Suppose that we want to test the following hypothesis


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 $V(b_{1}+b_{3}-2b_{2})=V(\sum_{i=1}^{3}c_{i}b_{i})$

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

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