The Relative Standard Uncertainty of homogeneity is given by the following calculation:

$$u_r(\text{Homogeneity}) = \frac{ \text{Standard Uncertainty} }{ \text{Grand Mean} } = \frac{ u(\text{Homogeneity}) }{ \overline{X}_T } $$

where \(\displaystyle u(\text{Homogeneity}) = \text{max}\{u_a,u_b\}\),

\(\displaystyle u_a(\text{Homogeneity}) = \sqrt{\frac{ MSS_B - MSS_W }{ n_0 }}\)and\(\displaystyle u_b(\text{Homogeneity}) = \sqrt{ \frac{ MSS_W }{ n_0 } } \times \sqrt{ \frac{ 2 }{ k(n_0-1) } }\)

 

A one-way analysis of variance (ANOVA) test is carried out to calculate the Mean Sum of Squares Between groups (\(MSS_B\)) and the Mean Sum of Squares Within groups (\(MSS_W\)) given by:

\(\displaystyle MSS_B = \frac{ \sum\limits_{j=1}^k n_j(\overline{X}_{j}-\overline{X}_T)^2 } { k-1 } \) and \(\displaystyle MSS_W = \frac{ \sum\limits_{j=1}^k\sum\limits_{i=1}^{n_j} (X_{ij}-\overline{X}_j)^2 } { N-k }\)

 

• \(k\) is the number of groups.
• \(n_j\) is the number of measurements/replicates in the group \(j\) where \(j=1\ldots k\).
• \(\displaystyle n_0 = \frac{1}{k-1} \left[\sum\limits_{j=1}^k n_j - \frac{ \sum\limits_{j=1}^k n_j^2 } { \sum\limits_{j=1}^k n_j }\right] = \frac{1}{k-1} \left[ N - \frac{A}{N} \right] \)
  Where all \(n_j\)'s are the same (i.e. \(n_1=n_2=\ldots=n_k=n\)) then this simplifies to \(n_0 = n\).
• \(N\) is the total number of measurements, i.e. \(N = \sum\limits_{j=1}^k n_j\).
• \(A\) is the sum of squared number of measurements/replicates in the group, i.e. \(A = \sum\limits_{j=1}^k n^2_j\)
• \(X_{ij}\) is the \(i^{th}\) measurement of the \(j^{th}\) group.
• \(\overline{X}_j\) is the mean of measurement in group \(j\).
• \(\overline{X}_T\) is the grand mean, calculated as the sum of all measurements \(\left(\sum\limits_{j=1}^k\sum\limits_{i=1}^{n_j} X_{ij}\right)\) divided by the number of measurements \((N)\).

Using the degrees of freedom and alpha value the F critical value is read from a F-Distribution Table.

All computations and details of formulas used for computing the uncertainty of the calibration curve are displayed here. A quadratic fit has been applied on the uploaded data.

The Method tab shows the main formulas used to compute the quadratic uncertainty of the calibration curve.

When using a quadratic fit currently it is not possible to specify weights for the regression model.

Where a linear fit is required, please return to the start page and upload your data in the "Linear Fit" tab.

Background

For a general function \(X=f(x_1,x_2,\ldots,x_n)\), the variance \(Var(X)\) by Taylor's theorem (first-order expansion) is given by:

\(\displaystyle Var(X) = \left(\frac{\partial X}{\partial x_1}\right)^2 Var(x_1) + \left(\frac{\partial X}{\partial x_2}\right)^2 Var(x_2) + \ldots + \left(\frac{\partial X}{\partial x_n}\right)^2 Var(x_n) + \\ \displaystyle \hspace{3em} 2\left(\frac{\partial X}{\partial x_1}\right) \left(\frac{\partial X}{\partial x_2}\right) Cov(x_1,x_2) + 2\left(\frac{\partial X}{\partial x_1}\right) \left(\frac{\partial X}{\partial x_3}\right) Cov(x_1,x_3) + \ldots \)

 

Using the approach described by D. Brynn Hibbert: The uncertainty of a result from a linear calibration (2006), a quadratic model of the form \(y=b_0 + b_1x + b_2x^2\) can be rewritten as \(y - \overline{y} = b_1(x-\overline{x}) + b_2(x^2 - \overline{x^2})\) to make the curve start from the origin. Doing this removes the covariance dependence of \(b_0\) with \(b_1\) and \(b_2\).

Given an instrument response of case sample peak area ratio \(y_s\), the level of concentration \(x_s\) is estimated by solving for \(x\) as:

\( \displaystyle \hat{x_s} = \frac{ -b_1 \sqrt{ b_1^2-4b_2( \overline{y}-y_s-b_1 \overline{x}-b_2 \overline{x^2}) } } {2b_2} \)

The standard uncertainty \(u\text{(CalCurve)}\) is then obtained by apply Taylor's theorem to the variance of \(\hat{x}_s\).

Using the approach described in D. Brynn Hibbert: The uncertainty of a result from a linear calibration (2006) the uncertainty of the quadratic curve is given by:

\(\displaystyle u\text{(CalCurve)}^2 = \left(\frac{\partial \hat{x_s}}{\partial b_1}\right)^2 Var(b_1) + \left(\frac{\partial \hat{x_s}}{\partial b_2}\right)^2 Var(b_2) + \left(\frac{\partial \hat{x_s}}{\partial \overline{y}}\right)^2 Var(\overline{y}) + \\ \displaystyle \hspace{3em}\left(\frac{\partial \hat{x_s}}{\partial y_s}\right)^2 Var(y_s) + 2\left(\frac{\partial \hat{x_s}}{\partial b_1}\right) \left(\frac{\partial \hat{x_s}}{\partial b_2}\right) Cov(b_1,b_2)\)

Note: As described in the overview, this formula removes the covariance dependency of \(b_0\) with the other parameters \(b_1\) and \(b_2\) by making the model start from the origin [Pg. 4/5 - D. Brynn Hibbert (2006)].

The Relative Standard Uncertainty is then given by:

\(\displaystyle u_r\text{(CalCurve)} = \frac{u\text{(CalCurve)}}{x_s}\)

The partial derivatives is obtained by differentiating the equation below with respect to \(b_1, b_2, \overline{y}, y_0\)

\(\displaystyle\hat{x_s} = \frac{-b_1\sqrt{b_1^2-4b_2(\overline{y}-y_s-b_1\overline{x}-b_2\overline{x^2})}}{2b_2}\)

Note: It is assumed that the regression parameters are independent and that \(Var(\overline{x}) = 0\) and \(Var(\overline{x^2}) = 0\).

The Covariance Matrix for \(Var(b_1)\), \(Var(b_2)\) and \(Cov(b_1,b_2)\) can be estimated as:

\(\sigma^2(\underline{X}^T\underline{X})^{-1} = \begin{Bmatrix} Var(b_0) & Cov(b_0,b_1) & Cov(b_0,b_2) \\ Cov(b_0,b_1) & Var(b_1) & Cov(b_1,b_2) \\ Cov(b_0,b_2) & Cov(b_1,b_2) & Var(b_2) \\ \end{Bmatrix}\)

where \(\sigma^2\) is the variance of \(y\) estimated by the Standard Error of Regression squared \(S_{y/x}^2\):

\(\displaystyle S_{y/x} = \sqrt{\frac{\sum\limits_{i=1}^n (y_i-\hat{y}_i)^2}{n-3}}\)

• \(x_i\) concentration at level \(i\).
• \(x_s\) is the mean concentration of the Case Sample.
• \(r_s\) is the number of replicates made on test sample to determine \(x_s\).
• \(y_i\) observed peak area ratio for a given concentration \(x_i\).
• \(\hat{y}_i\) predicted value of \(y\) for a given value \(x_i\).
• \(b_1\) is the Slope of the of regression line.
• \(y_s\) is the mean of Peak Area Ratio of the Case Sample.
• \(n\) is the number of measurements used to generate the Calibration Curve.
• \(\overline{x}\) is the mean values of the different calibration standards.
• \(Var(\overline{y})\) is the variance of the mean of \(y\), given as: \(\frac{S_{y/x}^2}{n}\).
• \(Var(y_s)\) is the variance of the mean of Case Sample Peak Area Ratios \((y_s)\), given as: \(\frac{S_{y/x}^2}{r_s}\)

The Relative Standard Uncertainty of method precision is given by: $$u_r(\text{MethodPrec})_{\text{(NV)}} = \frac{S_{p\text{(NV)}}}{\overline{x}_{\text{(NV)}}\sqrt{r_s}} = \frac{u(\text{MethodPrec})_{\text{(NV)}}}{\overline{x}_{\text{(NV)}}},$$ where $$u(\text{MethodPrec})_{\text{(NV)}} = \frac{S_{p\text{(NV)}}}{\sqrt{r_s}},$$ and $$S_{p(\text{NV})} = \sqrt{\frac{\sum{(S^2 \times {\large\nu})_{\text{(NV)}}}}{\sum {\large\nu}_{\text{(NV)}}}}.$$

• \(S\) is the standard deviation for each run.
• \({\large\nu}\) is the individual degrees of freedom.
• \(S_p\) is the pooled standard deviation.
• \(NV\) is the nominal value of concentration.
• \(\overline{x}_{\text{(NV)}}\) is the mean concentration for the nominal value \(\text{NV}\).

The Relative Standard Uncertainty (RSU) of each equipment is computed using: $$u_r(\text{Equipment}) = \frac{\frac{\text{Tolerance}}{\text{Coverage Factor}}}{\text{Volume}}$$ The RSU of reference compound is calculated using: $$u_r(\text{Reference Compound}) = \frac{\frac{\text{Tolerance}}{\text{Coverage Factor}}}{\text{Purity}}$$ The RSU of each solution is computed using: $$u_r\text{(WorkingSolution)} = \sqrt{u_r\text{(Parent Solution)}^2 + \sum{[u_r\text{(Equipment)}^2_{\text{(Vol,Tol)}} \times N\text{(Equipment)}_{\text{(Vol,Tol)}}]}}$$ The RSU of standard (spiking) solution is obtained by pooling the RSU's of the calibration curve spiking range. $$u_r(\text{CalStandard}) = \sqrt{\sum{u_r\text{(Calibration Curve Spiking Range)}^2}}$$

• \(\text{Parent Solution}\) is the solution from which a given solution is made.
• \(N\text{(Equipment)}\) is the number of times a piece of equipment is used in the preparation of a given solution.

The Relative Standard Uncertainty (RSU) of each equipment is computed using: $$u_r(\text{Equipment}) = \frac{\frac{\text{Tolerance}}{\text{Coverage Factor}}}{\text{Capacity}}$$ The RSU of sample preparation is given by: $$u_r\text{(SamplePreparation)} = \sqrt{\sum{[u_r(\text{Equipment})_{\text{(Cap,Tol)}}^2 \times N(\text{Equipment})_{\text{(Cap,Tol)}}}]}$$

• \(N\text{(Equipment)}\) is the number of times a piece of equipment is used in taking the preparation of a given sample.

The combined uncertainty is given by: $$\text{CombUncertainty} = x_s \sqrt{\sum{u_r\text{(Individual Uncertainty Component)}^2}}$$

• Where \(x_s\) is the Case Sample Mean Concentration.

The effective degrees of freedom \(({\LARGE\nu}_{\text{eff}})\) using Welch-Satterthwaite approximation for relative standard uncertainty is given by:

$${\LARGE\nu}_{\text{eff}} =\frac{(\frac{\text{CombUncertainty}}{x_s})^4}{\sum{\frac{u_r\text{(Individual Uncertainty Component)}^4}{{\LARGE\nu}_{\text{(Individual Uncertainty Component)}}}}}$$

The coverage factor \((k_{{\large\nu}_{\text{eff}}, {\small CI\%}})\) is read from the T-Distribution Table using the calculated \({\Large\nu}_{\text{eff}}\) and specified \({\small CI\%}\).

• \(x_s\) is the Case Sample Mean Concentration.
• \(\nu\) is the Degrees of Freedom for each uncertainty component.
• \({\small\text{CombUncertainty}}\) is the Combined Uncertainty of the individual uncertainty components.

The expanded uncertainty \(\text{(ExpUncertainty)}\) is given by:

$$\text{ExpUncertainty} = \text{CoverageFactor} \times \text{CombUncertainty}$$

Where the \(\text{CoverageFactor}\) is \(k\) when a value has been manually specified or \(k_{{\large\nu}_{\text{eff}}, {\small CL\%}}\) when a Confidence Level \(CL\%\) has been specified.

The percentage expanded uncertainty can then be given by:

$$\text{%ExpUncertainty} = \frac{\text{ExpUncertainty}}{x_s} \times 100$$

where:

• \(k_{{\large\nu}_{\text{eff}}, {\small CL\%}}\) is the coverage factor based on effective degrees of freedom and specified confidence level percentage.
• \({\small\text{CombUncertainty}}\) is the combined uncertainty.
• \(x_s\) case sample mean concentration.