Calibration Curves Data

R script for calculating Limit of Detection and Limit of Quantification

Note: the data and theory on calibration curve were with reference from: Statistics and Chemometrics for Analytical Chemistry, James N. Miller and Jane Charlotte Miller, 6th edition, Chapter 5

Loading required packages

library(pacman)
p_load(tidyverse, broom, chemCal)

Background

Chemists often work with calibration data using standards of known concentrations and putting them through instrumental analysis. When plotting a calibration curve, it is of interest to calculate the limit of detection (LOD) and limit of quantification (LOQ) of the method.

Import Example Dataset

The fluorescence intensities of standard aqueous fluorescein solutions were analysed with a spectrophotometer, and the fluorescence results are shown below:

# A tibble: 7 x 2
  conc_pgml  fluo
      <dbl> <dbl>
1         0   2.1
2         2   5  
3         4   9  
4         6  12.6
5         8  17.3
6        10  21  
7        12  24.7

Visualize

Model

Let’s fit a linear model to get the slope (b) and intercept(a).

\[ y = a + bx \]

Call:
lm(formula = fluo ~ conc_pgml, data = data)

Residuals:
       1        2        3        4        5        6        7 
 0.58214 -0.37857 -0.23929 -0.50000  0.33929  0.17857  0.01786 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   1.5179     0.2949   5.146  0.00363 ** 
conc_pgml     1.9304     0.0409  47.197 8.07e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4328 on 5 degrees of freedom
Multiple R-squared:  0.9978,    Adjusted R-squared:  0.9973 
F-statistic:  2228 on 1 and 5 DF,  p-value: 8.066e-08

From above, we can see that slope = 1.9304, and intercept = 1.5179.

Errors in slope and intercept of regression line

The limit of detection is defined as:

\[ LOD = \gamma_B + 3_{SB} \] where LOD is the analyte concentration wich gives a signal equal to the blank signal plus three standard deviations of the blank.

A function was created to calculate LOD and LOQ:

calcLOD_y <- function(model) {
  SSE <- sum(model$residuals**2)
  n <- length(model$residuals) -2
  Syx <- sqrt(SSE/n)
  intercept <- as.numeric(model$coefficients[1])
  calculated_y <- intercept + 3*Syx
  names(calculated_y) <- "calculated_y"
  print(calculated_y)
  
  chemCal::inverse.predict(model,
                  newdata = calculated_y,
                  alpha = 0.05) 
}

calcLOQ_y <- function(model) {
  SSE <- sum(model$residuals**2)
  n <- length(model$residuals) -2
  Syx <- sqrt(SSE/n)
  intercept <- as.numeric(model$coefficients[1])
  calculated_y <- intercept + 10*Syx
  names(calculated_y) <- "calculated_y"
  print(calculated_y)
  
  chemCal::inverse.predict(model,
                  newdata = calculated_y,
                  alpha = 0.05) 
}

Inserting the linear model from the fluorescence data:

LOD_x <- calcLOD_y(fl_mod)

calculated_y 
      2.8164 

LOD_x$Prediction 

[1] 0.6726958

LOQ_x <- calcLOQ_y(fl_mod)

calculated_y 
    5.846334 

LOQ_x$Prediction 

[1] 2.242319

Predict the value of x from y:

To predict the concentration of fluorescein that has fluorescence units of 2.9, we use the function inverse.predict():

chemCal::inverse.predict(fl_mod, 
                newdata = 2.9,
                alpha = 0.05)

$Prediction
[1] 0.7160037

$`Standard Error`
[1] 0.2645698

$Confidence
[1] 0.6800982

$`Confidence Limits`
[1] 0.03590545 1.39610195