 # Convert a quadratic L_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 low real settings ($$\sum_L$$). It is used for L_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.

$$L = \displaystyle\sum_{i=1}^q L_i$$

Rewrite the model substituting…

$$\frac{X_i\, -\, L_i\,}{1\, -\, L}$$ for each component, while replacing the X with the component ID being replaced.

The rewrite of the Pseudo Example is:

$Real = \beta_1\frac{A\, -\, A_L}{1\, -\, L}\, +\, \beta_2\frac{B\, -\, B_L}{1\, -\, L}\, +\, \beta_3\frac{B\, -\, B_L}{1\, -\, L}\, +\, \beta_{12}\frac{A\, -\, A_L}{1\, -\, L}\frac{B\, -\, B_L}{1\, -\, L}\, +\, \beta_{13}\frac{A\, -\, A_L}{1\, -\, L}\frac{C\, -\, C_L}{1\, -\, L}\, +\, \beta_{23}\frac{B\, -\, B_L}{1\, -\, L}\frac{C\, -\, C_L}{1\, -\, L}$

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.

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

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

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

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

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