Reference Point Proportional Calculations

The relationship between the response and signal factor for this case can be modeled as

y - {\bar{y}}_{s} = β (M - M_{s})

where $y$ is the response, ${\bar{y}}_{s}$ is the average from the reference standard data (data obtained with the signal factor at the reference value), $M$ is the signal factor, $M_{s}$ is the reference standard (the reference signal factor level), and $β$ is the slope of the line fit to the signal/response data.

The reference standard response average is calculated as follows:

y_{s} = \frac{(y_{1} + y_{2} + \dots + y_{j})}{r_{0}}

where $y_{1}$ to $y_{j}$ are the response values at the reference standard signal factor level for a given control experiment.

The reference standard average is then subtracted from each response data value, for a given control experiment number, and the reference standard signal factor level is subtracted from all signal factor levels. The steps for calculating the dynamic signal-to-noise ratio for the reference proportional relationship, using the reference standard adjusted data and signal factor levels, are then given as follows:

$r_{0}$ = number of noise experiments

$k$ = number of signal factor levels

Calculate the slope, $β$ :
$β = \frac{1}{r} ((M_{1} - M_{s}) y_{1} + (M_{2} - s) y_{2} + \dots + (M_{k} - M_{s}) y_{k})$

where
$r = r_{0} ({(M_{1} - M_{s})}^{2} + {(M_{2} - M_{s})}^{2} + \dots + {(M_{k} - M_{s})}^{2})$

and $y_{k}$ is the sum of the response values at signal level $k$ .
Calculate the total sum of the squares:
$S_{T} = \sum_{i = 1}^{k} \sum_{j = 1}^{y_{0}} {(y_{i j} - {\bar{y}}_{s})}^{2}$
Calculate the variation caused by the linear effect:
$S_{β} = \frac{1}{r} {((M_{1} - M_{s}) y_{1} + (M_{2} - M_{s}) y_{2} + \dots + (M_{k} - M_{s}) y_{k})}^{2}$
Calculate the variation associated with error and nonlinearity:
$S_{e} = S_{T} - S_{β}$
Calculate the error variance:
$V_{e} = \frac{1}{k r_{0} - 1} S_{e}$
Calculate the dynamic $S / N$ ratio:
$n = 10 \log \frac{1}{r} \frac{(S_{β} - V_{e})}{V_{e}}$