Zero Point Proportional Calculations

For the zero point proportional relationship, the dynamic signal-to-noise ratio is calculated as follows:

$r_0$ = number of noise experiments

$k$ = number of signal factor levels

1. Calculate the slope, $\beta$:
   $$\beta =\frac{1}{r}\left(M_1{y}_1+M_2{y}_2+\dots +M_k{y}_k\right)$$

   where

   $$r=r_0\left(M_1^2+M_2^2+\cdots +M_k^2\right)$$

   and $y_k$ is the sum of the response values at signal level $k$.

2. Calculate the total sum of the squares:
   $$S_T=y_{11}^2+y_{12}^2+\cdots +y_{k{y}_0}^2$$
3. Calculate the variation caused by the linear effect:
   $$S_{\beta }=\frac{1}{r}{\left(M_1{y}_1+M_2{y}_2+\cdots +M_k{y}_k\right)}^2$$
4. Calculate the variation associated with error and nonlinearity:
   $$S_e=S_T-S_{\beta }$$
5. Calculate the error variance:
   $$V_e=\frac{1}{k{r}_0-1}{S}_e$$
6. Calculate the dynamic $S/N$ ratio:
   $$\eta =10log\frac{1}{\text{r}}\frac{\left(S_{\beta }-V_e\right)}{V_e}$$

The dynamic $S/N$ ratio is calculated for each control experiment. The control experiment with the largest $S/N$ ratio value represents the best design (best combination of linear fit and low variability) of those executed. The main effect on the dynamic $S/N$ ratio can be used to identify other combinations of control factor settings that may produce even better designs.