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 
A_{L} 
A_{H} 
B 
B_{L} 
B_{H} 
C 
C_{L} 
C_{H} 
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:
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 \(\beta_{23}B_LC_L\ /\, (1\, \, L)^2\).
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.
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.