diff options
| author | ranke <ranke@5fad18fb-23f0-0310-ab10-e59a3bee62b4> | 2007-10-01 19:49:20 +0000 |
|---|---|---|
| committer | ranke <ranke@5fad18fb-23f0-0310-ab10-e59a3bee62b4> | 2007-10-01 19:49:20 +0000 |
| commit | 8e511e8c4bb9b6077b667aea808091774b1b84a6 (patch) | |
| tree | 66d9bec143f31bf227c9810391c059ff1fc3fd81 | |
| parent | 6865f34bfe02ceae7027fcb0bc7d074d84369cf1 (diff) | |
Imported revision 18, because this was last released on CRAN
with version number 0.1-18
git-svn-id: http://kriemhild.uft.uni-bremen.de/svn/chemCal@24 5fad18fb-23f0-0310-ab10-e59a3bee62b4
36 files changed, 7720 insertions, 0 deletions
diff --git a/branches/0.1/chemCal/DESCRIPTION b/branches/0.1/chemCal/DESCRIPTION new file mode 100644 index 0000000..07ee5a0 --- /dev/null +++ b/branches/0.1/chemCal/DESCRIPTION @@ -0,0 +1,19 @@ +Package: chemCal +Version: 0.1-18 +Date: 2006-06-23 +Title: Calibration functions for analytical chemistry +Author: Johannes Ranke <jranke@uni-bremen.de> +Maintainer: Johannes Ranke <jranke@uni-bremen.de> +Depends: R +Suggests: MASS +Description: chemCal provides simple functions for plotting linear + calibration functions and estimating standard errors for measurements + according to the Handbook of Chemometrics and Qualimetrics: Part A + by Massart et al. There are also functions estimating the limit + of detection (LOQ) and limit of quantification (LOD). + The functions work on model objects from - optionally weighted - linear + regression (lm) or robust linear regression (rlm from the MASS package). +License: GPL version 2 or newer +URL: http://www.r-project.org, + http://www.uft.uni-bremen.de/chemie/ranke, + http://kriemhild.uft.uni-bremen.de/viewcvs/?root=chemCal diff --git a/branches/0.1/chemCal/INDEX b/branches/0.1/chemCal/INDEX new file mode 100644 index 0000000..f880eb6 --- /dev/null +++ b/branches/0.1/chemCal/INDEX @@ -0,0 +1,9 @@ +calplot Plot calibration graphs from univariate linear + models +chemCal-package Calibration functions for analytical chemistry +din32645 Calibration data from DIN 32645 +inverse.predict Predict x from y for a linear calibration +lod Estimate a limit of detection (LOD) +loq Estimate a limit of quantification (LOQ) +massart97ex3 Calibration data from Massart et al. (1997), + example 3 diff --git a/branches/0.1/chemCal/R/calplot.R b/branches/0.1/chemCal/R/calplot.R new file mode 100644 index 0000000..6aed9c0 --- /dev/null +++ b/branches/0.1/chemCal/R/calplot.R @@ -0,0 +1,80 @@ +calplot <- function(object, + xlim = c("auto", "auto"), ylim = c("auto", "auto"), + xlab = "Concentration", ylab = "Response", alpha = 0.05, + varfunc = NULL) +{ + UseMethod("calplot") +} + +calplot.default <- function(object, + xlim = c("auto","auto"), ylim = c("auto","auto"), + xlab = "Concentration", ylab = "Response", + alpha=0.05, varfunc = NULL) +{ + stop("Calibration plots only implemented for univariate lm objects.") +} + +calplot.lm <- function(object, + xlim = c("auto","auto"), ylim = c("auto","auto"), + xlab = "Concentration", ylab = "Response", alpha=0.05, + varfunc = NULL) +{ + if (length(object$coef) > 2) + stop("More than one independent variable in your model - not implemented") + + if (alpha <= 0 | alpha >= 1) + stop("Alpha should be between 0 and 1 (exclusive)") + + m <- object + level <- 1 - alpha + y <- m$model[[1]] + x <- m$model[[2]] + if (xlim[1] == "auto") xlim[1] <- 0 + if (xlim[2] == "auto") xlim[2] <- max(x) + xlim <- as.numeric(xlim) + newdata <- list( + x = seq(from = xlim[[1]], to = xlim[[2]], length=250)) + names(newdata) <- names(m$model)[[2]] + if (is.null(varfunc)) { + varfunc <- if (length(m$weights)) { + function(variable) mean(m$weights) + } else function(variable) rep(1,250) + } + pred.lim <- predict(m, newdata, interval = "prediction", + level=level, weights.newdata = varfunc(m)) + conf.lim <- predict(m, newdata, interval = "confidence", + level=level) + yrange.auto <- range(c(0,pred.lim)) + if (ylim[1] == "auto") ylim[1] <- yrange.auto[1] + if (ylim[2] == "auto") ylim[2] <- yrange.auto[2] + plot(1, + type = "n", + xlab = xlab, + ylab = ylab, + xlim = as.numeric(xlim), + ylim = as.numeric(ylim) + ) + points(x,y, pch = 21, bg = "yellow") + matlines(newdata[[1]], pred.lim, lty = c(1, 4, 4), + col = c("black", "red", "red")) + if (length(object$weights) > 0) { + legend(min(x), + max(pred.lim, na.rm = TRUE), + legend = c("Fitted Line", "Confidence Bands"), + lty = c(1, 3), + lwd = 2, + col = c("black", "green4"), + horiz = FALSE, cex = 0.9, bg = "gray95") + } else { + matlines(newdata[[1]], conf.lim, lty = c(1, 3, 3), + col = c("black", "green4", "green4")) + legend(min(x), + max(pred.lim, na.rm = TRUE), + legend = c("Fitted Line", "Confidence Bands", + "Prediction Bands"), + lty = c(1, 3, 4), + lwd = 2, + col = c("black", "green4", "red"), + horiz = FALSE, cex = 0.9, bg = "gray95") + } +} diff --git a/branches/0.1/chemCal/R/inverse.predict.lm.R b/branches/0.1/chemCal/R/inverse.predict.lm.R new file mode 100644 index 0000000..927e672 --- /dev/null +++ b/branches/0.1/chemCal/R/inverse.predict.lm.R @@ -0,0 +1,97 @@ +# This is an implementation of Equation (8.28) in the Handbook of Chemometrics +# and Qualimetrics, Part A, Massart et al (1997), page 200, validated with +# Example 8 on the same page, extended as specified in the package vignette + +inverse.predict <- function(object, newdata, ..., + ws = "auto", alpha = 0.05, var.s = "auto") +{ + UseMethod("inverse.predict") +} + +inverse.predict.default <- function(object, newdata, ..., + ws = "auto", alpha = 0.05, var.s = "auto") +{ + stop("Inverse prediction only implemented for univariate lm objects.") +} + +inverse.predict.lm <- function(object, newdata, ..., + ws = "auto", alpha = 0.05, var.s = "auto") +{ + yname = names(object$model)[[1]] + xname = names(object$model)[[2]] + if (ws == "auto") { + ws <- ifelse(length(object$weights) > 0, mean(object$weights), 1) + } + if (length(object$weights) > 0) { + wx <- split(object$weights,object$model[[xname]]) + w <- sapply(wx,mean) + } else { + w <- rep(1,length(split(object$model[[yname]],object$model[[xname]]))) + } + .inverse.predict(object = object, newdata = newdata, + ws = ws, alpha = alpha, var.s = var.s, w = w, xname = xname, yname = yname) +} + +inverse.predict.rlm <- function(object, newdata, ..., + ws = "auto", alpha = 0.05, var.s = "auto") +{ + yname = names(object$model)[[1]] + xname = names(object$model)[[2]] + if (ws == "auto") { + ws <- mean(object$w) + } + wx <- split(object$weights,object$model[[xname]]) + w <- sapply(wx,mean) + .inverse.predict(object = object, newdata = newdata, + ws = ws, alpha = alpha, var.s = var.s, w = w, xname = xname, yname = yname) +} + +.inverse.predict <- function(object, newdata, ws, alpha, var.s, w, xname, yname) +{ + if (length(object$coef) > 2) + stop("More than one independent variable in your model - not implemented") + + if (alpha <= 0 | alpha >= 1) + stop("Alpha should be between 0 and 1 (exclusive)") + + ybars <- mean(newdata) + m <- length(newdata) + + yx <- split(object$model[[yname]], object$model[[xname]]) + n <- length(yx) + df <- n - length(object$coef) + x <- as.numeric(names(yx)) + ybar <- sapply(yx,mean) + yhatx <- split(object$fitted.values, object$model[[xname]]) + yhat <- sapply(yhatx, mean) + se <- sqrt(sum(w * (ybar - yhat)^2)/df) + + if (var.s == "auto") { + var.s <- se^2/ws + } + + b1 <- object$coef[[xname]] + + ybarw <- sum(w * ybar)/sum(w) + +# This is the adapted form of equation 8.28 (see package vignette) + sxhats <- 1/b1 * sqrt( + (var.s / m) + + se^2 * (1/sum(w) + + (ybars - ybarw)^2 * sum(w) / + (b1^2 * (sum(w) * sum(w * x^2) - sum(w * x)^2))) + ) + + if (names(object$coef)[1] == "(Intercept)") { + b0 <- object$coef[["(Intercept)"]] + } else { + b0 <- 0 + } + + xs <- (ybars - b0) / b1 + t <- qt(1-0.5*alpha, n - 2) + conf <- t * sxhats + result <- list("Prediction"=xs,"Standard Error"=sxhats, + "Confidence"=conf, "Confidence Limits"=c(xs - conf, xs + conf)) + return(result) +} diff --git a/branches/0.1/chemCal/R/lod.R b/branches/0.1/chemCal/R/lod.R new file mode 100644 index 0000000..f5bbb7d --- /dev/null +++ b/branches/0.1/chemCal/R/lod.R @@ -0,0 +1,53 @@ +lod <- function(object, ..., alpha = 0.05, beta = 0.05, method = "default") +{ + UseMethod("lod") +} + +lod.default <- function(object, ..., alpha = 0.05, beta = 0.05, method = "default") +{ + stop("lod is only implemented for univariate lm objects.") +} + +lod.lm <- function(object, ..., alpha = 0.05, beta = 0.05, method = "default") +{ + if (length(object$weights) > 0) { + stop(paste( + "\nThe detemination of a lod from calibration models obtained by", + "weighted linear regression requires confidence intervals for", + "predicted y values taking into account weights for the x values", + "from which the predictions are to be generated.", + "This is not supported by the internally used predict.lm method.", + sep = "\n" + )) + } + xname <- names(object$model)[[2]] + yname <- names(object$model)[[1]] + newdata <- data.frame(0) + names(newdata) <- xname + y0 <- predict(object, newdata, interval = "prediction", + level = 1 - 2 * alpha) + yc <- y0[[1,"upr"]] + if (method == "din") { + y0.d <- predict(object, newdata, interval = "prediction", + level = 1 - 2 * beta) + deltay <- y0.d[[1, "upr"]] - y0.d[[1, "fit"]] + lod.y <- yc + deltay + lod.x <- inverse.predict(object, lod.y)$Prediction + } else { + f <- function(x) { + newdata <- data.frame(x) + names(newdata) <- xname + pi.y <- predict(object, newdata, interval = "prediction", + level = 1 - 2 * beta) + yd <- pi.y[[1,"lwr"]] + (yd - yc)^2 + } + lod.x <- optimize(f,interval=c(0,max(object$model[[xname]])))$minimum + newdata <- data.frame(x = lod.x) + names(newdata) <- xname + lod.y <- predict(object, newdata) + } + lod <- list(lod.x, lod.y) + names(lod) <- c(xname, yname) + return(lod) +} diff --git a/branches/0.1/chemCal/R/loq.R b/branches/0.1/chemCal/R/loq.R new file mode 100644 index 0000000..5776096 --- /dev/null +++ b/branches/0.1/chemCal/R/loq.R @@ -0,0 +1,40 @@ +loq <- function(object, ..., alpha = 0.05, k = 3, n = 1, w.loq = "auto", + var.loq = "auto") +{ + UseMethod("loq") +} + +loq.default <- function(object, ..., alpha = 0.05, k = 3, n = 1, w.loq = "auto", + var.loq = "auto") +{ + stop("loq is only implemented for univariate lm objects.") +} + +loq.lm <- function(object, ..., alpha = 0.05, k = 3, n = 1, w.loq = "auto", + var.loq = "auto") +{ + if (length(object$weights) > 0 && var.loq == "auto" && w.loq == "auto") { + stop(paste("If you are using a model from weighted regression,", + "you need to specify a reasonable approximation for the", + "weight (w.loq) or the variance (var.loq) at the", + "limit of quantification")) + } + xname <- names(object$model)[[2]] + yname <- names(object$model)[[1]] + f <- function(x) { + newdata <- data.frame(x = x) + names(newdata) <- xname + y <- predict(object, newdata) + p <- inverse.predict(object, rep(y, n), ws = w.loq, + var.s = var.loq, alpha = alpha) + (p[["Prediction"]] - k * p[["Confidence"]])^2 + } + tmp <- optimize(f,interval=c(0,max(object$model[[2]]))) + loq.x <- tmp$minimum + newdata <- data.frame(x = loq.x) + names(newdata) <- xname + loq.y <- predict(object, newdata) + loq <- list(loq.x, loq.y) + names(loq) <- c(xname, yname) + return(loq) +} diff --git a/branches/0.1/chemCal/TODO b/branches/0.1/chemCal/TODO new file mode 100644 index 0000000..1fc44a7 --- /dev/null +++ b/branches/0.1/chemCal/TODO @@ -0,0 +1,5 @@ +- lod(): At the moment, it is not possible to calculate an lod for the + case of more than one replicates (m is fixed to 1). The formula + for the prediction of y from mean(x) in Massart et al, p. 433 + could be used to generalize this. +- Write methods for nonlinear calibration functions diff --git a/branches/0.1/chemCal/data/din32645.rda b/branches/0.1/chemCal/data/din32645.rda Binary files differnew file mode 100644 index 0000000..4e2913a --- /dev/null +++ b/branches/0.1/chemCal/data/din32645.rda diff --git a/branches/0.1/chemCal/data/massart97ex1.rda b/branches/0.1/chemCal/data/massart97ex1.rda Binary files differnew file mode 100644 index 0000000..1a6fd4b --- /dev/null +++ b/branches/0.1/chemCal/data/massart97ex1.rda diff --git a/branches/0.1/chemCal/data/massart97ex3.rda b/branches/0.1/chemCal/data/massart97ex3.rda Binary files differnew file mode 100644 index 0000000..7f60f3a --- /dev/null +++ b/branches/0.1/chemCal/data/massart97ex3.rda diff --git a/branches/0.1/chemCal/inst/doc/Makefile b/branches/0.1/chemCal/inst/doc/Makefile new file mode 100644 index 0000000..8eca69e --- /dev/null +++ b/branches/0.1/chemCal/inst/doc/Makefile @@ -0,0 +1,33 @@ +# Makefile for Sweave documents containing both Latex and R code +# Author: Johannes Ranke <jranke@uni-bremen.de> +# Last Change: 2006 Jun 23 +# based on the Makefile of Nicholas Lewin-Koh +# in turn based on work of Rouben Rostmaian +# SVN: $Id: Makefile.rnoweb 62 2006-05-24 08:30:59Z ranke $ + +RNWFILES = $(wildcard *.Rnw) +TARGETS = $(patsubst %.Rnw,%.tex,$(RNWFILE)) $(patsubst %.Rnw,%.pdf,$(RNWFILES)) + +RERUN = "(There were undefined references|Rerun to get (cross-references|the bars) right|Table widths have changed. 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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/cgi-bin/R/wishlst-fulfilled?id=8877;user=guest}{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}. + +\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} diff --git a/branches/0.1/chemCal/inst/doc/chemCal.aux b/branches/0.1/chemCal/inst/doc/chemCal.aux new file mode 100644 index 0000000..20bfc98 --- /dev/null +++ b/branches/0.1/chemCal/inst/doc/chemCal.aux @@ -0,0 +1,18 @@ +\relax +\ifx\hyper@anchor\@undefined +\global \let \oldcontentsline\contentsline +\gdef \contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global \let \oldnewlabel\newlabel +\gdef \newlabel#1#2{\newlabelxx{#1}#2} +\gdef \newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\let \contentsline\oldcontentsline +\let \newlabel\oldnewlabel} +\else +\global \let \hyper@last\relax +\fi + +\citation{massart97} +\citation{massart97} +\citation{massart97} +\citation{massart97} +\bibcite{massart97}{1} diff --git a/branches/0.1/chemCal/inst/doc/chemCal.bbl b/branches/0.1/chemCal/inst/doc/chemCal.bbl new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/branches/0.1/chemCal/inst/doc/chemCal.bbl diff --git a/branches/0.1/chemCal/inst/doc/chemCal.blg b/branches/0.1/chemCal/inst/doc/chemCal.blg new file mode 100644 index 0000000..72f5a0e --- /dev/null +++ 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calibration functions for analytical chemistry} +\author{Johannes Ranke} + +\usepackage{/usr/share/R/share/texmf/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/cgi-bin/R/wishlst-fulfilled?id=8877;user=guest}{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] 0.892611 + +$Confidence +[1] 2.478285 + +$`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 + +$Confidence +[1] 7.855012 + +$`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} diff --git a/branches/0.1/chemCal/man/calplot.lm.Rd b/branches/0.1/chemCal/man/calplot.lm.Rd new file mode 100644 index 0000000..6f6d584 --- /dev/null +++ b/branches/0.1/chemCal/man/calplot.lm.Rd @@ -0,0 +1,62 @@ +\name{calplot} +\alias{calplot} +\alias{calplot.default} +\alias{calplot.lm} +\title{Plot calibration graphs from univariate linear models} +\description{ + Produce graphics of calibration data, the fitted model as well + as confidence, and, for unweighted regression, prediction bands. +} +\usage{ + calplot(object, xlim = c("auto", "auto"), ylim = c("auto", "auto"), + xlab = "Concentration", ylab = "Response", alpha=0.05, varfunc = NULL) +} +\arguments{ + \item{object}{ + A univariate model object of class \code{\link{lm}} or + \code{\link[MASS:rlm]{rlm}} + with model formula \code{y ~ x} or \code{y ~ x - 1}. + } + \item{xlim}{ + The limits of the plot on the x axis. + } + \item{ylim}{ + The limits of the plot on the y axis. + } + \item{xlab}{ + The label of the x axis. + } + \item{ylab}{ + The label of the y axis. + } + \item{alpha}{ + The error tolerance level for the confidence and prediction bands. + } + \item{varfunc}{ + The variance function for generating the weights in the model. + Currently, this argument is ignored (see note below). + } +} +\value{ + A plot of the calibration data, of your fitted model as well as lines showing + the confidence limits. Prediction limits are only shown for models from + unweighted regression. +} +\note{ + Prediction bands for models from weighted linear regression require weights + for the data, for which responses should be predicted. Prediction intervals + using weights e.g. from a variance function are currently not supported by + the internally used function \code{\link{predict.lm}}, therefore, + \code{calplot} does not draw prediction bands for such models. +} +\examples{ +data(massart97ex3) +m <- lm(y ~ x, data = massart97ex3) +calplot(m) +} +\author{ + Johannes Ranke + \email{jranke@uni-bremen.de} + \url{http://www.uft.uni-bremen.de/chemie/ranke} +} +\keyword{regression} diff --git a/branches/0.1/chemCal/man/chemCal-package.Rd b/branches/0.1/chemCal/man/chemCal-package.Rd new file mode 100644 index 0000000..4456150 --- /dev/null +++ b/branches/0.1/chemCal/man/chemCal-package.Rd @@ -0,0 +1,17 @@ +\name{chemCal-package} +\alias{chemCal-package} +\docType{package} +\title{ + Calibration functions for analytical chemistry +} +\description{ + See \url{../DESCRIPTION} +} +\details{ + There is a package vignette located in \url{../doc/chemCal.pdf}. +} +\author{ + Author and Maintainer: Johannes Ranke <jranke@uni-bremen.de> +} +\keyword{manip} +} diff --git a/branches/0.1/chemCal/man/din32645.Rd b/branches/0.1/chemCal/man/din32645.Rd new file mode 100644 index 0000000..cacbf07 --- /dev/null +++ b/branches/0.1/chemCal/man/din32645.Rd @@ -0,0 +1,61 @@ +\name{din32645} +\docType{data} +\alias{din32645} +\title{Calibration data from DIN 32645} +\description{ + Sample dataset to test the package. +} +\usage{data(din32645)} +\format{ + A dataframe containing 10 rows of x and y values. +} +\examples{ +data(din32645) +m <- lm(y ~ x, data = din32645) +calplot(m) + +## Prediction of x with confidence interval +(prediction <- inverse.predict(m, 3500, alpha = 0.01)) + +# This should give 0.07434 according to test data from Dintest, which +# was collected from Procontrol 3.1 (isomehr GmbH) in this case +round(prediction$Confidence,5) + +## Critical value: +(crit <- lod(m, alpha = 0.01, beta = 0.5)) + +# According to DIN 32645, we should get 0.07 for the critical value +# (decision limit, "Nachweisgrenze") +round(crit$x, 2) +# and according to Dintest test data, we should get 0.0698 from +round(crit$x, 4) + +## Limit of detection (smallest detectable value given alpha and beta) +# In German, the smallest detectable value is the "Erfassungsgrenze", and we +# should get 0.14 according to DIN, which we achieve by using the method +# described in it: +lod.din <- lod(m, alpha = 0.01, beta = 0.01, method = "din") +round(lod.din$x, 2) + +## Limit of quantification +# This accords to the test data coming with the test data from Dintest again, +# except for the last digits of the value cited for Procontrol 3.1 (0.2121) +(loq <- loq(m, alpha = 0.01)) +round(loq$x,4) + +# A similar value is obtained using the approximation +# LQ = 3.04 * LC (Currie 1999, p. 120) +3.04 * lod(m,alpha = 0.01, beta = 0.5)$x +} +\references{ + DIN 32645 (equivalent to ISO 11843), Beuth Verlag, Berlin, 1994 + + Dintest. Plugin for MS Excel for evaluations of calibration data. Written + by Georg Schmitt, University of Heidelberg. + \url{http://www.rzuser.uni-heidelberg.de/~df6/download/dintest.htm} + + Currie, L. A. (1997) Nomenclature in evaluation of analytical methods including + detection and quantification capabilities (IUPAC Recommendations 1995). + Analytica Chimica Acta 391, 105 - 126. +} +\keyword{datasets} diff --git a/branches/0.1/chemCal/man/inverse.predict.Rd b/branches/0.1/chemCal/man/inverse.predict.Rd new file mode 100644 index 0000000..347d670 --- /dev/null +++ b/branches/0.1/chemCal/man/inverse.predict.Rd @@ -0,0 +1,69 @@ +\name{inverse.predict} +\alias{inverse.predict} +\alias{inverse.predict.lm} +\alias{inverse.predict.rlm} +\alias{inverse.predict.default} +\title{Predict x from y for a linear calibration} +\usage{inverse.predict(object, newdata, \dots, + ws, alpha=0.05, var.s = "auto") +} +\arguments{ + \item{object}{ + A univariate model object of class \code{\link{lm}} or + \code{\link[MASS:rlm]{rlm}} + with model formula \code{y ~ x} or \code{y ~ x - 1}. + } + \item{newdata}{ + A vector of observed y values for one sample. + } + \item{\dots}{ + Placeholder for further arguments that might be needed by + future implementations. + } + \item{ws}{ + The weight attributed to the sample. This argument is obligatory + if \code{object} has weights. + } + \item{alpha}{ + The error tolerance level for the confidence interval to be reported. + } + \item{var.s}{ + The estimated variance of the sample measurements. The default is to take + the residual standard error from the calibration and to adjust it + using \code{ws}, if applicable. This means that \code{var.s} + overrides \code{ws}. + } +} +\value{ + A list containing the predicted x value, its standard error and a + confidence interval. +} +\description{ + This function predicts x values using a univariate linear model that has been + generated for the purpose of calibrating a measurement method. Prediction + intervals are given at the specified confidence level. + The calculation method was taken from Massart et al. (1997). In particular, + Equations 8.26 and 8.28 were combined in order to yield a general treatment + of inverse prediction for univariate linear models, taking into account + weights that have been used to create the linear model, and at the same + time providing the possibility to specify a precision in sample measurements + differing from the precision in standard samples used for the calibration. + This is elaborated in the package vignette. +} +\note{ + The function was validated with examples 7 and 8 from Massart et al. (1997). +} +\references{ + Massart, L.M, Vandenginste, B.G.M., Buydens, L.M.C., De Jong, S., Lewi, P.J., + Smeyers-Verbeke, J. (1997) Handbook of Chemometrics and Qualimetrics: Part A, + p. 200 +} +\examples{ +# This is example 7 from Chapter 8 in Massart et al. (1997) +data(massart97ex1) +m <- lm(y ~ x, data = massart97ex1) +inverse.predict(m, 15) # 6.1 +- 4.9 +inverse.predict(m, 90) # 43.9 +- 4.9 +inverse.predict(m, rep(90,5)) # 43.9 +- 3.2 +} +\keyword{manip} diff --git a/branches/0.1/chemCal/man/lod.Rd b/branches/0.1/chemCal/man/lod.Rd new file mode 100644 index 0000000..e468e1d --- /dev/null +++ b/branches/0.1/chemCal/man/lod.Rd @@ -0,0 +1,83 @@ +\name{lod} +\alias{lod} +\alias{lod.lm} +\alias{lod.rlm} +\alias{lod.default} +\title{Estimate a limit of detection (LOD)} +\usage{ + lod(object, \dots, alpha = 0.05, beta = 0.05, method = "default") +} +\arguments{ + \item{object}{ + A univariate model object of class \code{\link{lm}} or + \code{\link[MASS:rlm]{rlm}} + with model formula \code{y ~ x} or \code{y ~ x - 1}, + optionally from a weighted regression. + } + \item{\dots}{ + Placeholder for further arguments that might be needed by + future implementations. + } + \item{alpha}{ + The error tolerance for the decision limit (critical value). + } + \item{beta}{ + The error tolerance beta for the detection limit. + } + \item{method}{ + The default method uses a prediction interval at the LOD + for the estimation of the LOD, which obviously requires + iteration. This is described for example in Massart, p. 432 ff. + The \dQuote{din} method uses the prediction interval at + x = 0 as an approximation. + } +} +\value{ + A list containig the corresponding x and y values of the estimated limit of + detection of a model used for calibration. +} +\description{ + The decision limit (German: Nachweisgrenze) is defined as the signal or + analyte concentration that is significantly different from the blank signal + with a first order error alpha (one-sided significance test). + The detection limit, or more precise, the minimum detectable value + (German: Erfassungsgrenze), is then defined as the signal or analyte + concentration where the probability that the signal is not detected although + the analyte is present (type II or false negative error), is beta (also a + one-sided significance test). +} +\note{ + - The default values for alpha and beta are the ones recommended by IUPAC. + - The estimation of the LOD in terms of the analyte amount/concentration + xD from the LOD in the signal domain SD is done by simply inverting the + calibration function (i.e. assuming a known calibration function). + - The calculation of a LOD from weighted calibration models requires + a weights argument for the internally used \code{\link{predict.lm}} + function, which is currently not supported in R. +} +\references{ + Massart, L.M, Vandenginste, B.G.M., Buydens, L.M.C., De Jong, S., Lewi, P.J., + Smeyers-Verbeke, J. (1997) Handbook of Chemometrics and Qualimetrics: Part A, + Chapter 13.7.8 + + J. Inczedy, T. Lengyel, and A.M. Ure (2002) International Union of Pure and + Applied Chemistry Compendium of Analytical Nomenclature: Definitive Rules. + Web edition. + + Currie, L. A. (1997) Nomenclature in evaluation of analytical methods including + detection and quantification capabilities (IUPAC Recommendations 1995). + Analytica Chimica Acta 391, 105 - 126. +} +\examples{ +data(din32645) +m <- lm(y ~ x, data = din32645) +lod(m) + +# The critical value (decision limit, German Nachweisgrenze) can be obtained +# by using beta = 0.5: +lod(m, alpha = 0.01, beta = 0.5) +} +\seealso{ + Examples for \code{\link{din32645}} +} +\keyword{manip} diff --git a/branches/0.1/chemCal/man/loq.Rd b/branches/0.1/chemCal/man/loq.Rd new file mode 100644 index 0000000..7541e77 --- /dev/null +++ b/branches/0.1/chemCal/man/loq.Rd @@ -0,0 +1,77 @@ +\name{loq} +\alias{loq} +\alias{loq.lm} +\alias{loq.rlm} +\alias{loq.default} +\title{Estimate a limit of quantification (LOQ)} +\usage{ + loq(object, \dots, alpha = 0.05, k = 3, n = 1, w.loq = "auto", + var.loq = "auto") +} +\arguments{ + \item{object}{ + A univariate model object of class \code{\link{lm}} or + \code{\link[MASS:rlm]{rlm}} + with model formula \code{y ~ x} or \code{y ~ x - 1}, + optionally from a weighted regression. If weights are specified + in the model, either \code{w.loq} or \code{var.loq} have to + be specified. + } + \item{alpha}{ + The error tolerance for the prediction of x values in the calculation. + } + \item{\dots}{ + Placeholder for further arguments that might be needed by + future implementations. + } + \item{k}{ + The inverse of the maximum relative error tolerated at the + desired LOQ. + } + \item{n}{ + The number of replicate measurements for which the LOQ should be + specified. + } + \item{w.loq}{ + The weight that should be attributed to the LOQ. Defaults + to one for unweighted regression, and to the mean of the weights + for weighted regression. See \code{\link{massart97ex3}} for + an example how to take advantage of knowledge about the + variance function. + } + \item{var.loq}{ + The approximate variance at the LOQ. The default value is + calculated from the model. + } +} +\value{ + The estimated limit of quantification for a model used for calibration. +} +\description{ + The limit of quantification is the x value, where the relative error + of the quantification given the calibration model reaches a prespecified + value 1/k. Thus, it is the solution of the equation + \deqn{L = k c(L)}{L = k * c(L)} + where c(L) is half of the length of the confidence interval at the limit L + (DIN 32645, equivalent to ISO 11843). c(L) is internally estimated by + \code{\link{inverse.predict}}, and L is obtained by iteration. +} +\note{ + - IUPAC recommends to base the LOQ on the standard deviation of the signal + where x = 0. + - The calculation of a LOQ based on weighted regression is non-standard + and therefore not tested. Feedback is welcome. +} +\examples{ +data(massart97ex3) +attach(massart97ex3) +m <- lm(y ~ x) +loq(m) + +# We can get better by using replicate measurements +loq(m, n = 3) +} +\seealso{ + Examples for \code{\link{din32645}} +} +\keyword{manip} diff --git a/branches/0.1/chemCal/man/massart97ex1.Rd b/branches/0.1/chemCal/man/massart97ex1.Rd new file mode 100644 index 0000000..44e1b85 --- /dev/null +++ b/branches/0.1/chemCal/man/massart97ex1.Rd @@ -0,0 +1,17 @@ +\name{massart97ex1} +\docType{data} +\alias{massart97ex1} +\title{Calibration data from Massart et al. (1997), example 1} +\description{ + Sample dataset from p. 175 to test the package. +} +\usage{data(massart97ex1)} +\format{ + A dataframe containing 6 observations of x and y data. +} +\source{ + Massart, L.M, Vandenginste, B.G.M., Buydens, L.M.C., De Jong, S., Lewi, P.J., + Smeyers-Verbeke, J. (1997) Handbook of Chemometrics and Qualimetrics: Part A, + Chapter 8. +} +\keyword{datasets} diff --git a/branches/0.1/chemCal/man/massart97ex3.Rd b/branches/0.1/chemCal/man/massart97ex3.Rd new file mode 100644 index 0000000..efdcf02 --- /dev/null +++ b/branches/0.1/chemCal/man/massart97ex3.Rd @@ -0,0 +1,51 @@ +\name{massart97ex3} +\docType{data} +\alias{massart97ex3} +\title{Calibration data from Massart et al. (1997), example 3} +\description{ + Sample dataset from p. 188 to test the package. +} +\usage{data(massart97ex3)} +\format{ + A dataframe containing 6 levels of x values with 5 + observations of y for each level. +} +\examples{ +data(massart97ex3) +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) +calplot(m) + +# The following concords with the book p. 200 +inverse.predict(m, 15, ws = 1.67) # 5.9 +- 2.5 +inverse.predict(m, 90, ws = 0.145) # 44.1 +- 7.9 + +# The LOD is only calculated for models from unweighted regression +# with this version of chemCal +m0 <- lm(y ~ x) +lod(m0) + +# Limit of quantification from unweighted regression +loq(m0) + +# For calculating the limit of quantification from a model from weighted +# regression, we need to supply weights, internally used for inverse.predict +# If we are not using a variance function, we can use the weight from +# the above example as a first approximation (x = 15 is close to our +# loq approx 14 from above). +loq(m, w.loq = 1.67) +# The weight for the loq should therefore be derived at x = 7.3 instead +# of 15, but the graphical procedure of Massart (p. 201) to derive the +# variances on which the weights are based is quite inaccurate anyway. +} +\source{ + Massart, L.M, Vandenginste, B.G.M., Buydens, L.M.C., De Jong, S., Lewi, P.J., + Smeyers-Verbeke, J. (1997) Handbook of Chemometrics and Qualimetrics: Part A, + Chapter 8. +} +\keyword{datasets} diff --git a/branches/0.1/chemCal/tests/din32645.R b/branches/0.1/chemCal/tests/din32645.R new file mode 100644 index 0000000..e5ffed7 --- /dev/null +++ b/branches/0.1/chemCal/tests/din32645.R @@ -0,0 +1,7 @@ +require(chemCal) +data(din32645) +m <- lm(y ~ x, data = din32645) +inverse.predict(m, 3500, alpha = 0.01) +lod <- lod(m, alpha = 0.01, beta = 0.5) +lod(m, alpha = 0.01, beta = 0.01) +loq <- loq(m, alpha = 0.01) diff --git a/branches/0.1/chemCal/tests/din32645.Rout.save b/branches/0.1/chemCal/tests/din32645.Rout.save new file mode 100644 index 0000000..c5ed5a7 --- /dev/null +++ b/branches/0.1/chemCal/tests/din32645.Rout.save @@ -0,0 +1,45 @@ + +R : Copyright 2006, The R Foundation for Statistical Computing +Version 2.3.1 (2006-06-01) +ISBN 3-900051-07-0 + +R is free software and comes with ABSOLUTELY NO WARRANTY. +You are welcome to redistribute it under certain conditions. +Type 'license()' or 'licence()' for distribution details. + +R is a collaborative project with many contributors. +Type 'contributors()' for more information and +'citation()' on how to cite R or R packages in publications. + +Type 'demo()' for some demos, 'help()' for on-line help, or +'help.start()' for an HTML browser interface to help. +Type 'q()' to quit R. + +> require(chemCal) +Loading required package: chemCal +[1] TRUE +> data(din32645) +> m <- lm(y ~ x, data = din32645) +> inverse.predict(m, 3500, alpha = 0.01) +$Prediction +[1] 0.1054792 + +$`Standard Error` +[1] 0.02215619 + +$Confidence +[1] 0.07434261 + +$`Confidence Limits` +[1] 0.03113656 0.17982178 + +> lod <- lod(m, alpha = 0.01, beta = 0.5) +> lod(m, alpha = 0.01, beta = 0.01) +$x +[1] 0.132904 + +$y +[1] 3764.977 + +> loq <- loq(m, alpha = 0.01) +> diff --git a/branches/0.1/chemCal/tests/massart97.R b/branches/0.1/chemCal/tests/massart97.R new file mode 100644 index 0000000..00f837f --- /dev/null +++ b/branches/0.1/chemCal/tests/massart97.R @@ -0,0 +1,25 @@ +require(chemCal) +data(massart97ex1) +m <- lm(y ~ x, data = massart97ex1) +inverse.predict(m, 15) # 6.1 +- 4.9 +inverse.predict(m, 90) # 43.9 +- 4.9 +inverse.predict(m, rep(90,5)) # 43.9 +- 3.2 + +data(massart97ex3) +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) +#calplot(m) + +inverse.predict(m, 15, ws = 1.67) # 5.9 +- 2.5 +inverse.predict(m, 90, ws = 0.145) # 44.1 +- 7.9 + +m0 <- lm(y ~ x) +lod(m0) + +loq(m0) +loq(m, w.loq = 1.67) diff --git a/branches/0.1/chemCal/tests/massart97.Rout.save b/branches/0.1/chemCal/tests/massart97.Rout.save new file mode 100644 index 0000000..cb113d0 --- /dev/null +++ b/branches/0.1/chemCal/tests/massart97.Rout.save @@ -0,0 +1,123 @@ + +R : Copyright 2006, The R Foundation for Statistical Computing +Version 2.3.1 (2006-06-01) +ISBN 3-900051-07-0 + +R is free software and comes with ABSOLUTELY NO WARRANTY. +You are welcome to redistribute it under certain conditions. +Type 'license()' or 'licence()' for distribution details. + +R is a collaborative project with many contributors. +Type 'contributors()' for more information and +'citation()' on how to cite R or R packages in publications. + +Type 'demo()' for some demos, 'help()' for on-line help, or +'help.start()' for an HTML browser interface to help. +Type 'q()' to quit R. + +> require(chemCal) +Loading required package: chemCal +[1] TRUE +> data(massart97ex1) +> m <- lm(y ~ x, data = massart97ex1) +> inverse.predict(m, 15) # 6.1 +- 4.9 +$Prediction +[1] 6.09381 + +$`Standard Error` +[1] 1.767278 + +$Confidence +[1] 4.906751 + +$`Confidence Limits` +[1] 1.187059 11.000561 + +> inverse.predict(m, 90) # 43.9 +- 4.9 +$Prediction +[1] 43.93983 + +$`Standard Error` +[1] 1.767747 + +$Confidence +[1] 4.908053 + +$`Confidence Limits` +[1] 39.03178 48.84788 + +> inverse.predict(m, rep(90,5)) # 43.9 +- 3.2 +$Prediction +[1] 43.93983 + +$`Standard Error` +[1] 1.141204 + +$Confidence +[1] 3.168489 + +$`Confidence Limits` +[1] 40.77134 47.10832 + +> +> data(massart97ex3) +> 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) +> #calplot(m) +> +> inverse.predict(m, 15, ws = 1.67) # 5.9 +- 2.5 +$Prediction +[1] 5.865367 + +$`Standard Error` +[1] 0.892611 + +$Confidence +[1] 2.478285 + +$`Confidence Limits` +[1] 3.387082 8.343652 + +> inverse.predict(m, 90, ws = 0.145) # 44.1 +- 7.9 +$Prediction +[1] 44.06025 + +$`Standard Error` +[1] 2.829162 + +$Confidence +[1] 7.855012 + +$`Confidence Limits` +[1] 36.20523 51.91526 + +> +> m0 <- lm(y ~ x) +> lod(m0) +$x +[1] 5.406637 + +$y +[1] 13.63822 + +> +> loq(m0) +$x +[1] 13.97767 + +$y +[1] 30.62355 + +> loq(m, w.loq = 1.67) +$x +[1] 7.346231 + +$y +[1] 17.90784 + +> |