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Diffstat (limited to 'inst/doc/chemCal.tex')
| -rw-r--r-- | inst/doc/chemCal.tex | 89 |
1 files changed, 58 insertions, 31 deletions
diff --git a/inst/doc/chemCal.tex b/inst/doc/chemCal.tex index a9e8495..8887496 100644 --- a/inst/doc/chemCal.tex +++ b/inst/doc/chemCal.tex @@ -1,9 +1,5 @@ \documentclass[a4paper]{article} %\VignetteIndexEntry{Short manual for the chemCal package} -\newcommand{\chemCal}{{\tt chemCal}} -\newcommand{\calplot}{{\tt calplot}} -\newcommand{\calpredict}{{\tt calpredict}} -\newcommand{\R}{{\tt R}} \usepackage{hyperref} \title{Basic calibration functions for analytical chemistry} @@ -13,20 +9,15 @@ \begin{document} \maketitle -The \chemCal{} package was first designed in the course of a lecture and lab -course on "analytics of organic trace contaminants" at the University of Bremen -from October to December 2004. In the fall 2005, an email exchange with -Ron Wehrens led to the belief that it could be heavily improved if the -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} 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 -will have to restrict ourselves to univariate, possibly weighted, linear -regression so far. +a suitable model. If we want to use the \texttt{chemCal} functions, we +will have to restrict ourselves to univariate, possibly weighted +\footnote{ +For the weighted case, the function \texttt{predict.lm} would have to be +adapted (Bug report PR\#8877), in order to allow for weights for the x values +used to predict the y values. This affects the functions \texttt{calplot} +and \texttt{lod}. +}, linear regression so far. Once such a model has been created, the calibration can be graphically shown by using the \texttt{calplot} function: @@ -36,20 +27,39 @@ shown by using the \texttt{calplot} function: > library(chemCal) > data(massart97ex3) > attach(massart97ex3) -> yx <- split(y, factor(x)) +> m0 <- lm(y ~ x) +> calplot(m0) +\end{Sinput} +\end{Schunk} +\includegraphics{chemCal-001} + +As we can see, the scatter increases with increasing x. This is also +illustrated by one of the diagnostic plots for linear models +provided by R: + +\begin{Schunk} +\begin{Sinput} +> plot(m0, which = 3) +\end{Sinput} +\end{Schunk} +\includegraphics{chemCal-002} + +Therefore, in Example 8 in \cite{massart97} weighted regression +is proposed which can be reproduced by + +\begin{Schunk} +\begin{Sinput} +> yx <- split(y, x) > ybar <- sapply(yx, mean) > s <- round(sapply(yx, sd), digits = 2) > w <- round(1/(s^2), digits = 3) > weights <- w[factor(x)] > m <- lm(y ~ x, w = weights) -> calplot(m) \end{Sinput} \end{Schunk} -\includegraphics{chemCal-001} -This is a reproduction of Example 8 in \cite{massart97}. We can -see the influence of the weighted regression on the confidence -and prediction bands of the calibration. +Unfortunately, \texttt{calplot} does not work on weighted linear models, +as noted in the footnote above. If we now want to predict a new x value from measured y values, we use the \texttt{inverse.predict} function: @@ -63,13 +73,29 @@ $Prediction [1] 5.865367 $`Standard Error` -[1] 0.7288138 +[1] 0.892611 + +$Confidence +[1] 2.478285 + +$`Confidence Limits` +[1] 3.387082 8.343652 +\end{Soutput} +\begin{Sinput} +> inverse.predict(m, 90, ws = 0.145) +\end{Sinput} +\begin{Soutput} +$Prediction +[1] 44.06025 + +$`Standard Error` +[1] 2.829162 $Confidence -[1] 2.023511 +[1] 7.855012 $`Confidence Limits` -[1] 3.841856 7.888878 +[1] 36.20523 51.91526 \end{Soutput} \end{Schunk} @@ -77,7 +103,7 @@ The weight \texttt{ws} assigned to the measured y value has to be 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} +\section*{Theory for \texttt{inverse.predict}} 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$: @@ -85,7 +111,8 @@ 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}} + \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) }} + {{b_1}^2 \left( \sum{w_i} \sum{w_i {x_i}^2} - + {\left( \sum{ w_i x_i } \right)}^2 \right) }} \end{equation} with @@ -105,9 +132,9 @@ $y_i$, 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 +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} + |