Stat 401 B – Lecture 111Multiple Regression A single numerical response variable, Y. Multiple numerical explanatory variables, X1, X2,…, Xk2Simple Linear Regressionεββεμ++=+=xYYxY10|3Multiple Regressionεββββεμ+++++=+=kkxxxYxxxYYk...22110,...,,|21Stat 401 B – Lecture 114Conditions The random error term, , is Independent Identically distributed Normally distributed with standard deviation, .εσ5Example Y, Response – Effectiveness score based on experienced teachers’ evaluations. Explanatory – Test 1, Test 2, Test 3, Test 4.6200300400500600700EVAL0 50 100 150Test1Bivariate Fit of EVAL By Test1Stat 401 B – Lecture 117200300400500600700EVAL110 120 130 140 150 160 170Test2Bivariate Fit of EVAL By Test28200300400500600700EVAL30 40 50 60 70 80 90Test3Bivariate Fit of EVAL By Test39200300400500600700EVAL35 40 45 50 55 60 65 70Test4Bivariate Fit of EVAL By Test4Stat 401 B – Lecture 1110Method of Least Squares Choose estimates of the various parameters in the multiple regression model so that the sum of squared residuals, (SSError), is the smallest it can be.11Method of Least Squares Finding the estimates involves solving k simultaneous equations with k unknowns (the estimates of the parameters). Do this with a statistical analysis computer package, like JMP.12JMP Analyze – Fit Model Pick Role Variables Y – EVAL Construct Model Effects Add – Test1, Test2, Test3, Test4Stat 401 B – Lecture 1113JMP Analyze – Fit Model Personality – Standard Least Squares Emphasis – Minimal Report14RSquareRSquare AdjRoot Mean Square ErrorMean of ResponseObservations (or Sum Wgts)0.8028610.75905237.53627444.478323Summary of FitModelErrorC. TotalSource41822DF103286.2525361.49128647.74Sum ofSquares25821.61409.0Mean Square18.3265F Ratio<.0001*Prob > FAnalysis of VarianceInterceptTest1Test2Test3Test4Term-193.49941.11585392.243267-1.3670016.0482387Estimate125.30740.3197460.6284490.5639651.202281Std Error-1.543.493.57-2.425.03t Ratio0.13990.0026*0.0022*0.0261*<.0001*Prob>|t|Parameter EstimatesResponse EVAL15Prediction Equation Predicted Evaluation = –193.50 + 1.116*Test1 + 2.243*Test2 – 1.367*Test3 +
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