Section 22 MLR: Coefficients

Multiple Linear Regression: Coefficients

22.1 Estimates: Effects

\[ \large fm \leftarrow lm(SBP \sim BMI + Age, \space data=BP) \]

\[ \large summary(fm) \]

Call:
lm(formula = SBP ~ BMI + Age, data = BP)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.6030 -2.0345  0.1196  1.9800  6.8630 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -10.83438    6.91013  -1.568    0.118    
BMI           2.33147    0.07108  32.799  < 2e-16 ***
Age           1.09032    0.15678   6.954 1.12e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.633 on 497 degrees of freedom
Multiple R-squared:  0.8175,    Adjusted R-squared:  0.8168 
F-statistic:  1113 on 2 and 497 DF,  p-value: < 2.2e-16

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	-10.8344	6.9101	-1.5679	0.1175
BMI	2.3315	0.0711	32.7990	0.0000
Age	1.0903	0.1568	6.9544	0.0000

22.2 Explanation

Statistical Model

\[ \large y_{i} = a + \beta_1 x_{1i} + \beta_2 x_{2i} + \epsilon_{i} \]

\[ \large \hat\sigma^2 = Var(\hat\epsilon) = RSS/(n-p-1) = MSE \]

22.3 Hypothesis testing

\[ \large H_O: \pmb\beta = 0 \] \[ \large H_A: \pmb\beta \ne 0 \]

Test Statistic under the Null Hypothesis

Regression coefficients for k-th predictors

\[ \large \hat\beta_k / SE(\hat\beta_k) \sim t_{df_{residual}} \]

95% Confidence Interval of k-th coefficients

\[ \large CI_{0.95}(\hat \beta_k) = \left[ \hat\beta_k \pm t_{0.025, df_{residual}} * SE(\hat\beta_k) \right]\]