OVERVIEW

The detection limit is a statistical estimate that is used to make the binary decision of whether or not the true concentration in a given sample is greater than zero. A frequent confusion regarding the detection limit is that measured concentrations exceeding the detection limit are quantifiable (i.e. the estimated concentration can be reliably determined). This is not the case. Measured concentrations above the detection limit only allow one to conclude that the analyte is present in the sample at a concentration greater than zero. Quantification limits (also known as quantitation limits) have been developed for the purpose of quantitative determination (Currie 1968 and Gibbons et. al., 1992) and are also provided in Detect.

Currie (1968) defined the "detection limit" LD as the true concentration "at which a given analytical procedure may be relied upon to lead to a detection." Note that the emphasis here is on "true concentration" and not measured concentration. Currie also defined the "critical level" LC as the measured concentration "at which one may decide whether or not the result of an analysis indicates detection". It is important to understand the difference between LC and LD. When the true concentration is equal to LC the probability of detecting it is only 50%. In contrast, when the true concentration is at the LD the probability of a measured concentration below LC (hence a non-detection) is 1%. The detection limit LD may be relied upon to lead to a detection in 99% of the cases, whereas the critical level leads to a detection only half of the time. As an analogy, the critical level for the federal speed limit is 55 mph, however, police will rarely identify an exceedance until a driver exceeds 60 mph and their confidence that the true speed (not just the measured speed) has exceeded the limit is high.

There are many different names for detection and quantification limits which can lead to confusion. Often, the detection limit is confused with the critical level. Furthermore, there are diverse views about the choice of statistical multiplier, use of blanks, and single concentration versus multiple concentration calibration design. Unfortunately, different investigators have also given different names to the various statistical approaches to estimating the same thing. While the distinction between the critical level and the detection limit are qualitative and are therefore deserving of different names, the other distinctions are not. Detect provides several statistical approaches to estimating both the detection limit LD and the quantification limit LQ. The choice among these different estimators should be based on the different assumptions made by these methods, the suitability of which can be determined from empirical data.

Single concentration based estimators such as USEPA's Method Detection Limit (MDL) and Minimum Level (ML) and Currie's LD are also provided in Detect for completeness; however, they generally should not be used in practice because the results are highly dependent on the concentration at which the samples are spiked. Similarly, the Hubaux and Vos (1970) procedure is also included in Detect because of its historical importance as the first calibration based detection limit. Copyright© Robert D. Gibbons Ltd. & Discerning Systems Inc. 3 estimator. Note, however, that the Hubaux-Vos method assumes constant variability over the entire calibration range and will therefore routinely lead to overestimates of the true detection limit since it must overestimate variability at low levels in order to accommodate variability at higher levels.

The remaining detection limit estimators in Detect can be categorized in terms of approximate versus iterative solution based on prediction limits versus tolerance limits. These four models all assume that variability is a function of concentration and as such direct solution for LD which is a function of variability at LD is not possible. Two solutions are possible. First, Gibbons et. al., (1991) (also see Gibbons 1994 and 1995 and Oppenheimer et. al., 1983) describe an approximate solution in which variability at the lowest spiking concentration is used in anchoring the detection limit. Second, using a model for the relationship between concentration and variability (e.g., an exponential model or the Rocke and Lorenzato model, 1995) we can iteratively solve for LD. This more exact solution should be used in practice.

The second choice involves use of a prediction limit (PL) versus a tolerance limit (TL). Detection limits based on prediction limits apply to the next detection decision only. By contrast, detection limits based on tolerance limits apply to the entire population of detection decisions and provide a specific level of confidence of including a specified proportion of all future detection decisions. In Detect, detection limits based on tolerance limits will cover 99% of all future detection decisions with 95% confidence. For this reason, detection limits based on tolerance limits are recommended for routine applications in which a large and potentially unknown number of future detection decisions are made on the basis of a single detection limit study.

In terms of quantitative determination, Detect provides three different estimators; the USEPA Minimum Level (ML), the Alternative Minimum Level (AML, Gibbons, Coleman and Maddalone, 1996) and an iterative version of Currie's (1968) Determination Limit (LQ). The ML is a single concentration based method and is provided in Detect for illustrative purposes only. The iterative LQ estimator is the most statistically appealing in that it solves for the true concentration at which the signal to noise ratio is 10:1 (i.e. a percent relative standard deviation (%RSD) of 10%). However, if the rate of change in SD as a function of concentration is greater than .1, the LQ is not defined and the iterative solution will not converge. Detect deals with this problem by testing for convergence and if the model does not converge, 2.5% is added to the required %RSD (i.e. 12.5% RSD) and the process is repeated until a %RSD is identified that leads to convergence. In this way, the lowest possible %RSD is identified and the corresponding concentration estimated. Finally, the AML provides a good compromise between the ML and iterative LQ by initially anchoring the estimate of the standard deviation at the estimate of LC (i.e. the critical level) which is the lowest concentration that is differentiable from zero. The AML is then estimated as the upper prediction limit for the true concentration that is 10 times the standard deviation at the critical level. Note that in estimating the prediction limit, we use the estimated standard deviation at the provisional estimate of the AML (i.e. 10SLC). As such, although the AML does not guarantee a %RSD of 10% it will typically provide a %RSD of approximately 10% and the actual.4 %RSD at the AML is automatically computed by the program. The AML is the quantification limit estimator that is recommended for routine application.

TECHNICAL CONSIDERATIONS

A major advantage of calibration-based methods relative to single concentration based methods, such as USEPA's MDL and ML, is that they are not dependent on spiking concentration whereas the MDL and ML can vary dramatically across different spiking concentrations. It should be noted that in an attempt to eliminate this bias, USEPA suggests that only those spiking concentrations that yield a ratio of five to one or less with the Method Detection Limit (MDL) (i.e. spike/MDL <= 5) should be considered. Unfortunately, even following this criterion, the MDL and ML can vary by orders of magnitude. The calibration based estimates of the detection limit (Ld) and quantification limit (Lq) do not suffer from this limitation because they directly model the relationship between concentration and variability. The user should note, however, that the range of spiking concentrations must be relevant for computing Ld and Lq.

Example
In several studies in distilled (DI) water, Lds for benzene have been found at approximately 1 ppb. Selecting spiking concentrations from 25 to 200 ppb will not provide the necessary information regarding variability at 1 ppb, therefore biased Lqs can result. However, if spiking concentrations cover the lower end of the range, (i.e. 1, 5, 10, 20 ppb) then unbiased estimates of the Lq will be obtained.

To model the relationship between concentration and variability, the Detect program provides three options:
	the Rocke and Lorenzato (R & L) model
	the exponential model
	the best fit of these two models

The R & L model is based on a weighted least squares (WLS) regression of the sample standard deviation on concentration so that variability at lower concentrations is given greater weight than variability at higher concentrations. As such, including higher concentrations in the calibration will typically have little or no effect on the computed AML. This is not always true for the exponential model which, in general, should only be used if the program identifies a problem with the results of the R & L model. In most cases, selecting best fit will result in fitting the R & L model.

The previous discussion pertains to only fitting the model that describes the relationship between variability and concentration. For the recovery curve (i.e. true concentration versus measured concentration), a WLS regression model is used in all cases. Similar to the R & L model, the WLS regression model for the recovery curve gives greater weight to the lower end of the concentration range.

In general, the user should attempt to cover the range from concentrations of zero through five times the hypothesized Lq. Note, however, that if blank samples are used (i.e. concentration equal to zero), approximately 50% of the measured concentrations should be negative. If the instrument censors these negative concentrations (i.e. sets them to zero), the estimated Ld and Lq (including the MDL and ML) will be too low since only half of the true variability is observed. If this is the case, the user should select the lowest concentration for which the analyte is identified by the instrument in all cases and the estimated concentration is greater than zero.

As a final note, the R & L model can yield variable results when used to extrapolate to low level concentrations when only concentrations in the linear range are available. In general, the user should attempt to include blank samples as the lowest point in the recovery curve. If these data are not available, Detect will use the standard deviation at the lowest available concentration as an estimate of the low-level constant variance component in the R & L model. This is not a problem for the exponential model.