9. MLR Hypothesis Testing

SLR Review

Recall our earlier F-tests for anova T tests with respect to correlation and our parameters
Where $F_{*} = \frac{M S R}{M S E} = (n - 2) \frac{r^{2}}{1 - r^{2}}$

MLR

ε_{n \times 1} \sim N_{n} (0_{n \times 1}, σ^{2} I_{n \times n})

Overall Test

y_{i} = X_{i}^{T} β + ε_{i} = β_{0} + x_{i, 1} β_{1} + x_{i, 2} β_{2} + \dots + x_{i, p} β_{p}

$H_{0} : β_{1} = β_{2} = \dots = β_{p} = 0$ , $H_{a} :$ at least one $β_{i} \neq 0$
$S S T = S S R + S S E$

R^{2} \equiv \frac{S S R}{S S T}

\begin{aligned} F_{*} & = \frac{M S R}{M S E} = \frac{\frac{S S R}{p}}{\frac{S S E}{n - (p + 1)}} = \frac{n - (p + 1)}{p} \frac{S S R}{S S E} \\ or F_{*} & = \frac{n - (p + 1)}{p} \frac{\frac{S S R}{S S T}}{\frac{S S E}{S S T}} = \frac{n - (p + 1)}{p} \frac{R^{2}}{1 - R^{2}} \end{aligned}

where df numerator = $p$ , df denominator = $n - (p + 1)$

(remember $S S R = S S Y - S S E$ )

RR = ${F_{*} such that F_{*} > F_{α, p, n - (p + 1)}}$

Addition of a Group of Variables

F_{*} = \frac{\frac{(S S E (R) - S S E (C)}{(complete parameters - reduced parameters))}}{\frac{S S E (C)}{n - complete parameters}}

R = reduced , C = complete
$H_{a} :$ At least one new parameter contributes information
RR = ${F_{*} > F_{α, C - R, n - C}}$