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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}
-
-\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:
-
-<<echo=TRUE,fig=TRUE>>=
-library(chemCal)
-data(massart97ex3)
-m0 <- lm(y ~ x, data = massart97ex3)
-calplot(m0)
-@
-
-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: 
-
-<<echo=TRUE,fig=TRUE>>=
-plot(m0,which=3)
-@
-
-Therefore, in Example 8 in \cite{massart97} weighted regression
-is proposed which can be reproduced by
-
-<<>>=
-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)
-@
-
-If we now want to predict a new x value from measured y values,
-we use the \texttt{inverse.predict} function:
-
-<<>>=
-inverse.predict(m, 15, ws=1.67)
-inverse.predict(m, 90, ws = 0.145)
-@
-
-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}.
-
-\section*{Fitting and using a variance function}
-
-In the R package \texttt{nlme} variance functions are used for weighted 
-regressions. But since the \texttt{predict.nlme} method does not calculate
-prediction intervals, this is not useful for the \texttt{calplot} function.
-
-Two approaches could be used for fitting variance functions, one based on
-residuals from an unweighted fit, and one based on just the variances
-of the different samples along the x axis. If we used the residuals for
-fitting, a bias of the model in a certain area would result in a higher
-variance, so it seems preferable to choose the second approach. Of course,
-a prerequisite is to have sufficient repetitions for each sample in any
-case.
-
-Let's take the above example and estimate a variance function
-
-<<>>=
-library(doBy)
-massart97ex3
-massart97ex3$x <- factor(massart97ex3$x)
-summary <- summaryBy(y~x, data = massart97ex3,FUN=c(mean,sd,var))
-summary$x <- as.numeric(as.vector((summary$x)))
-plot(summary$x, summary$y.var,
-	xlim=c(0,50), 
-	ylim=c(0,max(summary$y.var)))
-varModel <- lm(y.var ~ I(x^2) + x, data=summary)
-varCurve <- predict(varModel, newdata=data.frame(x=0:5000/100))
-lines(0:5000/100,varCurve)
-@
-
-
-\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}