Section 27 Estimates: Coefficients

27.1 Estimates: Effects

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

\[ \large summary(fm) \]

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

Residuals:
    Min      1Q  Median      3Q     Max 
-8.3636 -2.1681  0.1586  2.1492  6.5777 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 36.20368    1.48011   24.46   <2e-16 ***
BMI          2.63229    0.05903   44.59   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.756 on 498 degrees of freedom
Multiple R-squared:  0.7997,    Adjusted R-squared:  0.7993 
F-statistic:  1989 on 1 and 498 DF,  p-value: < 2.2e-16

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	36.2037	1.4801	24.4601	0
BMI	2.6323	0.0590	44.5940	0

27.2 Explanation

Statistical Model

\[ \large y_{i} = \beta_0 + \beta_1 x_{i} + \epsilon_{i} \]

\(\beta_1\) Estimates & SE

\[ \large \hat \beta_1 = \frac{\sum\limits_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)}{\sum\limits_{i=1}^{n}(x_i-\bar x)^2} = \frac{S_{xy}}{S_{xx}} \]

\[ \large Var(\hat \beta_1) = \frac{\hat\sigma^2}{\sum\limits_{i=1}^{n}(x_i-\bar x)^2} = \frac{\hat\sigma^2}{S_{xx}} \]

\[ \large SE(\hat \beta_1) = \sqrt {Var(\hat \beta_1)} \]

\(\beta_0\) Estimates & SE

\[ \large \hat \beta_0 = \bar y - \hat \beta_1 \bar x \]

\[ \large Var(\hat \beta_0) = \hat\sigma^2 \left[ \frac{1}{n} + \frac{\bar x^2}{\sum\limits_{i=1}^{n}(x_i-\bar x)^2} \right] = \hat\sigma^2 \left[ \frac{1}{n} + \frac{\bar x^2}{S_{xx}} \right] \]

\[ \large SE(\hat \beta_0) = \sqrt {Var(\hat \beta_0)} \]

Here,

\[ \large \bar{x} = \frac{1}{n}\sum\limits_{i=1}^{n} x_{i} \]

\[ \large \bar{y} = \frac{1}{n}\sum\limits_{i=1}^{n} y_{i} \]

\[ \large \hat\sigma^2 = Var(\hat\epsilon) \]