| Title: | Calibration Functions for Analytical Chemistry |
|---|---|
| Description: | 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. (1997) There are also functions estimating the limit of detection (LOD) and limit of quantification (LOQ). The functions work on model objects from - optionally weighted - linear regression (lm) or robust linear regression ('rlm' from the 'MASS' package). |
| Authors: | Anıl A. Tellbüscher [aut] (Linearity tests),
Johannes Ranke [aut, cre, cph]
|
| Maintainer: | Johannes Ranke <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.2.3.9000 |
| Built: | 2025-02-21 06:16:19 UTC |
| Source: | https://github.com/jranke/chemCal |
Produce graphics of calibration data, the fitted model as well as confidence, and, for unweighted regression, prediction bands.
calplot( object, xlim = c("auto", "auto"), ylim = c("auto", "auto"), xlab = "Concentration", ylab = "Response", legend_x = "auto", alpha = 0.05, varfunc = NULL )calplot( object, xlim = c("auto", "auto"), ylim = c("auto", "auto"), xlab = "Concentration", ylab = "Response", legend_x = "auto", alpha = 0.05, varfunc = NULL )
object |
A univariate model object of class |
xlim |
The limits of the plot on the x axis. |
ylim |
The limits of the plot on the y axis. |
xlab |
The label of the x axis. |
ylab |
The label of the y axis. |
legend_x |
An optional numeric value for adjusting the x coordinate of the legend. |
alpha |
The error tolerance level for the confidence and prediction
bands. Note that this includes both tails of the Gaussian distribution,
unlike the alpha and beta parameters used in |
varfunc |
The variance function for generating the weights in the model. Currently, this argument is ignored (see note below). |
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.
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 predict.lm,
therefore, calplot does not draw prediction bands for such models.
It is possible to compare the calplot prediction bands with
the lod values if the lod() alpha and beta parameters
are half the value of the calplot() alpha parameter.
Johannes Ranke
data(massart97ex3) m <- lm(y ~ x, data = massart97ex3) calplot(m)data(massart97ex3) m <- lm(y ~ x, data = massart97ex3) calplot(m)
Sample dataset to test the package.
din32645din32645
A dataframe containing 10 rows of x and y values.
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. Formerly available from the Website of the University of Heidelberg.
Currie, L. A. (1997) Nomenclature in evaluation of analytical methods including detection and quantification capabilities (IUPAC Recommendations 1995). Analytica Chimica Acta 391, 105 - 126.
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$x, 4) # A similar value is obtained using the approximation # LQ = 3.04 * LC (Currie 1999, p. 120) 3.04 * lod(m, alpha = 0.01, beta = 0.5)$xm <- 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
Example dataset B.1 from DIN 38402 with concentrations in µg/L and the extinction as response measured using continuous flow analysis (CFA) according to ISO 13395.
din38402b1din38402b1
A tibble containing 12 concentration levels with the respective instrument response values.
DIN 38402-51:2017-05, Beuth Verlag, Berlin. https://dx.doi.org/10.31030/2657448
Example dataset B.3 from DIN 38402. Cu was measured according to ISO 11885, using ICP-OES. The concentration are reported in mg/L and the response as counts/s, describing the count of photons that are detected by the photomultiplier detector of the device.
din38402b3din38402b3
A tibble containing 13 concentration levels and the respective instrument response values.
DIN 38402-51:2017-05, Beuth Verlag, Berlin. https://dx.doi.org/10.31030/2657448
Example dataset B.6 from DIN 38402 measured using LC-MS/MS. The concentrations are reported in in µg/L and the response in arbitrary units (AU).
din38402b6din38402b6
A tibble containing 12 concentration levels and the respective instrument response values.
DIN 38402-51:2017-05, Beuth Verlag, Berlin. https://dx.doi.org/10.31030/2657448
Example dataset C.3 from DIN 38402 determined by ion chromatography. Concentrations are reported in mg/L and the extinction as response.
din38402c3din38402c3
A tibble containing 10 concentration levels and the respective response values.
DIN 38402-51:2017-05, Beuth Verlag, Berlin. https://dx.doi.org/10.31030/2657448
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.
inverse.predict( object, newdata, ..., ws = "auto", alpha = 0.05, var.s = "auto" )inverse.predict( object, newdata, ..., ws = "auto", alpha = 0.05, var.s = "auto" )
object |
A univariate model object of class |
newdata |
A vector of observed y values for one sample. |
... |
Placeholder for further arguments that might be needed by future implementations. |
ws |
The weight attributed to the sample. This argument is obligatory
if |
alpha |
The error tolerance level for the confidence interval to be reported. |
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 |
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
A list containing the predicted x value, its standard error and a confidence interval.
The function was validated with examples 7 and 8 from Massart et al. (1997). Note that the behaviour of inverse.predict changed with chemCal version 0.2.1. Confidence intervals for x values obtained from calibrations with replicate measurements did not take the variation about the means into account. Please refer to the vignette for details.
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
# This is example 7 from Chapter 8 in Massart et al. (1997) 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 # For reproducing the results for replicate standard measurements in example 8, # we need to do the calibration on the means when using chemCal > 0.2 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) m3.means <- lm(y ~ x, w = weights, data = massart97ex3.means) inverse.predict(m3.means, 15, ws = 1.67) # 5.9 +- 2.5 inverse.predict(m3.means, 90, ws = 0.145) # 44.1 +- 7.9# This is example 7 from Chapter 8 in Massart et al. (1997) 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 # For reproducing the results for replicate standard measurements in example 8, # we need to do the calibration on the means when using chemCal > 0.2 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) m3.means <- lm(y ~ x, w = weights, data = massart97ex3.means) inverse.predict(m3.means, 15, ws = 1.67) # 5.9 +- 2.5 inverse.predict(m3.means, 90, ws = 0.145) # 44.1 +- 7.9
A function to create diagnostic plots for the assessment of the linearity of calibration data based on their point-to-point slope or the curvature. The underlying methods follow ISO 84 66-1:2021 and DIN 32 402-51:2017 (German Industrial Norm).
linearity(x, y, method = c("slope", "curvature"), tolerance = 0.1)linearity(x, y, method = c("slope", "curvature"), tolerance = 0.1)
x |
numeric vector of independent values (usually concentrations). |
y |
numeric vector of dependent values (usually the signal of the analytical device). |
method |
character string. Supported methods are "slope" and "curvature". |
tolerance |
numeric value between 0 and 1, describing the acceptable deviation from the median of the slopes or the signal-to-concentration ratio. The default tolerance is 10%. |
The point-to-point slope method is based on the assumption that the slope between two points should not vary greatly within the linear range.
The curvature method is similar to the point-to-point slope method. Here, the ratio between the instrument signal and the concentration of the calibration standard is assumed not to vary greatly within the linear range.
The use of the Mandel test is discouraged due to its limitations in the identification of non-linear behaviour of calibration curves (Andrade and Gomes-Carracedo, 2013).
returns a diagnostic plot
Anıl Axel Tellbüscher
ISO 8466-1:2021. Water quality — Calibration and evaluation of analytical methods — Part 1: Linear calibration function
J. M. Andrade and M. P. Gomez-Carracedo (2013) Notes on the use of Mandel's test to check for nonlinearity in laboratory calibrations. Analytical Methods 5(5), 1145 - 1149.
# Continuous Flow Analysis (CFA) data data(din38402b1) # Point-to-point slope plot linearity(din38402b1$conc, din38402b1$ext, method = "slope") # Curvature plot linearity(din38402b1$conc, din38402b1$ext, method = "curvature")# Continuous Flow Analysis (CFA) data data(din38402b1) # Point-to-point slope plot linearity(din38402b1$conc, din38402b1$ext, method = "slope") # Curvature plot linearity(din38402b1$conc, din38402b1$ext, method = "curvature")
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).
lod( object, ..., alpha = 0.05, beta = 0.05, method = "default", tol = "default" )lod( object, ..., alpha = 0.05, beta = 0.05, method = "default", tol = "default" )
object |
A univariate model object of class |
... |
Placeholder for further arguments that might be needed by future implementations. |
alpha |
The error tolerance for the decision limit (critical value). |
beta |
The error tolerance beta for the detection limit. |
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 “din” method uses the prediction interval at x = 0 as an approximation. |
tol |
When the “default” method is used, the default tolerance for the LOD on the x scale is the value of the smallest non-zero standard divided by 1000. Can be set to a numeric value to override this. |
A list containig the corresponding x and y values of the estimated limit of detection of a model used for calibration.
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 predict.lm
function, which is currently not supported in R.
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 for 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)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)
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
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
inverse.predict, and L is obtained by iteration.
loq( object, ..., alpha = 0.05, k = 3, n = 1, w.loq = "auto", var.loq = "auto", tol = "default" )loq( object, ..., alpha = 0.05, k = 3, n = 1, w.loq = "auto", var.loq = "auto", tol = "default" )
object |
A univariate model object of class |
... |
Placeholder for further arguments that might be needed by future implementations. |
alpha |
The error tolerance for the prediction of x values in the calculation. |
k |
The inverse of the maximum relative error tolerated at the desired LOQ. |
n |
The number of replicate measurements for which the LOQ should be specified. |
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 |
var.loq |
The approximate variance at the LOQ. The default value is calculated from the model. |
tol |
The default tolerance for the LOQ on the x scale is the value of the smallest non-zero standard divided by 1000. Can be set to a numeric value to override this. |
The estimated limit of quantification for a model used for calibration.
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 for din32645
m <- lm(y ~ x, data = massart97ex1) loq(m) # We can get better by using replicate measurements loq(m, n = 3)m <- lm(y ~ x, data = massart97ex1) loq(m) # We can get better by using replicate measurements loq(m, n = 3)
Sample dataset from p. 175 to test the package.
massart97ex1massart97ex1
A dataframe containing 6 observations of x and y data.
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.
Sample dataset from p. 188 to test the package.
massart97ex3massart97ex3
A dataframe containing 6 levels of x values with 5 observations of y for each level.
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.
# For reproducing the results for replicate standard measurements in example 8, # we need to do the calibration on the means when using chemCal > 0.2 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) m3.means <- lm(y ~ x, w = weights, data = massart97ex3.means) # The following concords with the book p. 200 inverse.predict(m3.means, 15, ws = 1.67) # 5.9 +- 2.5 inverse.predict(m3.means, 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, data = massart97ex3) 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(m3.means, 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.# For reproducing the results for replicate standard measurements in example 8, # we need to do the calibration on the means when using chemCal > 0.2 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) m3.means <- lm(y ~ x, w = weights, data = massart97ex3.means) # The following concords with the book p. 200 inverse.predict(m3.means, 15, ws = 1.67) # 5.9 +- 2.5 inverse.predict(m3.means, 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, data = massart97ex3) 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(m3.means, 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.
Dataset reproduced from Table 1 in Rocke and Lorenzato (1995).
rl95_cadmiumrl95_cadmium
A dataframe containing four replicate observations for each of the six calibration standards.
Rocke, David M. und Lorenzato, Stefan (1995) A two-component model for measurement error in analytical chemistry. Technometrics 37(2), 176-184.
Dataset reproduced from Table 4 in Rocke and Lorenzato (1995). The toluene amount in the calibration samples is given in picograms per 100 µL. Presumably this is the volume that was injected into the instrument.
rl95_toluenerl95_toluene
A dataframe containing four replicate observations for each of the six calibration standards.
Rocke, David M. und Lorenzato, Stefan (1995) A two-component model for measurement error in analytical chemistry. Technometrics 37(2), 176-184.
Dataset read into R from https://sites.chem.utoronto.ca/chemistry/coursenotes/analsci/stats/files/example14.xls.
utstats14utstats14
A tibble containing three replicate observations of the response for five calibration concentrations.
David Stone and Jon Ellis (2011) Statistics in Analytical Chemistry. Tutorial website maintained by the Departments of Chemistry, University of Toronto. https://sites.chem.utoronto.ca/chemistry/coursenotes/analsci/stats/index.html