As was mentioned in the first article on detection limits (DLs) (Part 26, American Laboratory, June/July 2007), many formulas exist for calculating these values. One of the most widely used methods is known as 3-Sigma (3σ). The basic approach is as follows. Seven or eight replicates of a blank are analyzed by the analytical method, the responses are converted into concentration units, and the standard deviation is calculated. This statistic is multiplied by 3, and the result is the detection limit. If blanks are not available, then a low-level standard may be used instead. However, the resulting detection limits must be greater than one-fifth of the spike concentration for the DL to be valid.
The advantages of this procedure are that it is easy to collect the data and it is easy to calculate the DL. On the opposite side of the coin is a serious disadvantage: the rate of false negatives (i.e., β) is unknown and uncontrolled. Beta can be calculated via Receiver Operating Characteristic (ROC) curves (see Parts 27 and 31 of this series, American Laboratory, Sep 2007 and Sep 2008, respectively) if the requisite data are available. In many instances, β is quite high. This reality is due to the fact that only a limited amount of variability (i.e., the noise in the blank data) is incorporated into the study and calculation. Thus, the 3σ DL is quite low. When the calculation is complete, the user has chosen values for two of the three variables in the ROC equation and must live with the calculated third value. To be specific, α is set to a nominal level of 1%, based on the 3 of 3σ. Three is approximately equal to the 99% quantile for the Student’s t distribution, when there are seven or eight degrees of freedom. (If eight replicates are analyzed for the 3σ method, then the degrees of freedom equal seven. Each data point supplies one degree, but one is lost when the standard deviation is calculated.) The DL has been calculated from a single-concentration data set, and the value of β is determined by the equation. If the DL is quite low, then β must be high to compensate.
Table 1 - Detection limits (in ppt) for chloride, using both the 3σ and the Hubaux-Vos approaches*
These trade-offs can best be understood via an actual example. Recall the chloride data, whose ROC curves were discussed in Part 27 of this series and whose Hubaux-Vos (H-V) DLs (at 95% confidence) were estimated in Part 30 (American Laboratory, June/ July 2008)*. These results were from a ppt-level study of deionized water. A standard-preparation blank and eight spike concentrations were prepared and analyzed on each of eight separate days. The lowest two non-zero concentrations were 25 and 37.5 ppt. For each anion, the 3σ DL can be calculated using, in turn, the blank and the lowest two non-zero concentrations. The values are given in Table 1. All 3σ DLs from non-blank data are valid, since each is greater than one-fifth of the spike concentration.
Note that the H-V DL is larger than any of the corresponding 3σ values. This result is not surprising. Remember that with the H-V method, both α and β are known and controlled (here, each is equal to 0.01 or 1%). Thus, the DL (the third “leg” of the detection world’s triangle) is determined by the DL relationship. However, the 3σ protocol controls only α. Once the DL is determined, β is whatever value the equation allows.
Table 2 - Values of β associated with the DLs given in Table 1
Using Eq. 13 from Part 31, β can be determined. Table 2 shows the associated values of β. For all of the 3σ limits, the values are low enough that β must be, at best, 0.263. While there is nothing “wrong” with claiming these low DLs, only if the user knows the associated value of β can he or she decide if such a false-negative rate is acceptable for the situation at hand.
By definition, in these cases where α and β are not equal, the prediction intervals will not be symmetrical about the regression line. If α = 0.01 and β is a higher value, then the lower prediction limit will be closer to the regression line than will be the upper prediction limit. Such a configuration must result in order to have the DL be the low value that was calculated.
Figure 1 - Prediction limits (blue line is upper limit, green line is lower limit, and red line is regression line) for the chloride data. This plot illustrates the case when α = 0.01 (as is the case for the 3σ protocol) and β = 0.263 (which results when the 3σ DL of 16.2 ppt is chosen). The lower part of the range is shown, but all of the chloride data were used to compute the prediction line and the limits. The vertical line is located at the 16.2-ppt detection limit.
Figure 2 - Same plot as above, except that β = 0.742, the value associated with a DL of 9.3 ppt (black vertical line). Note that the lower prediction limit (green line) has “crossed” the regression (red) line.
Figure 1 illustrates this “unequal” situation for the 3σ detection limit of 16.2 ppt, resulting in a β of 0.263. Note the closeness of the lower prediction limit to the regression line. If β becomes large enough, the lower prediction limit will actually cross the regression line. The crossover point is when β rises to 0.5 (a similar situation occurs for the upper prediction limit when α becomes 0.5). For this probability, Student’s t is zero, for all degrees of freedom. Recall from Part 4 (American Laboratory, Mar 2003) that the expression for a given prediction limit is [(RMSE/b) * (t) * (a square-root term)]. When t = 0, multiplying by zero forces the entire expression (which is added to the prediction line to generate the prediction limit) to disappear. Thus, the distance of the limit from the regression line is zero, and the limit corresponds to the line itself. Once β (or α) becomes larger than 0.5, the value for Student’s t changes sign, meaning that the sign of the prediction-limit expression changes sign also. As a result, the limit will cross to the other side of the regression line.
An example of a crossover situation occurs for these data when the 3σ DL of 9.3 ppt is chosen (see Figure 2). For this DL, Table 2 shows that β = 0.742, which is greater than the crossover value of 0.5. Another way to look at the situation is to consider the plot. This crossing had to occur in order to meet the criteria of α = 0.01 and the DL of 9.3 ppt. Only the user can decide if this type of result is acceptable for his or her application.
*In writing this article, L.V. discovered that the data set for Part 27 was inadvertently based on a subset of the data set used for Part 30. Part 27 used only 7 days’ worth of replicates, while Part 30 utilized the entire 8 days. The calculations and plots in this article are based on the entire set of data. L.V. apologizes for this discrepancy.
Mr. Coleman is an Applied Statistician, Alcoa Technical Center, MST-C, 100 Technical Dr., Alcoa Center, PA 15069, U.S.A.; e-mail: [email protected]. Ms. Vanatta is an Analytical Chemist, Air Liquide-Balazs™ Analytical Services, 13546 N. Central Expressway, Dallas, TX 75243-1108, U.S.A.; tel.: 972-995-7541; fax: 972-995-3204; e-mail: [email protected].