Assume $\beta_{U,T}$ is the underlying slope of straight line associating $U$ with $T$. We know that $X=U+f$ and $Z=T+e$ are measurable instead of $U$ and $T$, where $e$ and $f$ are uncorrelated random errors. $R_Z=\frac{\sigma_T^2}{\sigma_T^2+\sigma_e^2}$ and $R_X=\frac{\sigma_U^2}{\sigma_U^2+\sigma_f^2}$ are reliabilities of $Z$ and $X$.
I don't know how to convert the words into equations. Does this mean that $U=\alpha+\beta T$? Would the observed variables be $X=\hat{\alpha}+\hat{\beta}Z$ and then we have
$(observed-expected)^2=(\hat{\alpha}+\hat{\beta}Z-\alpha-\beta T)^2$?
Thank you
yes you're right and everything you said is true, just wonder why you asked, lol. It's like fitting a line to two observations $X$ and $Z$ that are subject to noise. I'd recommend you to take a look at this excelent paper from Jayenes. In the second part, you're calculating what is called residuals or square error. In statistical setting, you're having one than one observation, say $N$, and the sum of residuals is written as:
$\mathcal{X}^2 = \sum_{i=1}^{N}(x - \hat{x})^2 $
where, $\hat{x} = a + b. y$. And the task of model fitting is basically to minimize $\mathcal{X}^2$ with respect to the parameters $a$ and $b$.
was that your answer? or I'm missing something.