STATISTICS IN THE LABORATORY continued


calibration relationship and extends out to the measured value. Statisti- cal analysis (usually linear least squares) can be used to obtain the best fit of a two-parameter (slope and intercept) straight line to these noisy data. The y-intercept of the calibration line will be called the true mean of the blank μb


standard deviation of the blank σb


, where a blank is a sample containing no analyte. The can be obtained from the variation


of the data points about the fitted calibration line at small amounts of analyte. Alternatively, both μb


and σb and σb


measurements of a blank sample. In this column we’ll assume that the uncertainties in estimating μb


. can be obtained from repetitive are sufficiently small that we may


represent them as population parameters. For generality, the calibration relationship shown in Figure 1 has a non-zero μb


Figure 2 shows the results of multiple measurements of a blank. Note that the horizontal x-axis is now the measurement number, not the amount. The Gaussian distribution in the new panel at the left sum- marizes the measurements—the Gaussian is centered at μb standard deviation of σb


and has a . Here’s the way the game is played. Given a limit of detection LD , if y repre-


sents the measured signal for a sample containing an unknown amount of analyte, the rules are:


if y ≥ LD if y ≤ LD


, analyte is said to be detected (present) , analyte is said to be not detected


Not detected doesn’t necessarily mean absent; it just means that if there is any analyte present, there isn’t enough of it to give a signal strong enough to say that it’s been detected.


You get to decide what value you’re going to use for LD


couple of possibilities. Suppose you decide to set the limit of detection LD


mean of the blank μb . Let’s look at a equal to the true as shown in Figure 2. If you then make multiple


measurements on a blank, half the time you’ll get a signal y that’s less than LD


(the red measurements)


(the green measurements) and conclude that analyte has not been detected. So far, so good: there is no analyte to detect. But half the time you’ll get a signal y that’s greater than LD


and conclude that analyte has been detected. This does not represent the truth. This is a positive statement (“analyte is present”) that’s clearly false; in fact, it’s called a false positive, and in this case the false positive risk α is equal to 0.5 (half the measurements lie in the red part of the Gaussian).


For most applications, α = 0.5 is probably an unacceptable false positive risk. Consider a drug-testing laboratory. If LD


= μb for each drug test, then


a non-drug user stands a 50% chance of being wrongly accused of hav- ing a specific drug in his or her body. It’s worse if k drugs are being tested at the same time. Remember αEW


making at least one false positive statement: (1)


For k = 10 simultaneous drug tests, the overall risk of getting at least one false positive result is 0.9990234375, or greater than 99.9%. This means that if 1000 totally clean applicants were each tested for a battery of 10 illicit drugs, it is expected that 999 of them would be falsely accused of having one or more of those drugs in their bodies. Are you beginning to see that the false positive risk α should be appropriate for the applica- tion? You and your client need to decide what α should be, and then adjust LD


appropriately.


So how do you do that? Let’s say you get together with your client and decide that an acceptable αEW


would be 0.005 (0.5%; now only 5 nailed


out of 1000 innocents … but you’ll let them take the drug test again to further decrease their chances of being wrongly accused). You can re- arrange Eq. (1) to give


(2) , the overall “experiment-wise” risk of


Figure 2 – The false positive risk α of setting LD


equal to μb


. AMERICAN LABORATORY 42


Figure 3 – The false positive risk α of setting LD JUNE/JULY 2017


equal to μb


+ 3.3×σb


.


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