2.5.6.3. Sensitivity coefficients for measurements from a 2-level design

Sensitivity coefficients from a 2-level design

If the temporal components are estimated from a 2-level nested design, and the reported value for a test item is an average over

• \(N\) short-term repetitions
• \(M\) (\(M = 1\) is permissible) days

of measurements on the test item, the standard deviation for the reported value is: $$ s_{reported \, value} = \sqrt{\frac{1}{M}s_{days}^2 + \frac{1}{MN} s_1^2} $$ See the relationships in the section on 2-level nested design for definitions of the standard deviations and their respective degrees of freedom.

Problem with estimating degrees of freedom

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

Sensitivity coefficients

The sensitivity coefficients are: \(a_1 = \sqrt{\frac{(J-N)}{MNJ}} ; \, a_2 = \sqrt{\frac{1}{M}} \).

Specific sensitivity coefficients are shown in the table below for selections of \(N, \, M\).

  Sensitivity coefficients for two components 
              of uncertainty

Number short-term \(N\)	Number day-to-day \(M\)	Short-term sensitivity coefficient \( a_1\)	Day-to-day sensitivity coefficient \( a_2\)
\(1\)	\(1\)	\( \sqrt{(J-1)/J}\)	\(1\)
\(N\)	\(1\)	\(\sqrt{(J-N)/(NJ)}\)	\(1\)
\(N\)	\(M\)	\(\sqrt{(J-N)/(MNJ)}\)	\(\sqrt{1/M}\)