TMA4267 Linear Statistical Models Part 2: Linear regression by Mette Langaas https://padlet.com/mette_langaas/7xp70hykdfkv What are the important results in this part of the course? en-us 2017-01-31 09:59:52 UTC 2026-03-01 07:07:09 UTC hello@padlet.com https://padlet-assets.s3.amazonaws.com/icons/Thunder.png Classical linear model mette_langaas https://padlet.com/mette_langaas/7xp70hykdfkv/wish/152981585 $$Y=X\beta +\varepsilon$$
Y=response, X=design matrix, \(\beta\)=regression parameters and \(\varepsilon\)=error vector.
1) E\( (\varepsilon)\)=0
2) Cov(\(\varepsilon)=\sigma^2 I\)
3) Design matrix X has full rank.
For the normal linear model in addition assume that 
4) errors \(\varepsilon\) are normal.]]> 2017-02-10 07:31:29 UTC https://padlet.com/mette_langaas/7xp70hykdfkv/wish/152981585 Regression user8715 https://padlet.com/mette_langaas/7xp70hykdfkv/wish/156989383 Sums of Squares Regression:
$$ SSR = \sum_{i=1}^n (\hat{y}_i - \bar{y})^2$$
Sums of Squares Error:
$$ SSE = \sum_{i=1}^n (y_i - \hat{y}_i)^2 $$
Sums of Squared Total:
$$ SST = SSE + SSR $$

$$ R^2 = \frac{SSR}{SST} = 1 - \frac{SSE}{SST} $$]]> 2017-03-01 12:33:13 UTC https://padlet.com/mette_langaas/7xp70hykdfkv/wish/156989383 p-value for hypothesis for \(\beta_j\) user8715 https://padlet.com/mette_langaas/7xp70hykdfkv/wish/156991876 H_0: \beta_j = 0, \text{ vs. } H_1: \beta_j \neq 0

Statistic (assuming (H0) is true): 
T_{j0} = \frac{\hat{\beta_j} - 0}{\sqrt{C_{jj}} \hat{\sigma}}
where (Cjj) is the corresponding diagonal element of \((X^TX)^{-1}\)
p-value:
P(|T_{j0}| \geq |t_{j0}|, \text{when } H_0 \text{is true})

 where \(t_{j0}\) is the observed value of \(T_{j0}\).
]]> 2017-03-01 12:46:49 UTC https://padlet.com/mette_langaas/7xp70hykdfkv/wish/156991876