |
4 2.0000 2.70004 2.0000 2.7000 -0.7000 -1.382 | **| | -0.7000 -1.382 | **| |
5 4.0000 3.4000 5 4.0000 3.4000 0.6000 1.567 | |*** | 0.6000 1.567 | |*** |
Plot of standardized (student) residuals
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Testing ParametersTesting Parameters
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Evaluating Multiple Evaluating Multiple Regression Model Regression Model
StepsSteps
1.1. Examine Variation MeasuresExamine Variation Measures
2.2. Do Residual AnalysisDo Residual Analysis
3.3. Test Parameter SignificanceTest Parameter Significance Overall ModelOverall Model Individual CoefficientsIndividual Coefficients
4.4. Test for MulticollinearityTest for Multicollinearity
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Testing Overall Testing Overall SignificanceSignificance
1.1. Shows If There Is a Linear Relationship Shows If There Is a Linear Relationship Between Between AllAll XX Variables Variables TogetherTogether & & YY
2.2. Uses F Test StatisticUses F Test Statistic
3.3. HypothesesHypotheses HH00: : 1 1 = = 22 = ... = = ... = kk = 0 = 0
No Linear RelationshipNo Linear Relationship
HHaa: At Least One Coefficient Is Not 0 : At Least One Coefficient Is Not 0 At Least One At Least One XX Variable Affects Variable Affects YY
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Testing Overall Testing Overall SignificanceSignificance
Computer OutputComputer Output
Analysis of VarianceAnalysis of Variance
Sum of Mean Sum of Mean Source DF Squares Square F Value Prob>FSource DF Squares Square F Value Prob>F
Model 2 9.2497 4.6249 55.440 0.0043Model 2 9.2497 4.6249 55.440 0.0043
Error 3 0.2503 0.0834Error 3 0.2503 0.0834
C Total 5 9.5000C Total 5 9.5000
k n - k -1
n - 1 P-Value
MS(Model) MS(Error)
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MulticollinearityMulticollinearity
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Evaluating Multiple Evaluating Multiple Regression Model Regression Model
StepsSteps
1.1. Examine Variation MeasuresExamine Variation Measures
2.2. Do Residual AnalysisDo Residual Analysis
3.3. Test Parameter SignificanceTest Parameter Significance Overall ModelOverall Model Individual CoefficientsIndividual Coefficients
4.4. Test for MulticollinearityTest for Multicollinearity
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MulticollinearityMulticollinearity
1.1. High Correlation Between High Correlation Between XX Variables Variables
2.2. Coefficients Measure Combined EffectCoefficients Measure Combined Effect
3.3. Leads to Unstable Coefficients Leads to Unstable Coefficients Depending on Depending on XX Variables in Model Variables in Model
4.4. Always Exists -- Matter of DegreeAlways Exists -- Matter of Degree
5.5. Example: Using Both Age & Height as Example: Using Both Age & Height as Explanatory Variables in Same Model Explanatory Variables in Same Model
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Detecting Detecting MulticollinearityMulticollinearity
1.1. Examine Correlation MatrixExamine Correlation Matrix Correlations Between Pairs of Correlations Between Pairs of XX Variables Variables
Are More than With Are More than With YY Variable Variable
2.2. Examine Variance Inflation Factor (VIF)Examine Variance Inflation Factor (VIF) If VIFIf VIFjj > 5, Multicollinearity Exists > 5, Multicollinearity Exists
3.3. Few RemediesFew Remedies Obtain New Sample DataObtain New Sample Data Eliminate One Correlated Eliminate One Correlated X X VariableVariable
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Correlation Matrix Correlation Matrix Computer OutputComputer Output
Correlation AnalysisCorrelation Analysis
Pearson Corr Coeff /Prob>|R| under HO:Rho=0/ N=6Pearson Corr Coeff /Prob>|R| under HO:Rho=0/ N=6
RESPONSE ADSIZE CIRCRESPONSE ADSIZE CIRC
RESPONSE RESPONSE 1.00000 1.00000 0.90932 0.931170.90932 0.93117
0.0 0.0120 0.0069 0.0 0.0120 0.0069
ADSIZE ADSIZE 0.909320.90932 1.00000 1.00000 0.741180.74118
0.0120 0.0 0.0918 0.0120 0.0 0.0918
CIRC CIRC 0.931170.93117 0.741180.74118 1.000001.00000
0.0069 0.0918 0.0 0.0069 0.0918 0.0
rY1 rY2
All 1’sr12
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Variance Inflation Variance Inflation Factors Factors
Computer OutputComputer Output
Parameter Standard T for H0:Parameter Standard T for H0:Variable DF Estimate Error Param=0 Prob>|T|Variable DF Estimate Error Param=0 Prob>|T|
INTERCEP 1 0.0640 0.2599 0.246 0.8214INTERCEP 1 0.0640 0.2599 0.246 0.8214
ADSIZE 1 0.2049 0.0588 3.656 0.0399ADSIZE 1 0.2049 0.0588 3.656 0.0399
CIRC 1 0.2805 0.0686 4.089 0.0264CIRC 1 0.2805 0.0686 4.089 0.0264
VarianceVarianceVariable Variable DF DF InflationInflation
INTERCEP INTERCEP 1 1 0.00000.0000
ADSIZE ADSIZE 1 1 2.2190 2.2190
CIRC CIRC 1 1 2.2190 2.2190
VIF1 5
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Regression CautionsRegression Cautions
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Regression CautionsRegression Cautions
1.1. Violated AssumptionsViolated Assumptions
2.2. Relevancy of Relevancy of Historical DataHistorical Data
3.3. Level of SignificanceLevel of Significance
4.4. ExtrapolationExtrapolation
5.5. Cause & EffectCause & Effect
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YY
InterpolationInterpolation
XX
ExtrapolationExtrapolation ExtrapolationExtrapolation
Relevant RangeRelevant Range
ExtrapolationExtrapolation
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Cause & EffectCause & Effect
Liquor Liquor ConsumptionConsumption
# Teachers# Teachers
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ConclusionConclusion
1.1. Explained the Linear Multiple Regression Explained the Linear Multiple Regression ModelModel
2.2. Explained Residual AnalysisExplained Residual Analysis
3.3. Tested Overall SignificanceTested Overall Significance
4.4. Explained MulticollinearityExplained Multicollinearity
5.5. Interpreted Linear Multiple Regression Interpreted Linear Multiple Regression Computer OutputComputer Output