Mat 465/565 Extra sums of squares SSER=SSEF+SSR(F|R) For any given model, we always have the key decomposition SST=SSE+SSR For a case with 3 variables, we are going to decompose SST adding one variable at a time. We start with: SST=SSE(X1)+SSR(X1) Then we decompose SSE(X1) by adding X2 to the model: SSE(X1)= SSE(X1,X2) + SSR(X2| X1) We obtain: SST=SSE(X1)+SSR(X1)= SSE(X1,X2)+SSR(X2| X1) +SSR(X2| X1) where the last 2 terms on the right sum to SSR(X1,X2) (because SST= SSE(X1,X2)+ SSR(X1,X2)). Now let's add the last variable: SST=SSE(X1,X2)+SSR(X2| X1) +SSR(X2| X1) = SSE(X1,X2,X3)+ SSR(X3| X1, X2)+SSR(X2| X1)+SSR(X2| X1) Again, we notice that since SST= SSE(X1,X2, X3)+ SSR(X1,X2, X3), the last three terms sum to SSR(X1,X2, X3): SSR(X3| X1, X2)+SSR(X2| X1)+SSR(X2| X1)=SSR(X1,X2, X3) Source df SS X1 1 SSR(X1) Regression X2|X1 1 SSR(X2| X1) X3|X1,X2 1 SSR(X3| X1, X2) Residual 21 SSE(X1,X2,X3) Total 24 SST The sum of the three rows in blue is SSE(X1) If we look at a model with X1,X2, then we split the table as: Source df SS X1 1 SSR(X1) Regression X2|X1 1 SSR(X2| X1) X3|X1,X2 1 SSR(X3| X1, X2) Residual 21 SSE(X1,X2,X3) Total 24 SST Then the sum of the rows in orange is SSR(X1,X2) and the sum of the rows in blue is SSE(X1,X2). Lastly, if we look at the model with the three variables, then we split the table as: Source df SS X1 1 SSR(X1) Regression X2|X1 1 SSR(X2| X1) X3|X1,X2 1 SSR(X3| X1, X2) Residual 21 SSE(X1,X2,X3) Total 24 SST Now the sum of the rows in red is SSR(X1,X2,X3) and the rows in blue are SSE(X1,X2,X3). 1. The following table considers he regression of a response variable Y on three variables X1, X2, X3. Source df SS X1 1 18953.04 Regression X3|X1 1 7010.03 X2|X1,X3 1 10.93 Residual 21 2248.23 Total 24 28222.23 a. Provide a test to compare the following two models: = ! + ! ! + ! ! + ! ! + and = ! + ! ! + . b. Provide a test to compare the following two models: = ! + ! ! + ! ! + and = ! + . c. Which 2 models are being compared in the computation: 18953.04 + 7010.03 + 10.93 /3 = 2248.23/21