# Convert a quadratic U_Pseudo mixture model to Real¶

The following information is required for conversion from pseudo to real.

Convert Pseudo limits (0 = minimum, 1 = maximum) to proportion (real) limits.

 Low High A AL AH B BL BH C CL CH

Compute the sum of the high real settings ($$\sum_U$$). It is used for U_pseudo conversion.

The Pseudo model example:

$$Quadratic\, Pseudo = \beta_1A\, +\, \beta_2B\, +\, \beta_3C\, +\, \beta_{12}AB\, +\, \beta_{13}AC\, +\, \beta_{23}BC$$

Insignificant terms with a near zero coefficient should be included in this model.

$$U = \displaystyle\sum_{i=1}^q U_i$$

Rewrite the model substituting…

$$\frac{X_U\, -\, X}{U\, -\, 1}$$ for each component, change the X’s to the component ID being replaced.

The rewrite of the Quadratic Pseudo Example is:

$$Real = \beta_1\frac{A_U\, -\, A}{U\, -\, 1}\, +\, \beta_2\frac{B_U\, -\, B}{U\, -\, 1}\, +\, \beta_3\frac{C_U\, -\, C}{U\, -\, 1}\, +\, \beta_{12}\frac{A_U\, -\, A}{U\, -\, 1}\frac{B_U\, -\, B}{U\, -\, 1}\, +\, \beta_{13}\frac{A_U\, -\, A}{U\, -\, 1}\frac{C_U\, -\, C}{U\, -\, 1}\, +\, \beta_{23}\frac{B_U\, -\, B}{U\, -\, 1}\frac{C_U\, -\, C}{U\, -\, 1}$$

Expand all terms and combine like terms, starting with higher order terms first.

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

• J. Cornell. Experiments with Mixtures. Wiley, 3rd edition, 2002.