I am using Slope ratio method for estimation of relative potency in a drug. The rows in matrices correspond to number of parallels and columns are doses.
I have a two methods for calculating the slope ratio: Case 1 and Case 2.
Case 1
Let, $$ X = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{22} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix} , \quad T_1 = \begin{bmatrix} t_{11} & t_{12} & t_{13} \\ t_{22} & t_{22} & t_{23}\\ t_{31} & t_{32} & t_{33} \end{bmatrix} , \quad T_2 = \begin{bmatrix} y_{11} & y_{12} & y_{13} \\ y_{22} & y_{22} & y_{23}\\ y_{31} & y_{32} & y_{33} \end{bmatrix}, \quad T_3 = \begin{bmatrix} z_{11} & z_{12} & z_{13} \\ z_{22} & z_{22} & z_{23}\\ z_{31} & z_{32} & z_{33} \end{bmatrix} $$
let $\bar{x_i}$ be the average of columns of $X$, $\bar{t}$ of $T_1$, $\bar{y}$ of $T_2$ and $\bar{z}$ of $T_3$.
Define, $$ S = \begin{bmatrix} s_1 \\ s_2 \\ s_3 \end{bmatrix}, \quad K = \begin{bmatrix} k_1 \\ k_2 \\ k_3 \end{bmatrix} $$
Regression: $$ \left. \begin{align} \bar{x}' &= \alpha + \beta_S S, & &\\ \bar{t}' &= \alpha_1 + \beta_{t} K,&\lambda_1 &= \frac{\beta_{t}}{\beta_S} \\ \bar{y}' &= \alpha_2 + \beta_{y} K,& \lambda_2 &= \frac{\beta_{y}}{\beta_S} \\ \bar{z}' &= \alpha_3 + \beta_{z} K & \lambda_3 &= \frac{\beta_{z}}{\beta_S} \end{align} \quad \right\rbrace \quad \lambda = \frac{\lambda_1 + \lambda_2 + \lambda_3}{3} $$
Case 2
Averages of columns are not taken. Say, $$ X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} ,\quad \text{where}: x_1 = \{a_{11}, a_{12}, a_{13}\}, \quad x_2= \{a_{21}, a_{22}, a_{23}\} \dots $$ Similarly for rows of $T_1$, $T_2$, and $T_3$. So that, e.g. $t_1 = \{t_{11}, t_{12}, t_{13}\}, t_2=\{t_{21}, t_{22}, t_{23}\} \quad \dots$
Now the slope ratios are: $$ \begin{align} x_1' &= \alpha_1 + \beta_{x_1} S, & t_1' &= \alpha_4 + \beta_{t_1} K,\quad t_2' = \dots\quad, & \lambda_1 &=\frac{(\beta_{t_1} + \beta_{t_2} + \beta_{t_3})/3}{\beta_{x_1}} \\ x_2' &= \alpha_2 + \beta_{x_2} S, & y_1' &= \alpha_5 + \beta_{y_1} K,\quad y_2' = \dots \quad,& \lambda_2 &=\frac{(\beta_{y_1} + \beta_{y_2} + \beta_{y_3})/3}{\beta_{x_2}} \\ x_3' &= \alpha_3 + \beta_{x_3} S, & z_1' &= \alpha_6 + \beta_{z_1} K,\quad z_2' = \dots\quad, & \lambda_3 &=\frac{(\beta_{z_1} + \beta_{z_2} + \beta_{z_3})/3}{\beta_{x_3}} \end{align} $$ Then, the overall slope ratio is $\bar{\lambda}$.
For those wondering why I am complicating the calculations, the regression coefficients for individual tests, i.e. $T_i$ must be calculated and then taken average of.
Questions
Intuitively, I believe that Case 1 is not representative of the sample population, but I am not sure. How can I prove it? Or are both methods the same?
My question was regarding bioassays. I needed to find parallelism between $S$ and $T_i$. Both cases given in the question were wrong. The correct way to do this is described in European Pharmacopoeia, chapter "5.3 Statistical Analysis of Results of Biological Assays and Tests" and USP <1210> " Statistical Tools for Procedure Validation". For parallelism test can be conducted by the equivalence test (TOST procedure).