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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 students
- Response (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 inches
- Predictor (X3): Weight in pounds

The regression equation is PIQ = 4.7 + 1.18 MRI

Predictor Coef SE Coef T P

Constant 4.65 43.71 0.11 0.916

MRI 1.1766 0.4806 2.45 0.019

Analysis of Variance

Source DF SS MS F P

Regression 1 2697.1 2697.1 5.99 0.019

Error 36 16197.5 449.9

Total 37 18894.6

The regression equation is

PIQ = 111 + 2.06 MRI - 2.73 Height

Predictor Coef SE Coef T P

Constant 111.28 55.87 1.99 0.054

MRI 2.0606 0.5466 3.77 0.001

Height -2.7299 0.9932 -2.75 0.009

Analysis of Variance

Source DF SS MS F P

Regression 2 5572.7 2786.4 7.32 0.002

Residual 35 13321.8 380.6

Total 37 18894.6

Source DF Seq SS

MRI 1 2697.1

Height 1 2875.6

The regression equation is

PIQ = 111 + 2.06 MRI - 2.73 Height + 0.001 Weight

Predictor Coef SE Coef T P

Constant 111.35 62.97 1.77 0.086

MRI 2.0604 0.5634 3.66 0.001

Height -2.732 1.229 -2.22 0.033

Weight 0.0006 0.1971 0.00 0.998

Analysis of Variance

Source DF SS MS F P

Regression 3 5572.7 1857.6 4.74 0.007

Error 34 13321.8 391.8

Total 37 18894.6

Source DF Seq SS

MRI 1 2697.1

Height 1 2875.6

Weight 1 0.0

Sequential sums of squares: definition using SSE notation

- SSR(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 notation

- SSR(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 squares

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

The regression equation is

PIQ = 111 + 2.06 MRI - 2.73 Height

Predictor Coef SE Coef T P

Constant 111.28 55.87 1.99 0.054

MRI 2.0606 0.5466 3.77 0.001

Height -2.7299 0.9932 -2.75 0.009

Analysis of Variance

Source DF SS MS F P

Regression 2 5572.7 2786.4 7.32 0.002

Residual 35 13321.8 380.6

Total 37 18894.6

Source DF Seq SS

MRI 1 2697.1

Height 1 2875.6

The regression equation is

PIQ = 111 - 2.73 Height + 2.06 MRI

Predictor Coef SE Coef T P

Constant 111.28 55.87 1.99 0.054

Height -2.7299 0.9932 -2.75 0.009

MRI 2.0606 0.5466 3.77 0.00

Analysis of Variance

Source DF SS MS F P

Regression 2 5572.7 2786.4 7.32 0.002

Error 35 13321.8 380.6

Total 37 18894.6

Source DF Seq SS

Height 1 164.0

MRI 1 5408.8

A sequential sum of squares involving one extra predictor variable has one degree of freedom associated with it:

A sequential sum of squares involving two extra predictor variables has two degrees of freedom associated with it:

Degrees of freedom and regression mean squaresSequential sums of squares in Minitab

- The 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.

The regression equation is

PIQ = 111 + 2.06 MRI - 2.73 Height + 0.001 Weight

Predictor Coef SE Coef T P

Constant 111.35 62.97 1.77 0.086

MRI 2.0604 0.5634 3.66 0.001

Height -2.732 1.229 -2.22 0.033

Weight 0.0006 0.1971 0.00 0.998

Analysis of Variance

Source DF SS MS F P

Regression 3 5572.7 1857.6 4.74 0.007

Error 34 13321.8 391.8

Total 37 18894.6

Source DF Seq SS

MRI 1 2697.1

Height 1 2875.6

Weight 1 0.0

The regression equation is

PIQ = 111 - 2.73 Height + 0.001 Weight + 2.06 MRI

Predictor Coef SE Coef T P

Constant 111.35 62.97 1.77 0.086

Height -2.732 1.229 -2.22 0.033

Weight 0.0006 0.1971 0.00 0.998

MRI 2.0604 0.5634 3.66 0.001

Analysis of Variance

Source DF SS MS F P

Regression 3 5572.7 1857.6 4.74 0.007

Error 34 13321.8 391.8

Total 37 18894.6

Source DF Seq SS

Height 1 164.0

Weight 1 169.5

MRI 1 5239.2

Testing one slope β1= βMRI is 0

Predictor Coef SE Coef T P

Constant 111.35 62.97 1.77 0.086

Height -2.732 1.229 -2.22 0.033

Weight 0.0006 0.1971 0.00 0.998

MRI 2.0604 0.5634 3.66 0.001

Analysis of Variance

Source DF SS MS F P

Regression 3 5572.7 1857.6 4.74 0.007

Error 34 13321.8 391.8

Total 37 18894.6

Source DF Seq SS

Height 1 164.0

Weight 1 169.5

MRI 1 5239.2

Testing one slope β2= βHT is 0

Predictor Coef SE Coef T P

Constant 111.35 62.97 1.77 0.086

MRI 2.0604 0.5634 3.66 0.001

Weight 0.0006 0.1971 0.00 0.998

Height -2.732 1.229 -2.22 0.033

Analysis of Variance

Source DF SS MS F P

Regression 3 5572.7 1857.6 4.74 0.007

Error 34 13321.8 391.8

Total 37 18894.6

Source DF Seq SS

MRI 1 2697.1

Weight 1 940.9

Height 1 1934.7

Testing one slope β3= βWT is 0

Predictor Coef SE Coef T P

Constant 111.35 62.97 1.77 0.086

MRI 2.0604 0.5634 3.66 0.001

Height -2.732 1.229 -2.22 0.033

Weight 0.0006 0.1971 0.00 0.998

Analysis of Variance

Source DF SS MS F P

Regression 3 5572.7 1857.6 4.74 0.007

Error 34 13321.8 391.8

Total 37 18894.6

Source DF Seq SS

MRI 1 2697.1

Height 1 2875.6

Weight 1 0.0

The regression equation is

PIQ = 111 + 2.06 MRI - 2.73 Height + 0.001 Weight

Predictor Coef SE Coef T P

Constant 111.35 62.97 1.77 0.086

MRI 2.0604 0.5634 3.66 0.001

Height -2.732 1.229 -2.22 0.033

Weight 0.0006 0.1971 0.00 0.998

Analysis of Variance

Source DF SS MS F P

Regression 3 5572.7 1857.6 4.74 0.007

Error 34 13321.8 391.8

Total 37 18894.6

Source DF Seq SS

MRI 1 2697.1

Height 1 2875.6

Weight 1 0.0

Cumulative Distribution Function

F distribution with 2 DF in numerator and 34 DF in denominator

x P( X <= x )

3.6700 0.9640

Getting P-value for F-statistic in Minitab

- Select Calc >> Probability Distributions >> F…
- Select 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 β1 = β3 = 0

Analysis of Variance

Source DF SS MS F P

Regression 3 5572.7 1857.6 4.74 0.007

Error 34 13321.8 391.8

Total 37 18894.6

Source DF Seq SS

Height 1 164.0

Weight 1 169.5

MRI 1 5239.2

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