Stat 401 B – Lecture 231Home Gas Consumption Interaction? Should there be a different slope for the relationship between Gas and Temp after insulation than before insulation?2Home gas consumption Create a new explanatory variable. Temp*Insul This will allow for possibly different relationships between Gas and Temp.3Interaction Model Predicted Gas = 6.854 –0.3932*Temp – 2.263*Insul + 0.1436*Temp*Insul R2= 0.936, 93.6% of the variation in gas consumption can be explained by the interaction model with Temp, Insul and Temp*Insul.Stat 401 B – Lecture 234Un-Insulated House Predicted Gas = 6.854 –0.3932*Temp – 2.263*Insul+ 0.1436*Temp*Insul Insul = 0 Predicted Gas = 6.854 –0.3932*Temp5Interpretation For an un-insulated house (Insul = 0) when the average outside temperature is 0 oC, the predicted amount of gas used is 6.854 (1000 cubic feet).6Interpretation Holding Insul constant at 0 (an un-insulated house), gas consumption drops, on average, 393.2 cubic feet for every 1 oCincrease in average outside temperature.Stat 401 B – Lecture 237Insulated House Predicted Gas = 6.854 –0.3932*Temp – 2.263*Insul+ 0.1436*Temp*Insul Insul = 1 Predicted Gas = 4.591 –0.2496*Temp8Interpretation For an insulated house (Insul = 1) when the average outside temperature is 0 oC, the predicted amount of gas used is 4.591 (1000 cubic feet).9Interpretation Holding Isul constant at 1 (an insulated house), gas consumption drops, on average, 249.6 cubic feet for every 1 oCincrease in average outside temperature.Stat 401 B – Lecture 2310Summary Before adding insulation, there is a higher predicted gas use when Temp = 0 oC and a steeper average decline for every 1 oC increase in outdoor temperature. 11Summary After adding insulation, there is a lower predicted gas use when Temp = 0 oC and a slower average decline for every 1 oCincrease in outdoor temperature. 122345678Gas-5 0 5 10 15TempLinear Fit Insul==0Linear Fit Insul==1Stat 401 B – Lecture 2313Statistical Significance Model Utility F = 194.77, P-value < 0.0001 The model with Temp and Insul is useful. The P-value for the test of model utility is very small. RMSE = 0.27014Statistical Significance Temp t = –20.93, P-value < 0.0001 Because the P-value is small, Temp adds significantly to the model with Insul.15Statistical Significance Insul t = –13.10, P-value < 0.0001 Because the P-value is small, Insul adds significantly to theStat 401 B – Lecture 2316Statistical Significance Temp*Insul (Interaction) t = 3.22, P-value = 0.0025 Because the P-value is small, there is a statistically significant interaction between Temp and Insul17-1-0.500.51InteractionResidual-5 0 5 10 15TempBivariate Fit of Interaction Residual By Temp18Statistical Significance Temperature by itself is statistically significant. Adding the dummy variable for insulation adds significantly. Adding the interaction term adds significantly.Stat 401 B – Lecture 2319Change in R2 Temp: R2= 32.8% Temp, Insul: R2= 91.9% Temp, Insul, Temp*Insul: R2= 32.8% Each change is statistically significant.20Interaction Model The interaction model prediction equation can be split into two separate prediction equations by substituting in the two values for Insul (0 and 1).21Two separate regressions. Suppose instead of a multiple regression model with interaction we fit a simple linear regression for the un-insulated house and separate simple linear regression for the insulated house?Stat 401 B – Lecture 2322JMP – Fit Y by X Put Gas in for the Y, Response. Put Temp in for the X, Factor. Click on OK Group by Insul Fit Line232345678Gas-5 0 5 10 15TempLinear Fit Insul==0Linear Fit Insul==124Before Insulation Predicted Gas = 6.854 –0.3932*Temp R2= 0.944 RMSE = 0.281 Statistically significant t = –20.08, P-value < 0.0001Stat 401 B – Lecture 2325After Insulation Predicted Gas = 4.591 –0.2496*Temp R2= 0.733 RMSE = 0.252 Statistically significant t = –6.62, P-value < 0.000126Comment The prediction equations from the two separate models are exactly the same as the separate prediction equations from the interaction model.27Comment The interaction model pools all of the data together. The MSErrorfor the interaction is actually the weighted average of the two MSErrorvalues for the two separate
View Full Document
Unlocking...