chemCal - Calibration functions for analytical chemistry

Static documentation of this R package can be found at https://pkgdown.jrwb.de/chemCal

diff --git a/README.md b/README.md index 7091d0d..e72e464 100644 --- a/README.md +++ b/README.md @@ -2,6 +2,7 @@ [![](https://www.r-pkg.org/badges/version/chemCal)](https://cran.r-project.org/package=chemCal) [![Build Status](https://travis-ci.com/jranke/chemCal.svg?branch=master)](https://travis-ci.com/jranke/chemCal) +[![codecov](https://codecov.io/github/jranke/chemCal/branch/master/graphs/badge.svg)](https://codecov.io/github/jranke/chemCal) Static documentation of this R package can be found at https://pkgdown.jrwb.de/chemCal diff --git a/docs/articles/chemCal.html b/docs/articles/chemCal.html index 3da10fd..e301880 100644 --- a/docs/articles/chemCal.html +++ b/docs/articles/chemCal.html @@ -1,18 +1,18 @@ - + Introduction to chemCal • chemCal - - - + + + - @@ -22,14 +22,15 @@

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Introduction to chemCal

Johannes Ranke

2018-07-17

2019-02-21

chemCal.Rmd

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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)

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)

plot(m0, which=3)

Therefore, in Example 8 in Massart et al. (1997), 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)

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, 15, ws=1.67)

## $Prediction
 ## [1] 5.865367
 ## 
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 ## 
 ## $`Confidence Limits`
 ## [1] 3.387082 8.343652

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

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

## $Prediction
 ## [1] 44.06025
 ## 
@@ -138,27 +139,27 @@ m <- lm(y ~<
 Background for inverse.predict
 
 Equation 8.28 in Massart et al. (1997) 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}
+\[\begin{equation}
 s_{\hat{x_s}} = \frac{s_e}{b_1} \sqrt{\frac{1}{w_s m} + \frac{1}{\sum{w_i}} +
     \frac{(\bar{y_s} - \bar{y_w})^2 \sum{w_i}}
         {{b_1}^2 \left( \sum{w_i} \sum{w_i {x_i}^2} -
             {\left( \sum{ w_i x_i } \right)}^2 \right) }}
-\end{equation}\]
+\end{equation}\]
 with
-\[\begin{equation}
+\[\begin{equation}
 s_e = \sqrt{ \frac{\sum w_i (y_i - \hat{y_i})^2}{n - 2}}
-\end{equation}\]
+\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  I am using
-\[\begin{equation}
+\[\begin{equation}
 s_{\hat{x_s}} = \frac{1}{b_1} \sqrt{\frac{{s_s}^2}{w_s m} +
     {s_e}^2 \left( \frac{1}{\sum{w_i}} +
         \frac{(\bar{y_s} - \bar{y_w})^2 \sum{w_i}}
             {{b_1}^2 \left( \sum{w_i} \sum{w_i {x_i}^2} - {\left( \sum{ w_i x_i } \right)}^2 \right) } \right) }
-\end{equation}\]
+\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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 chemCal - Calibration functions for analytical chemistry
 
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 Developers
 
-Johannes Ranke 
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+    Cadmium concentrations measured by AAS as reported by Rocke and Lorenzato (1995)
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+    rl95_cadmium.Rd
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+    Dataset reproduced from Table 1 in Rocke and Lorenzato (1995).
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+    Format
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+    A dataframe containing four replicate observations for each 
+  of the six calibration standards.
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+    Source
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+    Rocke, David M. und Lorenzato, Stefan (1995) A two-component model for
+  measurement error in analytical chemistry. Technometrics 37(2), 176-184.
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diff --git a/docs/reference/rl95_toluene.html b/docs/reference/rl95_toluene.html
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