From b561996c588b6602b24aa1d236dee88f7c1d85a1 Mon Sep 17 00:00:00 2001
From: ranke <ranke@5fad18fb-23f0-0310-ab10-e59a3bee62b4>
Date: Tue, 11 Sep 2012 23:56:14 +0000
Subject: - Fixed the link to the bug report in the vignette - thanks to John
 Thaden for the hint - Moved the vignette to the vignettes/ folder in order to
 avoid to write an .Rinstignore file - Made the vignette compile in trunk, the
 trunk contains unfinished stuff however

git-svn-id: http://kriemhild.uft.uni-bremen.de/svn/chemCal@29 5fad18fb-23f0-0310-ab10-e59a3bee62b4
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+\documentclass[a4paper]{article}
+%\VignetteIndexEntry{Short manual for the chemCal package}
+\usepackage{hyperref}
+
+\title{Basic calibration functions for analytical chemistry}
+\author{Johannes Ranke}
+
+\usepackage{Sweave}
+\begin{document}
+\maketitle
+
+The \texttt{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 would be desirable to implement the
+inverse prediction method given in \cite{massart97} since it also covers the
+case of weighted regression. Studies of the IUPAC orange book and of DIN 32645
+as well as publications by Currie and the Analytical Method Committee of the 
+Royal Society of Chemistry and a nice paper by Castillo and Castells provided
+further understanding of the matter.
+
+At the moment, the package consists of four functions, working on univariate
+linear models of class \texttt{lm} or \texttt{rlm}, plus to datasets for
+validation.
+
+A \href{http://bugs.r-project.org/bugzilla3/show_bug.cgi?id=8877}{bug
+report (PR\#8877)} and the following e-mail exchange on the r-devel mailing list about
+prediction intervals from weighted regression entailed some further studies
+on this subject. However, I did not encounter any proof or explanation of the
+formula cited below yet, so I can't really confirm that Massart's method is correct.
+
+When calibrating an analytical method, the first task is to generate a suitable
+model. If we want to use the \texttt{chemCal} functions, we will have to restrict
+ourselves to univariate, possibly weighted, linear regression so far.
+
+Once such a model has been created, the calibration can be graphically
+shown by using the \texttt{calplot} function:
+
+\begin{Schunk}
+\begin{Sinput}
+> library(chemCal)
+> data(massart97ex3)
+> m0 <- lm(y ~ x, data = massart97ex3)
+> 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}
+> attach(massart97ex3)
+> 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)
+\end{Sinput}
+\end{Schunk}
+
+If we now want to predict a new x value from measured y values,
+we use the \texttt{inverse.predict} function:
+
+\begin{Schunk}
+\begin{Sinput}
+> inverse.predict(m, 15, ws=1.67)
+\end{Sinput}
+\begin{Soutput}
+$Prediction
+[1] 5.865367
+
+$`Standard Error`
+[1] 0.8926109
+
+$Confidence
+[1] 2.478285
+
+$`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
+
+$Confidence
+[1] 7.855012
+
+$`Confidence Limits`
+[1] 36.20523 51.91526
+\end{Soutput}
+\end{Schunk}
+
+The weight \texttt{ws} assigned to the measured y value has to be 
+given by the user in the case of weighted regression, or alternatively,
+the approximate variance \texttt{var.s} at this location.
+
+\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$:
+
+\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) }}
+\end{equation}
+
+with
+
+\begin{equation}
+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}
+
+where I interpret $\frac{{s_s}^2}{w_s}$ as an estimator of the variance at location
+$\hat{x_s}$, which can be replaced by a user-specified value using the argument
+\texttt{var.s} of the function \texttt{inverse.predict}.
+
+\begin{thebibliography}{1}
+\bibitem{massart97}
+Massart, L.M, Vandenginste, B.G.M., Buydens, L.M.C., De Jong, S., Lewi, P.J.,
+Smeyers-Verbeke, J. 
+\newblock Handbook of Chemometrics and Qualimetrics: Part A,
+\newblock Elsevier, Amsterdam, 1997
+\end{thebibliography}
+
+\end{document}
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