*Breakthrough Management GroupMULTIPLE REGRESSION天马行空官方博客：http:/t.qq.com/tmxk_docin ；QQ:1318241189；QQ群：175569632* *, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. Module ObjectivesqReview of Multiple RegressionqMultiCollinearity and Related Topics qQualitative Independent VariablesqWeighted Least SquaresqExamples to Demonstrate Concepts天马行空官方博客：http:/t.qq.com/tmxk_docin ；QQ:1318241189；QQ群：1755696322* *, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. Multiple Regression Review - ExampleqWork-related performance may depend upon many variables such as:mSchool historymExtra hours workedmSocial participationmMotivationmAttendanceLet us develop a best fit equation which models “Work Rating” as a linear combination of the three other variables.SchoolWork.mtw3* *, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. Steps In Regression Model DevelopmentqCorrelation MatrixqCorrelation Calculations with P-ValuesqTheoretical ModelqImportant Factor Identification (Stepwise Regression)qResidual AnalysisqTesting4* *, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. Correlation MatrixqComplete the correlation matrix generated by Minitab:mWhich variable correlates highest with Work Rating?mWhich variable correlates least with Work Rating?mWhich two variables show the least correlation?mWhich variable(s) might be used to predict Work Rating?5* *, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. Correlation Values With P-ValuesCorrelations: School History, Hours Extra, Motivation, Work RatingStats Basic Statistics CorrelationSchool H Hours Ex Motivati Hours Ex -0.2380.392Motivati -0.103 0.5440.716 0.036Work Rat 0.313 0.594 0.7680.257 0.020 0.001Cell Contents: Pearson correlation coefficient ()P-Value are associated with H0: =06* *, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. ResultsqMotivation correlates highest with Work Rating (0.768).qSchool History correlates least with Work Rating (0.313).qMotivation and School History show least correlation among the predictor variables.qMotivation and Extra Hours worked might be used to predict Work Rating, however both of these variables are themselves moderately correlated (0.544). Stats Basic Statistics CorrelationSchool Hours MotivHours -0.238 Motiv-0.103 0.544 Work 0.313 0.594 0.7687* *, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. Correlation and CausationqCommon sense as well as statistical reasoning is used to determine if an observed correlation can be practically interpreted or is spurious.qCorrelation does not imply causation.qWe may identify a relationship by observing a process. For example, two variables may tend to increase and decrease together, or the opposite might occur. qThis does not necessarily mean that we can adjust one variable through manipulation of another.8* *, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. Correlation vs. CausationqAny observed association or relationship between two or more variables may be:mReal-one variable directly affects the other.mSpurious-a lurking variable is present.mDue to chance.9* *, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. ErrorqError resulting from using the regression model is given by the statistic: Se = “standard error” of the estimate.mMeasures the “standard” distance data points lie from the regression line.mSe (also known as MSE1/2 )should be as small as possible as a measure of a good fit.qError is part of the following theoretical model:mCoefficients are solved by Least Squares Estimation.10* *, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. Stepwise RegressionqOpen Filename: SchoolWork.mtw Stat Regression Stepwise Response: Work RatingSchoolWork.mtw11* *, Breakthrough Management Group. Unpublished proprietary work available only under license. All rights reserved. Stepwise RegressionAlpha-to-Enter: 0.15 Alpha-to-Remove: 0.15Response is Work Rat on 3 predictors, with N = 15Step 1 2 3 Constant 34.314 9.332 -2.460Motivati 0.58 0.61 0.46 T-Value 4.33 5.52 4.23 P-Value 0.001 0.000 0.001School H 0.218 0.255 T-Value 2.70 3.70 P-Value 0.019 0.004Hours Ex 0.90 T-Value 2.48 P-Value