# Orthogonal contrasts for coefficients of regression

Suppose that we want to test the following hypothesis

$$H_{0}:b_{1}+b_{3}-2b_{2}=0$$

where $$b_{1},b_{2},b_{3}$$ are coefficient derived from a linear regression.We can see that $$H_{0}$$ is similar to orthogonal contrasts and I say similar because until know I only know that orthogonal contrasts are used for comparison of means.

My goal here is to find the t-test of $$H_{0}$$ and a confident interval for the linear combinatio $$b_{1}+b_{3}-2b_{2}$$.

Where $$t=\frac{b_{1}+b_{3}-2b_{2}}{s.e(b_{1}+b_{3}-2b_{2})}$$

I would like to asks if we can use the variance formula that we use in orthogonal contrasts (original version for means) $$V(C)=V(\sum_{i=1}^{n}c_{i}\mu_{i})$$??

Is it correct to write that $$V(b_{1}+b_{3}-2b_{2})=V(\sum_{i=1}^{3}c_{i}b_{i})$$

• $V(b_{1}+b_{3}-2b_{2})=V(\sum_{i=1}^{3}c_{i}b_{i})$, it is correct forever given that $c_1 = 1$, $c_2=-2$, and $c_3=1$. But need to be $\hat b$, instead of $b$. – user158565 Nov 21 '18 at 18:15
• Yes that's true , but we can still use that form of variance for the t-test ?? – G1I2A Nov 21 '18 at 18:24
• You mean for linear model? Yes, if $\epsilon \sim N(0,\sigma^2)$ – user158565 Nov 21 '18 at 18:26
• I mean that $t=\frac{b_{1}+b_{3}-2b_{2}}{s.e(b_{1}+b_{3}-2b_{2})}$,can we use $V(\sum_{i=1}^{3}c_{i}b_{i})$ in the place of $s.e(b_{1}+b_{3}-2b_{2})$ – G1I2A Nov 21 '18 at 18:29
• Yes. $s.e(b_1+b_3-2b_2) = \sqrt{V(\sum_{i=1}^3c_ib_i)}$. – user158565 Nov 21 '18 at 18:42