In chemical analysis, the instrument's signal are plotted as a function of chemical concentration. In general, higher the concentration higher is the response and the relationship is linear. At elementary level, textbooks teach to do ordinary least squares. In reality, the standard deviation of the instrument's signal increases with concentration, thus a proper way is to apply weighted least squares regression. The following is an adapted protocol from a regulatory body (ISO).
The standard of deviation of $n$ replicates of $Y_i$ measurements (instrument response corresponding to the $i^{th}$ concentration) trends linearly with the $x$ variable (analyte concentration) so each $Y_i$ value must be weighted ($w_i$) according to the inverse of variance at that concentration $(\sigma^2(x_i))$
$$ w_i=\frac{1}{\sigma^2\left(x_i\right)}=\frac{1}{\left(c+d x_i\right)^2}$$
However, since the $\sigma^2\left(x_i\right)$ is not known in the above equation, it must be modeled. We can apply a linear model here $(\sigma_i=c+bx_i)$. Since the intercept $(c)$ and slope $(d)$ of this linear model are not known, they must estimated as $\hat{c}$ and $\hat{d}$. A regulatory body recommends the following iterative procedure with weights of its own $(\hat{w}_{q_i})$:
$$\hat{\sigma}_{q i}=\hat{c}_q+\hat{d}_q x_i$$
$$\hat{w}_{q i}=\frac{1}{\left(\hat{\sigma}_{q i}\right)^2}$$
To calculate the weights $(\hat{w})$ used to estimate $\hat{c}$ and $\hat{d}$ we start the zeroth iteration $(q=0)$ assuming that $\hat{\sigma}_{0 i}=s_i$ where $s_i$ is the empirical standard deviation of the measured replicate $Y_i$ values. Then $\hat{\sigma}_{0 i}$ is used to calculate $\hat{w}_0$ through the above equation. The zeroth iteration ends with $\hat{w}_0$ being used to calculate $\hat{c}_1$ and $\hat{d}_1$ through the following “auxiliary values” $T_{1-5}$:
\begin{aligned} & T_{q+1,1}=\sum_{i=1}^I \hat{w}_{q i} \\ & T_{q+1,2}=\sum_{i=1}^I \hat{w}_{q i} x_i \\ & T_{q+1,3}=\sum_{i=1}^I \hat{w}_{q i} x_i^2 \\ & T_{q+1,4}=\sum_{i=1}^I \hat{w}_{q i} s_i \\ & T_{q+1,5}=\sum_{i=1}^I \hat{w}_{q i} x_i s_i \\ & \hat{c}_{q+1}=\frac{T_{q+1,3} T_{q+1,4}-T_{q+1,2} T_{q+1,5}}{T_{q+1,1} T_{q+1,3}-T_{q+1,2}^2} \\ & \hat{d}_{q+1}=\frac{T_{q+1,3} T_{q+1,5}-T_{q+1,2} T_{q+1,4}}{T_{q+1,1} T_{q+1,3}-T_{q+1,2}^2} \end{aligned}
These values are used to update the weights in the second and third equation until $\hat{c}_{q+1}$ and $\hat{d}_{q+1}$ satisfactory converge. Finally, the following equation is applied to calculate the weight needed for each $Y_i$ in the WLS regression we ultimately desired:
$$\hat{w}_{i}=\frac{1}{\hat{c}_{q+1}+\hat{d}_{q+1} x_i}$$
Does anyone know a book or paper that has the origin of this procedure? Why do we need to do a weighted least squares techniques on the regression used to calculate the weights in the desired least squares problem? How is this statistically different than using ordinary least squares to calculate the $\hat{c}$ and $\hat{d}$?
