Stat 401 B – Lecture 131Multiple Regression A single numerical response variable, Y. Multiple numerical explanatory variables, X1, X2,…, Xk2Multiple Regressionεββββεμ+++++=+=kkxxxYxxxYYk...22110,...,,|213Example Y, Response – Effectiveness score based on experienced teachers’ evaluations. Explanatory – Test 1, Test 2, Test 3, Test 4.Stat 401 B – Lecture 134Test of Model Utility Is there any explanatory variable in the model that is helping to explain significant amounts of variation in the response?5Conclusion At least one of the tests is providing statistically significant information about the evaluation score. The model is useful. Maybe not the best, but useful.6Individual Slope Parameters In order to see what tests for the various parameters in multiple regression mean, we need to go back to simple linear regression.Stat 401 B – Lecture 137SLR – EVAL on Test 1 Predicted Eval = 329.23 + 1.424*Test1 For each additional point scored on Test 1, the Evaluation score increases by 1.424 points, on average.8Explained Variation R2= 0.295, only 29.5% of the variation in Evaluation is explained by the linear relationship with Test 1.9Inference on t-Ratio = 2.97 P-value = 0.0074 Reject the null hypothesis that=0, because the P-value is so small. There is a statistically significant linear relationship between EVAL and Test 1.1β1βStat 401 B – Lecture 1310Model with Test 1 If Test 1 is the only explanatory variable in the model, then the other tests are ignored by this model. What happens if we add Test 2 to the model with Test 1?11RSquareRSquare AdjRoot Mean Square ErrorMean of ResponseObservations (or Sum Wgts)0.3673550.3040963.79201444.478323Summary of FitModelErrorC. TotalSource22022DF47259.3481388.40128647.74Sum ofSquares23629.74069.4Mean Square5.8066F Ratio0.0103*Prob > FAnalysis of VarianceInterceptTest1Test2Term129.376391.22146251.5114559Estimate138.34520.4851811.00162Std Error0.942.521.51t Ratio0.36090.0205*0.1469Prob>|t|Parameter EstimatesResponse EVAL12Model with Test 1, Test 2 Predicted EVAL = 129.38 + 1.221*Test1 + 1.511*Test2 For each additional point on Test 1, while holding Test 2 constant, the Evaluation score increases by 1.221 points, on average.Stat 401 B – Lecture 1313Model with Test 1, Test 2 Predicted EVAL = 129.38 + 1.221*Test1 + 1.511*Test2 For each additional point on Test 2, while holding Test 1 constant, the Evaluation score increases by 1.511 points, on average.14Explained Variation R2= 0.367, 36.7% of the variation in Evaluation is explained by the linear relationship with Test 1 and Test 2.15Explained Variation R2= 0.367 – Test 1 and Test 2. R2= 0.295 – Test 1 alone. 0.367 – 0.295 = 0.072, 7.2% of the variation in EVAL is explained by the addition of Test 2 to Test 1.Stat 401 B – Lecture 1316Parameter Estimates – Test 2 Has Test 2 added significantly to the relationship between Test 1 and Evaluation? Note that this is different from asking if Test 2 is linearly related to Evaluation!17Parameter Estimates – Test 2 t-Ratio = 1.51 P-value = 0.1469 Because the P-value is not small, Test 2’s addition to the model with Test 1 is not statistically significant.18Parameter Estimates – Test 2 Although R2has increased by adding Test 2, that increase could have happened just by chance. The increase is not large enough to be deemed statistically significant.Stat 401 B – Lecture 1319Parameter Estimates – Test 1 Does Test 1 add significantly to the relationship between Test 2 and Evaluation? Note that this is different from asking if Test 1 is linearly related to Evaluation!20Parameter Estimates – Test 1 t-Ratio = 2.52 P-value = 0.0205 Because the P-value is small, Test 1’s addition to the model with Test 2 is statistically significant.21Parameter Estimates – Test 1 If we had started with a model relating Test 2 to EVAL, adding Test 1 would result in an increase in R2. That increase is large enough to be deemed statistically
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