Ch. 7. The Analysis of Variance Table
I. Situation
A. Want to test if the regression line as a whole is a
significant prediction model for Y.
B. In simple linear regression, it is the same as the
significance test for b1.
II. Partitioning Variability
A. In the population
Yi = μY + (μY|X - μY) + (Yi - μY|X)
Yi - μY = (μY|X - μY) + (Yi - μY|X)
Σ(Yi - μY)² = Σ[(μY|X - μY) + (Yi - μY|X)]²
= Σ(μY|X - μY)² + Σ(Yi - μY|X)²
+ 2Σ(μY|X - μY)(Yi - μY|X)
We know the underlined part = 0.
Therefore,
Σ(Yi - μY)² = Σ(μY|X - μY)² + Σ(Yi - μY|X)²
B. In the sample
Σ(Yi -
Y
)² = Σ(
Y
ˆ
-
Y
)² + Σ(Yi -
Y
ˆ
)²
SSY = SSR + SSE
III. Procedure
Source df SS MS F(obs)
Model (Reg) p SSR SSR/p
MSR/MSE
Error (Res) n-p-1 SSE SSE/(n-p-1)
──────────────────────────────────────
Total n-1 SSY
A. Notes
1. If F(obs) ≥ F(crit) [α=.05 or.01, df(num)=p,
df(den)=n-p-1], then reject Ho.
2. In simple regression, F(obs) = t² (for b1 test).
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