Convert a quadratic L_Pseudo mixture model to Real

Quadratic term expansion

Showing the BC quadratic term as an example. Use this procedure for each quadratic term.

$\beta_{23}\frac{B\, -\, B_L}{1\, -\, L}\frac{C\, -\, C_L}{1\, -\, L}$

$\beta_{23}\frac{BC\, -\, B_LC\, -\, C_LB\, +\, B_LC_L}{(1\, -\, L)^2}$

$\beta_{23}\begin{bmatrix}\frac{BC}{(1\, -\, L)^2}\, +\, \frac{-B_LC}{(1\, -\, L)^2}\, +\, \frac{-C_LB}{(1\, -\, L)^2}\, +\, \frac{B_LC_L}{(1\, -\, L)^2} \end{bmatrix}$

The BC coefficient is changed to $\beta_{23}\,/\,(1\, -\, L)^2$ in the real model.

$-\beta_{23}B_L\,/\,(1\, -\, L)^2$ is a correction that will be applied to the C coefficient.

$-\beta_{23}C_L\,/\,(1\, -\, L)^2$ is a correction that will be applied to the B coefficient.

$B_LC_L\,/\,(1\, -\, L)^2$ is a constant which requires special handling.

From the mixture design property of a constant total, we know that $A\, +\, B\, +\, C\, =\, 1$ in terms of the reals. Rewrite $B_LC_L$ as $B_LC_L\, \cdot\, 1$ and substitute $[A\, +\, B\, +\, C]$ for 1, yielding $B_LC_L\, \cdot\, [A\, +\, B\, +\, C]$. When expanded, the result is an adjustment to all the linear coefficients of $\beta_{23}B_LC_L\ /\, (1\, -\, L)^2$.

Linear term expansion

Showing the C term as the example. Use this procedure for all the linear terms.

$\beta_3\frac{C\, -\, C_L}{1\, -\, L}\, =\, \beta_3\begin{bmatrix}\frac{C}{1\, -\, L}\, +\, \frac{-C_L}{1\, -\, L}\end{bmatrix}$

$\beta_3\,/\,(1\, -\, L)$ is the base coefficient for the C linear effect. This will be adjusted by quadratic and other linear effect adjustments.

$-C_L\,/\,(1\, -\, L)$ is a constant which is treated the same as the quadratic term’s constant becoming, $-\beta_3C_L\,/\,(1\, -\, L)\, \cdot\, [A\, +\, B\, +\, C]$. Each linear term creates an adjustment to all linear terms.

Combine Like Terms

After working through each term in the model, combine like terms into new coefficients for the real model.

$\beta_A = \frac{\beta_1}{1\, -\, L}\, -\, \frac{\beta_1A_L\, +\, \beta_2B_L\, +\, \beta_3C_L}{1\, -\, L}\, +\, \frac{\beta_{12}(A_LB_L\, -\, B_L)\, +\, \beta_{13}(A_LC_L\, -\, C_L)\, +\, \beta_{23}(B_LC_L)}{(1\, -\, L)^2}$

$\beta_B = \frac{\beta_2}{1\, -\, L}\, -\, \frac{\beta_1A_L\, +\, \beta_2B_L\, +\, \beta_3C_L}{1\, -\, L}\, +\, \frac{\beta_{12}(A_LB_L\, -\, A_L)\, +\, \beta_{23}(B_LC_L\, -\, C_L)\, +\, \beta_{13}(A_LC_L)}{(1\, -\, L)^2}$

$\beta_C = \frac{\beta_3}{1\, -\, L}\, -\, \frac{\beta_1A_L\, +\, \beta_2B_L\, +\, \beta_3C_L}{1\, -\, L}\, +\, \frac{\beta_{13}(A_LC_L\, -\, A_L)\, +\, \beta_{23}(B_LC_L\, -\, B_L)\, +\, \beta_{23}(A_LB_L)}{(1\, -\, L)^2}$

$\beta_{AB} = \frac{\beta_{12}}{(1\, -\, L)^2}$

$\beta_{AC} = \frac{\beta_{13}}{(1\, -\, L)^2}$

$\beta_{BC} = \frac{\beta_{23}}{(1\, -\, L)^2}$

References