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Diffstat (limited to 'inst/doc/chemCal.Rnw')
-rw-r--r-- | inst/doc/chemCal.Rnw | 31 |
1 files changed, 26 insertions, 5 deletions
diff --git a/inst/doc/chemCal.Rnw b/inst/doc/chemCal.Rnw index 2c902ab..26b224f 100644 --- a/inst/doc/chemCal.Rnw +++ b/inst/doc/chemCal.Rnw @@ -20,7 +20,7 @@ inverse prediction method given in \cite{massart97} would be implemented, since it also covers the case of weighted regression. At the moment, the package only consists of two functions, working -on univariate linear models of class \texttt{lm}. +on univariate linear models of class \texttt{lm} or \texttt{rlm}. When calibrating an analytical method, the first task is to generate a suitable model. If we want to use the \chemCal{} functions, we @@ -59,9 +59,9 @@ given by the user in the case of weighted regression. By default, the mean of the weights used in the linear regression is used. \section*{Theory} -Equation 8.28 in \cite{massart97} gives a general equation for predicting x -from measurements of y according to the linear calibration function -$ y = b_0 + b_1 \cdot x$: +Equation 8.28 in \cite{massart97} gives a general equation for predicting the +standard error $s_{\hat{x_s}}$ for an x value predicted from measurements of y +according to the linear calibration function $ y = b_0 + b_1 \cdot x$: \begin{equation} s_{\hat{x_s}} = \frac{s_e}{b_1} \sqrt{\frac{1}{w_s m} + \frac{1}{\sum{w_i}} + @@ -72,9 +72,30 @@ s_{\hat{x_s}} = \frac{s_e}{b_1} \sqrt{\frac{1}{w_s m} + \frac{1}{\sum{w_i}} + with \begin{equation} -s_e = \sqrt{ \frac{\sum w_i (y_i - \hat{y})^2}{n - 2}} +s_e = \sqrt{ \frac{\sum w_i (y_i - \hat{y_i})^2}{n - 2}} \end{equation} +where $w_i$ is the weight for calibration standard $i$, $y_i$ is the mean $y$ +value (!) observed for standard $i$, $\hat{y_i}$ is the estimated value for +standard $i$, $n$ is the number calibration standards, $w_s$ is the weight +attributed to the sample $s$, $m$ is the number of replicate measurements of +sample $s$, $\bar{y_s}$ is the mean response for the sample, +$\bar{y_w} = \frac{\sum{w_i y_i}}{\sum{w_i}}$ is the weighted mean of responses +$y_i$, and $x_i$ is the given $x$ value for standard $i$. + +The weight $w_s$ for the sample should be estimated or calculated in accordance +to the weights used in the linear regression. + +I adjusted the above equation in order to be able to take a different precisions +in standards and samples into account. In analogy to Equation 8.26 from \cite{massart97} +we get + +\begin{equation} +s_{\hat{x_s}} = \frac{1}{b_1} \sqrt{\frac{{s_s}^2}{w_s m} + + {s_e}^2 \left( \frac{1}{\sum{w_i}} + + \frac{(\bar{y_s} - \bar{y_w})^2 \sum{w_i}} + {{b_1}^2 \left( \sum{w_i} \sum{w_i {x_i}^2} - {\left( \sum{ w_i x_i } \right)}^2 \right) } \right) } +\end{equation} \begin{thebibliography}{1} \bibitem{massart97} |