Univariate Calibration Curve

Fits a univariate calibration curve (signal against concentration), with its figures of merit, limits of detection and quantification, and tests of linearity. The curve predicts the concentration of new samples from their signal, with confidence intervals.

Usage

calibration_curve(
  data,
  signal,
  concentration,
  model = "linear",
  weights = NULL,
  blank = NULL,
  lod_method = NULL,
  level = 0.95
)

Arguments

data: A data frame with the calibration samples.
signal, concentration: The columns of data holding the signal and the reference concentration, unquoted or as strings.
model: "linear" (default) or "quadratic".
weights: NULL (default, unweighted), "1/x", "1/x2", or a numeric vector of weights, one per row of data.
blank: An optional numeric vector of replicate signals of a blank sample.
lod_method: The estimate of the blank standard deviation: "residual", "intercept" or "blank" (see details). By default, "blank" when blank is given, "residual" otherwise.
level: The confidence level of the intervals of the coefficients. Default is 0.95.

Value

An object of class specproc_calibration, a list with

coefficients: a tibble of the estimates, standard errors and confidence intervals (lower, upper) at level;
figures_of_merit: a tibble with the number of standards n, the sensitivity, r_squared, the residual standard deviation sigma, the blank standard deviation used (sigma_blank), lod and loq (in concentration units);
linearity: a tibble with the statistic, degrees of freedom and p-value of Mandel's test and of the lack-of-fit test (when they apply);
fit: the lm fit, and data, the calibration data.

Use predict() for new samples and plot_calibration() to draw the curve.

Details

The signal, such as a line intensity from line_intensities() or a line ratio, is modeled as a straight line or a second-degree polynomial of the concentration, fitted by (weighted) least squares. Weights "1/x" or "1/x2" suit signals whose variance grows with concentration, which is common in LIBS.

Figures of merit. The sensitivity is the slope of the curve at zero concentration. The limits of detection and quantification are \(LOD = 3.3\,\sigma / S\) and \(LOQ = 10\,\sigma / S\) (ICH, 2005), where \(S\) is the sensitivity and \(\sigma\) the standard deviation of the signal of a blank, estimated from

"residual": the residual standard deviation of the fit (default without blank);
"intercept": the standard error of the intercept;
"blank": the standard deviation of replicate blank signals (default when given).

With weights, the residual standard deviation depends on the scale of the weights, so prefer "blank" or "intercept".

Linearity. For a straight line, Mandel's test compares its residual variance with that of a quadratic fit; a small p-value means that the curvature is significant (as with self-absorption or detector saturation). When concentrations are replicated, the lack-of-fit test compares the residuals with the pure error of the replicates.

Intervals. The confidence band of the curve shows where the mean signal lies at each concentration; the prediction band, wider, where a single new measurement is expected to fall, since it adds the noise of the measurement. plot_calibration() draws either or both, and predict() computes them for given concentrations (type = "signal").

Inverse prediction. The concentration of a sample is the solution of the calibration equation for its signal (for a quadratic curve, the root within or nearest to the calibration range). Its standard error, by the delta method, combines the uncertainty of the curve with that of the new signal, the mean of replicates measurements.

References

ICH (2005). Validation of analytical procedures: text and methodology Q2(R1). International Conference on Harmonisation.
Mandel, J. (1964). The Statistical Analysis of Experimental Data. Interscience, New York.
Miller, J.N., Miller, J.C. (2018). Statistics and Chemometrics for Analytical Chemistry, 7th ed. Pearson, Harlow.

Author

Christian L. Goueguel

Examples

data(forageLIBS)
# the K I 769.90 nm line, normalized to the C I 247.86 nm line of the matrix
lines <- line_intensities(forageLIBS[-(1:14)], c(C = 247.856, K = 769.896), baseline = TRUE)
standards <- data.frame(
  K = forageLIBS$K,
  signal = lines$intensity[lines$line == "K"] / lines$intensity[lines$line == "C"]
)
cal <- calibration_curve(standards[1:300, ], signal, K)
cal
#> Linear calibration curve (300 standards)
#> 
#>       term estimate std_error  lower  upper
#>  intercept   0.2708   0.03240 0.2070 0.3345
#>      slope   0.1384   0.01556 0.1078 0.1690
#> 
#> R-squared:    0.20976
#> Sensitivity:  0.1384
#> LOD:          3.38  (residual)
#> LOQ:          10.2
#> Mandel:        F = 4.03, p = 0.0457
#> Lack of fit:   F = 1.13, p = 0.239
# the concentrations of new samples, from their signals
head(predict(cal, standards$signal[301:368]))
#> # A tibble: 6 × 7
#>   signal concentration    se   lower upper below_lod below_loq
#>    <dbl>         <dbl> <dbl>   <dbl> <dbl> <lgl>     <lgl>    
#> 1  0.459          1.36  1.03 -0.665   3.38 TRUE      TRUE     
#> 2  0.543          1.97  1.03 -0.0538  3.99 TRUE      TRUE     
#> 3  0.580          2.23  1.03  0.212   4.25 TRUE      TRUE     
#> 4  0.492          1.59  1.03 -0.427   3.62 TRUE      TRUE     
#> 5  0.481          1.52  1.03 -0.507   3.54 TRUE      TRUE     
#> 6  0.501          1.67  1.03 -0.355   3.69 TRUE      TRUE