Topic 04

# Topic 04 - Topic 4 Extra Sums of Squares and the General...

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1 Topic 4 – Extra Sums of Squares and the General Linear Test Using Partial F Tests in Multiple Regression Analysis (Chapter 9)

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2 Recall: Types of Tests 1. ANOVA F Test : Does the group of predictor variables explain a significant percentage of the variation in the response? 2. Variable Added Last T-tests : Does a given variable explain a significant part of the variation remaining after all other variables have been included in the model? 3. Partial F Tests : Does a group of variables explain significant variation in the response over and above that already explained by another group of variables already in the model?
3 Hypotheses In Topic 3, we learned how to test whether a particular slope was zero (Variable Added Last T- tests). We also learned how to test whether all slopes are zero (Model F-test) We may also want to test other “groups” of variables. For example we might want to consider: 0 : 0 : 0 or 0 age size a age size H H β = =

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4 Hypotheses Using Models Such hypotheses as comparing two models Corresponding Example: 0 0 0 : : i smk smk a i smk smk age age size size H Y X H Y X X X β ε = + + = + + + +
5 Hypotheses Using Models (2) The alternative hypothesis is the “full model”, or the biggest model currently under consideration. The null hypothesis is what we get when we set the group of parameters we want to test to zero (that is, we apply the null hypothesis in terms of the betas to the regression equation). In the example, we are setting 0 : 0 age size H β = =

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6 Hypotheses in “English” Our null hypothesis is that the “additional group of variables” is not useful explaining the “remaining variation” in the response. “Additional Group of Variables” refers to those variables specified in the null. We are considering to add these to the model. “Remaining Variation” refers to variation not already explained by other variables in the model that are not listed in the null. The alternative is that there is at least one of the “additional variables” that is important.
7 Extra Sums of Squares In order to perform these hypothesis tests, we further partitioning of the model sums of squares into pieces.

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8 Review: Sums of Squares Total sums of squares represents the variation in the response variable that has a chance to be explained by predictors. Model (or regression) SS represents the variation that is explained Error SS represents the variation left unexplained.
9 And Remember! TOT R E SS SS SS = +

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Extra Sums of Squares (ESS) Extra sums of squares breaks down the model/regression sums of squares into pieces corresponding to each variable. Because of collinearity, the ESS are
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Topic 04 - Topic 4 Extra Sums of Squares and the General...

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