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2 2
κ ˆ
κ ˆ
parameter values are σ ˆ b = 0.267 and = 0.016. These give σ ˆ b = 0.52 and = 0.13. The MDL = 3.0σ ˆ b
= 3(0.52) = 1.56. This should be rounded to 1.6 µg /L.
The EPA method gave MDL = 1.7 µg/L. Pallesen’s method gave MDL = 1.6 µg/L. The difference
has no practical importance, despite the following differences in the two methods.
1. The z or t statistic multipliers are different: z α =0.01 = 2.33 for the EPA method and z α = 0.0013 =
3.00 for the Pallesen method. On purely statistical grounds, the EPA’s specification of α =
0.01 is no more justified than Pallesen’s recommended multiplier of 3.00.
2. Users of the EPA method will often estimate the MDL using seven, or perhaps fourteen,
replicate measurements because the procedure requires a minimum of seven replicates. In
Example 14.2, the MDL (EPA method) for Laboratory B was based on 34 measurements.
Pallesen’s method requires replicated measurements at several concentration levels (how many
levels or replicates are not specified). The case study used 50 measured values, divided among
five concentration levels. Each of these values contributes to the estimation of the MDL.
3. The EPA method estimates the MDL from measurements in a given matrix containing the
analyte and then assumes that the background noise observed is applicable to specimens that
are blank (zero concentration). The alternate method extrapolates to zero analyte concentration
to estimate the background noise (σ b ) that serves as the basis for computing the MDL. As
shown in Table 14.3, the total error variance increases as the concentration increases and
Pallesen’s method accounts for this property of the data. If the data have this property, the
EPA approach of pooling replicates from two different concentrations will tend to inflate the
MDL over the estimate of the alternate method.
4. The EPA’s MDL gives some information about the precision of measurements at low con-
centrations, but it says nothing about bias, which can be a major problem. Pallesen’s model
distinguishes between error contributions from the analytical method and from background
noise. It also retains the average values, which could be compared with the known concen-
trations to make an assessment of bias.

Calibration Designs
Another way to estimate MDLs is to use a calibration design (Gibbons, 1994). A series of samples are
spiked at known concentrations in the expected range of the MDL. Prediction limits (or tolerance limits
in some methods) are determined for the very low concentrations and these define a limit of detection.
Frequently, calibration data have nonconstant variance over a range of concentrations. If this is the case,
then weighted least squares must be used to fit the calibration curve or else the variability is overestimated
at low concentrations and the detection limit will be overestimated. Zorn et al. (1997) explain how this
is done. Calibration is discussed in Chapter 37 and weighted least squares is discussed in Chapter 38.

Comments
The limit of detection is a troublesome concept. It causes difficulties for the chemist who must determine
its value, for the analyst who must work with data that are censored by being reported as <MDL, and
for the regulator and discharger who must make important decisions with incomplete information. Some
statisticians and scientists think we would do better without it (Rhodes, 1981; Gilbert,1987). Nevertheless,
it is an idea that seems firmly fixed in environmental measurement methods, so we need to understand
what it is and what it is not. It is an attempt to prevent deciding that a blank sample is contaminated
with an analyte.
One weakness of the usual definitions of the MDL is that it is defined for a single analyst. ASTM
D2777 (1988) is very clear that instead of repeated single-operator MDLs, the correct approach is to
use the interlaboratory standard deviation from round-robin studies. This can change the MDL by a
factor of 0.5 to 3.0 (Maddelone et al., 1988, 1991, 1993).
© 2002 By CRC Press LLC