Statistics 350
Lecture 19
Today
• Last Day: R2 and Start Chapter 7
• Today: Partial sums of squares and related tests
Example
• Consider Example on page 257
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Y = Percent Body Fat
X1= Triceps Skinfold Thickness
X2 = Thigh Circumference
X3 = Midarm Circumference
Example
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Suppose only consider first two explanatory variables
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Test hypothesis:
Ho: 1=0 (assuming X2 already in model)
HA: 1≠0 (assuming X2 already in model)
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Important lesson is that the presence or absence of X2 in the model may
change depending upon which variables are in the model
Extra Sum of Squares
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Some Notation:
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SSR(X1)=
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SSR(X1,X2) =
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SSR(X1|X2) =
Extra Sum of Squares
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Some Notation:
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SSR(X1, X2 , X3)=
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SSR(X3| X1 , X2)=
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SSR(X2, X3 | X1)=
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And so on…
Extra Sum of Squares
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Decomposition:
Back to Example
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Look at the sums of squares for this problem
Back to Example
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Why doesn’t the regression sum of squares for the first two models not sum to
that for the third?
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SSR(X1,X2) -SSR(X2) =3.47…what is this difference and what does it mean?
Back to Example
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What is SSR(X2,X1) =
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In words:
SSR(X2 ,X3 | X1) =
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In words:
Back to Example
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Who cares?
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For example, Thigh Circumference (X2) and Midarm Circumference (X3) are
easily measured with precision using an ordinary tape measure, but Triceps
Skinfold Thickness (X1) requires a trained technician and an expensive
instrument. Then a natural question is whether we even need to bother with
X1 once we've already measured X2 and X3?
Back to Example
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Who cares?
•
For example, Thigh Circumference (X2) and Midarm Circumference (X3) are
easily measured with precision using an ordinary tape measure, but Triceps
Skinfold Thickness (X1) requires a trained technician and an expensive
instrument. Then a natural question is whether we even need to bother with
X1 once we've already measured X2 and X3?