Determine the detection limit based on using a calibration line (or curve) and inverse regression.
detectionLimitCalibrate(object, coverage = 0.99, simultaneous = TRUE)an object of class "calibrate" that is the result of calling the function
calibrate.
optional numeric scalar between 0 and 1 indicating the confidence level associated with
the prediction intervals used in determining the detection limit.
The default value is coverage=0.99.
optional logical scalar indicating whether to base the prediction intervals on
simultaneous or non-simultaneous prediction limits. The default value is
simultaneous=TRUE.
A numeric vector of length 2 indicating the signal detection limit and the concentration
detection limit. This vector has two attributes called coverage
and simultaneous indicating the values of these arguments that were used in the
call to detectionLimitCalibrate.
The idea of a decision limit and detection limit is directly related to calibration and can be framed in terms of a hypothesis test, as shown in the table below. The null hypothesis is that the chemical is not present in the physical sample, i.e., \(H_0: C = 0\), where C denotes the concentration.
| Your Decision | \(H_0\) True (\(C = 0\)) | \(H_0\) False (\(C > 0\)) |
| Reject \(H_0\) | Type I Error | |
| (Declare Chemical Present) | (Probability = \(\alpha\)) | |
| Do Not Reject \(H_0\) | Type II Error | |
| (Declare Chemical Absent) | (Probability = \(\beta\)) |
Ideally, you would like to minimize both the Type I and Type II error rates. Just as we use critical values to compare against the test statistic for a hypothesis test, we need to use a critical signal level \(S_D\) called the decision limit to decide whether the chemical is present or absent. If the signal is less than or equal to \(S_D\) we will declare the chemical is absent, and if the signal is greater than \(S_D\) we will declare the chemical is present.
First, suppose no chemical is present (i.e., the null hypothesis is true). If we want to guard against the mistake of declaring that the chemical is present when in fact it is absent (Type I error), then we should choose \(S_D\) so that the probability of this happening is some small value \(\alpha\). Thus, the value of \(S_D\) depends on what we want to use for \(\alpha\) (the Type I error rate), and the true (but unknown) value of \(\sigma\) (the standard deviation of the errors assuming a constant standard deviation) (Massart et al., 1988, p. 111).
When the true concentration is 0, the decision limit is the (1-\(\alpha\))100th percentile of the distribution of the signal S. Note that the decision limit is on the scale of and in units of the signal S.
Now suppose that in fact the chemical is present in some concentration C (i.e., the null hypothesis is false). If we want to guard against the mistake of declaring that the chemical is absent when in fact it is present (Type II error), then we need to determine a minimal concentration \(C_DL\) called the detection limit (DL) that we know will yield a signal less than the decision limit \(S_D\) only a small fraction of the time (\(\beta\)).
In practice we do not know the true value of the standard deviation of the errors (\(\sigma\)), so we cannot compute the true decision limit. Also, we do not know the true values of the intercept and slope of the calibration line, so we cannot compute the true detection limit. Instead, we usually set \(\alpha = \beta\) and estimate the decision and detection limits by computing prediction limits for the calibration line and using inverse regression.
The estimated detection limit corresponds to the upper confidence bound on concentration given that the signal is equal to the estimated decision limit. Currie (1997) discusses other ways to define the detection limit, and Glaser et al. (1981) define a quantity called the method detection limit.
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# NOT RUN {
# The data frame EPA.97.cadmium.111.df contains calibration
# data for cadmium at mass 111 (ng/L) that appeared in
# Gibbons et al. (1997b) and were provided to them by the U.S. EPA.
#
# The Example section in the help file for calibrate shows how to
# plot these data along with the fitted calibration line and 99%
# non-simultaneous prediction limits.
#
# For the current example, we will compute the decision limit (7.68)
# and detection limit (12.36 ng/L) based on using alpha = beta = 0.01
# and a linear calibration line with constant variance. See
# Millard and Neerchal (2001, pp.566-575) for more details on this
# example.
calibrate.list <- calibrate(Cadmium ~ Spike, data = EPA.97.cadmium.111.df)
detectionLimitCalibrate(calibrate.list, simultaneous = FALSE)
# Decision Limit (Signal) Detection Limit (Concentration)
# 7.677842 12.364670
#attr(,"coverage")
#[1] 0.99
#attr(,"simultaneous")
#[1] FALSE
#----------
# Clean up
#---------
rm(calibrate.list)
# }
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