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

## Sensitivity coefficients for measurements from a 3-level design

Sensitivity coefficients from a 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 $$a_3$$ $$1$$ $$1$$ $$1$$ $$\sqrt{(J-1)/J}$$ $$\sqrt{(K-1)/K}$$ $$1$$ $$N$$ $$1$$ $$1$$ $$\sqrt{(J-N)/(NJ)}$$ $$\sqrt{(K-1)/K}$$ $$1$$ $$N$$ $$M$$ $$1$$ $$\sqrt{(J-N)/(MNJ)}$$ $$\sqrt{(K-M)/(MK)}$$ $$1$$ $$N$$ $$M$$ $$P$$ $$\sqrt{(J-N)/(PMNJ)}$$ $$\sqrt{(K-M)/(MPK)}$$ $$\sqrt{1/P}$$