[R] Standard error of coefficient in linear regression

[Maciej Bliziński]

Hello R users,

I have a substantial question about statistics, not about R itself, but
I would love to have an answer from an R user, in form of an example in
R syntax. I have spent whole Sunday searching in Google and browsing the
books. I've been really close to the answer but there are at least three
standard errors you can talk about in the linear regression and I'm
really confused. The question is:

How exactly are standard errors of coefficients calculated in the linear
regression?

Here's an example from a website I've read [1]. A company wants to know
if there is a relationship between its advertising expenditures and its
sales volume. 

========================================================
> exped <- c(4.2, 6.1, 3.9, 5.7, 7.3, 5.9)
> sales <- c(27.1, 30.4, 25.0, 29.7, 40.1, 28.8)
> S <- data.frame(exped, sales)
> summary(lm(sales ~ exped, data = S))

Call:
lm(formula = sales ~ exped, data = S)

Residuals:
      1       2       3       4       5       6
 1.7643 -1.9310  0.7688 -1.1583  3.3509 -2.7947

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)   9.8725     5.2394   1.884   0.1326
exped         3.6817     0.9295   3.961   0.0167 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.637 on 4 degrees of freedom
Multiple R-Squared: 0.7968,     Adjusted R-squared: 0.7461
F-statistic: 15.69 on 1 and 4 DF,  p-value: 0.01666
========================================================

I can calculate the standard error of the estimate, according to the
equation [2]...

> S.m <- lm(sales ~ exped, data = S)
> S$pred <- predict(S.m)
> S$ye <- S$sales - S$pred
> S$ye2 <- S$ye ^ 2
> Se <- sqrt(sum(S$ye2)/(length(S$sales) - 1 - 1))
> Se
[1] 2.636901

...which matches the "Residual standard error" and I'm on the right
track. Next step would be to use the equation [3] to calculate the
standard error of the regression coefficient (here: exped). The equation
[3] uses two variables, meaning of which I can't really figure out. As
the calculated value Sb is scalar, all the parameters need also to be
scalars. I've already calculated Se, so I'm missing x and \bar{x}. The
latter could be the estimated coefficient. What is x then?

Regards,
Maciej

[1] http://www.statpac.com/statistics-calculator/correlation-regression.htm
[2] http://www.answers.com/topic/standard-error-of-the-estimate
[3] http://www.answers.com/topic/standard-error-of-the-regression-coefficient

-- 
Maciej Bliziński <m.blizinski at wit.edu.pl>
http://automatthias.wordpress.com