# Thread: Simultaneous confidence interval and constrasts

1. ## Simultaneous confidence interval and constrasts

Hi guys !

I have a quick question :
We don't understand why it is better to consider contrasts (linear combination of parameters $(\beta_j)_{j\in J}$ : $\sum_{j=1}^J c_j \beta_j$, where $\sum_{j=1}^J c_j =0$) in the Scheffé method for simultaneous confidence intervals...
Why do we need the latter sum to be 0 ? I read in a lecture note that if we consider contrasts instead of any linear combination, we're considering a Fisher distribution with a degree of freedom that is J-1 instead of J. But is it really the only reason ?

Thanks for any input, even if it doesn't exactly answer the question... if it casts a light on some point of the Scheffé method, I would also be pleased

2. I believe it has to do with the column space of your design matrix
https://netfiles.uiuc.edu/dgs/www/st...tes/042804.pdf
I used to teach that, but it's been a few years.

3. Well, sorry, I didn't really find the answer in this paper... I'm quite abstruse to algebra, so I don't understand the thing with column space :s

Thanks anyway