Ch. 9. Testing Hypotheses in Multiple Regression
I. Situation
Testing hypotheses about a whole model, a single variable,
or a group of variables.
II. Model Comparison Approach in General Linear Model
A. Situation: All hypothesis testing can be viewed as a
model comparison in General Linear Model which
include all multiple regressions, ANOVAs, and more.
B. Procedure
1. Develop two math models; Full and reduced models.
2. Full model includes all X variables of interest.
3. Reduced model includes all X variables of
interest except the X variable(s) we want to test.
4. Example: CGPA (Y), ACT (X1), HGPA (X2), CRED
(X3).
Want to test all three Xs (whole model).
Full: y = ฮฒ0 + ฮฒ1X1 + ฮฒ2X2 + ฮฒ3X3 + ฮตi
Reduced: y = ฮฒ0 + ฮตi
(SSER-SSEF)/(dfR-dfF) SSR/p
F = โโโโโโโโโโโโโโโโโโโโโโโโ = โโโโโโโโโโโโโโ
SSEF/dfF SSE/(n-p-1)
III. Testing Overall Regression
A. Same as the ANOVA test in Ch. 8.
B. Since SSR = SSY*Rยฒ and SSE = SSY*(1-Rยฒ),
SSR/p SSY*Rยฒ/p Rยฒ/p
F = โโโโโโโโโโโ = โโโโโโโโโโโโโโโโโโ = โโโโโโโโโโโโโ
SSE/(n-p-1) SSY*(1-Rยฒ)/(n-p-1) (1-Rยฒ)/(n-p-1)
C. ANOVA Table (From p. 120, table 8-2)
Source df SS MS F(obs)
Model (Reg) 3 693.06 231.02 9.47
Error (Res) 8 195.19 24.40
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
Total 11 888.25
Rยฒ = .7802
D. Model Comparison Procedure (Handout p. 2)
1. Run two Reg. models; Full and Reduced
2. Full: WGT = ฮฒ0 + ฮฒ1HGT + ฮฒ2AGE + ฮฒ3AGEยฒ + ฮตi
Reduced: WGT = ฮฒ0 + ฮตi
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