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Sequential sums of squares … or … extra sums of squares

# Sequential sums of squares … or … extra sums of squares

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• Sequential sums of squares or extra sums of squares

• Sequential sums of squares: what are they?The reduction in the error sum of squares when one or more predictor variables are added to the regression model.Or, the increase in the regression sum of squares when one or more predictor variables are added to the regression model.

• Sequential sums of squares:why?They can be used to test whether one slope parameter is 0.They can be used to test whether a subset (more than two, but less than all) of the slope parameters are 0.

• Example: Brain and body size predictive of intelligence?Sample of n = 38 college studentsResponse (Y): intelligence based on the PIQ (performance) scores from the (revised) Wechsler Adult Intelligence Scale.Predictor (X1): Brain size based on MRI scans (given as count/10,000)Predictor (X2): Height in inchesPredictor (X3): Weight in pounds

• OUTPUT #1The regression equation is PIQ = 4.7 + 1.18 MRI

Predictor Coef SE Coef T PConstant 4.65 43.71 0.11 0.916MRI 1.1766 0.4806 2.45 0.019

Analysis of Variance

Source DF SS MS F PRegression 1 2697.1 2697.1 5.99 0.019Error 36 16197.5 449.9Total 37 18894.6

• OUTPUT #2The regression equation is PIQ = 111 + 2.06 MRI - 2.73 Height

Predictor Coef SE Coef T PConstant 111.28 55.87 1.99 0.054MRI 2.0606 0.5466 3.77 0.001Height -2.7299 0.9932 -2.75 0.009

Analysis of VarianceSource DF SS MS F PRegression 2 5572.7 2786.4 7.32 0.002Residual 35 13321.8 380.6Total 37 18894.6

Source DF Seq SSMRI 1 2697.1Height 1 2875.6

• OUTPUT #3The regression equation isPIQ = 111 + 2.06 MRI - 2.73 Height + 0.001 Weight

Predictor Coef SE Coef T PConstant 111.35 62.97 1.77 0.086MRI 2.0604 0.5634 3.66 0.001Height -2.732 1.229 -2.22 0.033Weight 0.0006 0.1971 0.00 0.998

Analysis of VarianceSource DF SS MS F PRegression 3 5572.7 1857.6 4.74 0.007Error 34 13321.8 391.8Total 37 18894.6

Source DF Seq SSMRI 1 2697.1Height 1 2875.6Weight 1 0.0

• Sequential sums of squares: definition using SSE notationSSR(X2|X1) = SSE(X1) - SSE(X1,X2) In general, you subtract the error sum of squares due to all of the predictors both left and right of the bar from the error sum of squares due to the predictor to the right of the bar.SSR(X2,X3|X1) = SSE(X1) - SSE(X1,X2,X3)

• Sequential sums of squares: definition using SSR notationSSR(X2|X1) = SSR(X1,X2) SSR(X1)In general, you subtract the regression sum of squares due to the predictor to the right of the bar from the regression sum of squares due to all of the predictors both left and right of the bar.SSR(X2,X3|X1) = SSR(X1,X2,X3)-SSR(X1)

• Decomposition of regression sum of squaresIn multiple regression, there is more than one way to decompose the regression sum of squares. For example:

• OUTPUT #2The regression equation is PIQ = 111 + 2.06 MRI - 2.73 Height

Predictor Coef SE Coef T PConstant 111.28 55.87 1.99 0.054MRI 2.0606 0.5466 3.77 0.001Height -2.7299 0.9932 -2.75 0.009

Analysis of VarianceSource DF SS MS F PRegression 2 5572.7 2786.4 7.32 0.002Residual 35 13321.8 380.6Total 37 18894.6

Source DF Seq SSMRI 1 2697.1Height 1 2875.6

• OUTPUT #4The regression equation isPIQ = 111 - 2.73 Height + 2.06 MRI

Predictor Coef SE Coef T PConstant 111.28 55.87 1.99 0.054Height -2.7299 0.9932 -2.75 0.009MRI 2.0606 0.5466 3.77 0.00

Analysis of VarianceSource DF SS MS F PRegression 2 5572.7 2786.4 7.32 0.002Error 35 13321.8 380.6Total 37 18894.6

Source DF Seq SSHeight 1 164.0MRI 1 5408.8

• Decomposition of SSR: how?

• Decomposition of SSR: how?

• Even more ways to decompose SSR when 3 or more predictors

• Degrees of freedom and regression mean squares

• Sequential sums of squares in MinitabThe SSR is automatically decomposed into one-degree-of-freedom sequential sums of squares, in the order in which the predictor variables are entered into the model.To get sequential sum of squares involving two or more predictor variables, sum the appropriate one-degree-of-freedom sequential sums of squares.

• OUTPUT #3The regression equation isPIQ = 111 + 2.06 MRI - 2.73 Height + 0.001 Weight

Predictor Coef SE Coef T PConstant 111.35 62.97 1.77 0.086MRI 2.0604 0.5634 3.66 0.001Height -2.732 1.229 -2.22 0.033Weight 0.0006 0.1971 0.00 0.998

Analysis of VarianceSource DF SS MS F PRegression 3 5572.7 1857.6 4.74 0.007Error 34 13321.8 391.8Total 37 18894.6

Source DF Seq SSMRI 1 2697.1Height 1 2875.6Weight 1 0.0

• OUTPUT #5The regression equation isPIQ = 111 - 2.73 Height + 0.001 Weight + 2.06 MRI

Predictor Coef SE Coef T PConstant 111.35 62.97 1.77 0.086Height -2.732 1.229 -2.22 0.033Weight 0.0006 0.1971 0.00 0.998MRI 2.0604 0.5634 3.66 0.001

Analysis of VarianceSource DF SS MS F PRegression 3 5572.7 1857.6 4.74 0.007Error 34 13321.8 391.8Total 37 18894.6

Source DF Seq SSHeight 1 164.0Weight 1 169.5MRI 1 5239.2

• Testing one slope 1= MRI is 0Predictor Coef SE Coef T PConstant 111.35 62.97 1.77 0.086Height -2.732 1.229 -2.22 0.033Weight 0.0006 0.1971 0.00 0.998MRI 2.0604 0.5634 3.66 0.001

Analysis of VarianceSource DF SS MS F PRegression 3 5572.7 1857.6 4.74 0.007Error 34 13321.8 391.8Total 37 18894.6

Source DF Seq SSHeight 1 164.0Weight 1 169.5MRI 1 5239.2

• Testing one slope 2= HT is 0Predictor Coef SE Coef T PConstant 111.35 62.97 1.77 0.086MRI 2.0604 0.5634 3.66 0.001Weight 0.0006 0.1971 0.00 0.998Height -2.732 1.229 -2.22 0.033

Analysis of VarianceSource DF SS MS F PRegression 3 5572.7 1857.6 4.74 0.007Error 34 13321.8 391.8Total 37 18894.6

Source DF Seq SSMRI 1 2697.1Weight 1 940.9Height 1 1934.7

• Testing one slope 3= WT is 0Predictor Coef SE Coef T PConstant 111.35 62.97 1.77 0.086MRI 2.0604 0.5634 3.66 0.001Height -2.732 1.229 -2.22 0.033Weight 0.0006 0.1971 0.00 0.998

Analysis of VarianceSource DF SS MS F PRegression 3 5572.7 1857.6 4.74 0.007Error 34 13321.8 391.8Total 37 18894.6

Source DF Seq SSMRI 1 2697.1Height 1 2875.6Weight 1 0.0

• Testing one slope k is 0: why it works?

• Testing one slope k is 0: why it works? (contd)

• Testing whether 2 = 3 = 0

• Testing whether 2 = 3 = 0 (contd)

• OUTPUT #3The regression equation isPIQ = 111 + 2.06 MRI - 2.73 Height + 0.001 Weight

Predictor Coef SE Coef T PConstant 111.35 62.97 1.77 0.086MRI 2.0604 0.5634 3.66 0.001Height -2.732 1.229 -2.22 0.033Weight 0.0006 0.1971 0.00 0.998

Analysis of VarianceSource DF SS MS F PRegression 3 5572.7 1857.6 4.74 0.007Error 34 13321.8 391.8Total 37 18894.6

Source DF Seq SSMRI 1 2697.1Height 1 2875.6Weight 1 0.0

• Getting P-value for F-statistic in MinitabSelect Calc >> Probability Distributions >> FSelect Cumulative Probability. Use default noncentrality parameter of 0.Type in numerator DF and denominator DF.Select Input constant. Type in F-statistic. Answer appears in session window.P-value is 1 minus the number that appears.

• Test whether 1 = 3 = 0Analysis of VarianceSource DF SS MS F PRegression 3 5572.7 1857.6 4.74 0.007Error 34 13321.8 391.8Total 37 18894.6

Source DF Seq SSHeight 1 164.0Weight 1 169.5MRI 1 5239.2

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