Convert a quadratic U_Pseudo mixture model to Real

Quadratic term expansion

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

$\beta_{23}\frac{B_U\, -\, B}{U\, -\, 1}\frac{C_U\, -\, C}{U\, -\, 1}$

$\beta_{23}\frac{B_UC_U\, -\, BC_U\, -\, CB_U\, +\, BC}{(U\, -\, 1)^2}$

$\beta_{23}\begin{bmatrix}\frac{B_UC_U}{(U\, -\, 1)^2}\, +\, \frac{-BC_U}{(U\, -\, 1)^2}\, +\, \frac{-CB_U}{(U\, -\, 1)^2}\, +\, \frac{BC}{(U\, -\, 1)^2}\end{bmatrix}$

$B_UC_U\, /\, (U\, -\, 1)^2$ 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 $B_UC_U$ as $B_UC_U\, \cdot\, 1$ and substitute $[A\, +\, B\, +\, C]$ for 1, yielding $B_UC_U\, \cdot\, [A\, +\, B\, +\, C]$. When expanded, the result is an adjustment to all the linear coefficients of $\beta_{23}B_UC_U\, /\, (U\, -\, 1)^2$.

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

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

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

Linear term expansion

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

$\beta_3\frac{C_U\, -\, C}{U\, -\, 1}\, =\, \beta_3\begin{bmatrix}\frac{C_U}{U\, -\, 1}\, +\, \frac{-C}{U\, -\, 1} \end{bmatrix}$

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

$C_U\, /\, (U\, -\, 1)$ is a constant which is treated the same as the quadratic term’s constant becoming, $\beta_3C_U\, /\, (U\, -\, 1)\, \cdot\, [A\, +\, B\, +\, C]$ . Each linear term in the pseudo model creates an adjustment to all linear terms in the real model.

Combine Like Terms

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

$\beta_A = \frac{-\beta_1}{U\, -\, 1}\, +\, \frac{\beta_1A_U\, +\, \beta_2B_U\, +\, \beta_3C_U}{U\, -\, 1}\, + \frac{\beta_{12}(A_UB_U\, -\, B_U)\,+\, \beta_{13}(A_UC_U\, -\, C_U)\, +\, \beta_{23}(B_UC_U)}{(U\, -\, 1)^2}$

$\beta_B = \frac{-\beta_2}{U\, -\, 1}\, +\, \frac{\beta_1A_U\, +\, \beta_2B_U\, +\, \beta_3C_U}{U\, -\, 1}\, +\, \frac{\beta_{12}(A_UB_U\, -\, A_U)\, +\, \beta_{23}(B_UC_U\, -\, B_U)\, +\, \beta_{13}(A_UC_U)}{(U\, -\, 1)^2}$

$\beta_C = \frac{-\beta_3}{U\, -\, 1}\, +\, \frac{\beta_1A_U\, +\, \beta_2B_U\, +\, \beta_3C_U}{U\, -\, 1}\, +\, \frac{\beta_{13}(A_UC_U\, -\, A_U)\, +\, \beta_{23}(B_UC_U\, -\, B_U)\, +\, \beta_{13}(A_UC_U)}{(U\, -\, 1)^2}$

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

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

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

References