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Partial Sums of Squares and Related Tests

This lecture covers the concept of partial sums of squares and related tests, including the use of extra sums of squares to develop tests for sets of variables. The procedure for computing extra sums of squares and conducting F-tests is explained, along with an example for better understanding. Coefficient of partial determination is also discussed.

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Partial Sums of Squares and Related Tests

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  1. Statistics 350 Lecture 20

  2. Today • Last Day: Partial sums of squares • Today: related tests

  3. Extra Sum of Squares and Tests • Can use the extra sums of squares to develop tests for sets of variables • So, starting from the full model: • we can test to see whether any combination of variables is needed in a model that already contains the others; i.e., we can test whether a reduced model suffices

  4. Extra Sum of Squares and Tests • Procedure: • Fit full and reduced model • Find SSR, SSE for full model and associated degrees of freedom • Find SSR, SSE for reduced model and associated degrees of freedom • Compute extra sum of squares due to additional variables: • Compute corresponding degrees of freedom: • Compute the mean sqaures due to the adding of variables: • Compute F-Statistic:

  5. Extra Sum of Squares and Tests • Hypothesis: • F-test:

  6. Back to Example • Consider Example on page 257 • Y = Percent Body Fat • X1= Triceps Skinfold Thickness • X2 = Thigh Circumference • X3 = Midarm Circumference

  7. Back to Example • In body fat example, test whether a model containing all three variables is significantly better than a model containing just Thigh Circumference (X2).

  8. Coefficient of Partial Determination • Extra Sums of Squares measure the effect of variable(s) after accounting for other variable(s) in the model • Because the total sum of squares for a data set does not change, the Extra Sum of Squares for a set of variables is bounded by whatever amount is left in the SSE from the first set of variables in the model • We saw in the body fat example: • SSR(X1) = 352.27 SSE(X1) = 143.12 • SSR(X1 X2) = 384.44 SSE(X1 X2) = 109.95 • SSR(X1 X2 X3) = 396.98 SSE(X1 X2 X3) = 98.41 • So SSR(X2|X1) could not have been larger than 143.12, the SSE left after the X1 model fit

  9. Coefficient of Partial Determination • So. after you fit a model, how much of the remaining variability in the responses is accounted for by including a set of additional variables? • You can measure the proportion of the remaining variability explained by X2 after the fit of X1 as: • Coefficient of Partial Determination:

  10. Coefficient of Partial Determination • Back to example:

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