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 would be desirable to implement the
inverse prediction method given in @massart97 since it also covers the
case of weighted regression. Studies of the IUPAC orange book and of DIN 32645
(equivalent to ISO 11843), publications by @currie97 and the Analytical
Method Committee of the Royal Society of Chemistry [@amc89] and a nice paper by
Castells and Castillo [@castells00] provided some further understanding of the matter.

At the moment, the package consists of four functions
(calplot,
lod,
loq and
inverse.predict),
working on univariate linear models of class `lm`

or `rlm`

, plus several
datasets for validation.

A bug report 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.

In fact, in June 2018 I was made aware of the fact that the inverse prediction method implemented in chemCal version 0.1.37 and before did not take the variance of replicate calibration standards about their means into account, nor the number of replicates when calculating the degrees of freedom. Thanks to PhD student Anna Burniol Figols for reporting this issue!

As a consequence, I rewrote `inverse.predict`

not to automatically work with
the mean responses for each calibration standard any more. The example
calculations from @massart97 can still be reproduced when the regression model
is calculated using the means of the calibration data as shown below.

When calibrating an analytical method, the first task is to generate a suitable
model. If we want to use the `chemCal`

functions, we 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 `calplot`

function:

library(chemCal) 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:

plot(m0, which=3)

Therefore, in Example 8 in @massart97, weighted regression is proposed which can be reproduced by the following code. Note that we are building the model on the mean values for each standard in order to be able to reproduce the results given in the book with the current version of chemCal.

weights <- with(massart97ex3, { yx <- split(y, x) ybar <- sapply(yx, mean) s <- round(sapply(yx, sd), digits = 2) w <- round(1 / (s^2), digits = 3) }) massart97ex3.means <- aggregate(y ~ x, massart97ex3, mean) m <- lm(y ~ x, w = weights, data = massart97ex3.means)

If we now want to predict a new x value from measured y values,
we use the `inverse.predict`

function:

inverse.predict(m, 15, ws=1.67) inverse.predict(m, 90, ws = 0.145)

The weight `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 `var.s`

at this location.

`inverse.predict`

Equation 8.28 in @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}

In chemCal version before 0.2, I interpreted $w_i$ to be the weight for
calibration standard $i$, $y_i$ to be the mean value observed for standard $i$,
and $n$ to be the number of calibration standards. With this implementation
I was able to reproduce the examples given in the book. However, as noted above,
I was made aware of the fact that this way of calculation does not take the
variation of the y values about the means into account. Furthermore, I noticed
that for the case of unweighted linear calibration with replicate standards,
`inverse.predict`

produced different results than `calibrate`

from the
`investr`

package when using the Wald method.

Both issues are now addressed in chemCal starting from version 0.2.1. Here, $y_i$ is calibration measurement $i$, $\hat{y_i}$ is the estimated value for calibration measurement $i$ and $n$ is the total number of calibration measurements.

$w_s$ 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 had also 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} I am using

\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
`var.s`

of the function `inverse.predict`

.

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