2.5.6.4. Sensitivity coefficients for measurements from a 3-level design

2. Measurement Process Characterization
2.5. Uncertainty analysis
2.5.6. Uncertainty budgets and sensitivity coefficients

2.5.6.4. Sensitivity coefficients for measurements from a 3-level design

Sensitivity coefficients from a 3-level design

Case study showing sensitivity coefficients for 3-level design

If the temporal components are estimated from a 3-level nested design and the reported value is an average over

$N$ short-term repetitions
$M$ days
$P$ runs

of measurements on the test item, the standard deviation for the reported value is: $$ s_{reported \, value} = \sqrt{ \frac{1}{P} s_{runs}^2 + \frac{1}{PM} s_{days}^2 + \frac{1}{PMN} s_1^2 } $$ See the section on analysis of variability for definitions and relationships among the standard deviations shown in the equation above.

Problem with estimating degrees of freedom

If degrees of freedom are required for the uncertainty, the formula above cannot be used directly and must be rewritten in terms of the standard deviations $s_1$, $s_2$, and $s_3$. $$ s_{reported \, value} = \sqrt{ \frac{1}{P} s_3^2 + \frac{K-M}{PMK} s_2^2 + \frac{J-N}{PMNJ} s_1^2 } $$

Sensitivity coefficients

The sensitivity coefficients are: $$ \begin{eqnarray*} a_1 & = & \sqrt{\frac{(J-N)}{PMNJ}} \\ a_2 & = & \sqrt{\frac{(K-M)}{PMK}} \\ a_3 & = & \sqrt{\frac{1}{P}} \\ \end{eqnarray*} $$

Specific sensitivity coefficients are shown in the table below for selections of $N, \, M, \, P$. In addition, the following constraints must be observed:

$J$ must be $\ge$ $N$ and $K$ must be $\le$ $M$

Sensitivity coefficients for three components of uncertainty

Number short-term $N$	Number day-to-day $M$	Number run-to-run $P$	Short-term sensitivity coefficient $a_1$	Day-to-day sensitivity coefficient $a_2$	Run-to-run sensitivity coefficient $a_3$
$1$	$1$	$1$	$\sqrt{(J-1)/J}$	$\sqrt{(K-1)/K}$	$1$
$N$	$1$	$1$	$\sqrt{(J-N)/(NJ)}$	$\sqrt{(K-1)/K}$	$1$
$N$	$M$	$1$	$\sqrt{(J-N)/(MNJ)}$	$\sqrt{(K-M)/(MK)}$	$1$
$N$	$M$	$P$	$\sqrt{(J-N)/(PMNJ)}$	$\sqrt{(K-M)/(MPK)}$	$\sqrt{1/P}$

Number short-term \(N\)	Number day-to-day \(M\)	Number run-to-run \(P\)	Short-term sensitivity coefficient \(a_1\)	Day-to-day sensitivity coefficient \(a_2\)	Run-to-run sensitivity coefficient \(a_3\)
\(1\)	\(1\)	\(1\)	\(\sqrt{(J-1)/J}\)	\(\sqrt{(K-1)/K}\)	\(1\)
\(N\)	\(1\)	\(1\)	\(\sqrt{(J-N)/(NJ)}\)	\(\sqrt{(K-1)/K}\)	\(1\)
\(N\)	\(M\)	\(1\)	\(\sqrt{(J-N)/(MNJ)}\)	\(\sqrt{(K-M)/(MK)}\)	\(1\)
\(N\)	\(M\)	\(P\)	\(\sqrt{(J-N)/(PMNJ)}\)	\(\sqrt{(K-M)/(MPK)}\)	\(\sqrt{1/P}\)