AL


Looking at a statistical table of z-values, you find that you need to go out about 3.3 stan- dard deviations from the mean to exclude this fractional area α from one tail of a Gaussian curve. This is shown in Figure 3, where LD 3.3×σb


= μb + . If the sample contains no analyte, then


the probability of getting a measured signal greater than μb


+ 3.3×σb is 0.0005; thus, the


risk of making a false positive statement that analyte has been detected is 0.05%, 1/20th of one percent.


It’s curious that this value of LD = μb + 3.3×σb


is often found in the literature. Each applica- tion should require a different false positive risk, so why do so many applications seem to require 1/20th of one percent as the false positive risk? It doesn’t make sense. But here’s some history: someone once asked Sir Ronald A. Fisher (for whom the F test was named) what the false positive risk should be for scientists trying to discover the laws of the universe, and he said something like, “Oh, one time in twenty would be OK, I guess,” which gives us our common 95% level of con- fidence.4


I can only speculate that to some


persons, “one-twentieth of one percent” sounds like something Fisher might recom- mend; it sounds kind of scientific, somehow.


You and your client shouldn’t automatically adopt a value of 0.0005, or 0.005, or any other specific value for α. Think about your applica- tion and set an appropriate value for the false positive risk. Then you can use α, μb


, σb z-table to find the corresponding value for LD


, and a .


Let’s go back and consider again the state- ment that “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 it’s been detected.” Look at Figure 4. In this example, with LD


= μb + 3.3×σb , the probability of a false


positive is 0.0005 for a blank. For a sample that contains just a tiny amount of analyte (mov- ing to the right in the figure), the truth is that analyte is present, but the probability of not detecting it is very large (a bit less than 0.9995, but still very large). This is the false negative risk β. Clearly, the false negative risk depends on the amount of analyte actually present.


PTFE Solenoid-Operated Isolation


& Pinch Valves


Figure 4 – The concept of a false negative.


In the next column, we’ll explore the relationship between the false negative risk and the amount of analyte. This will lead us to the minimum con- sistently detectable amount, the MCDA.


Remember: the limit of detection LD is on the


vertical signal axis of a calibration plot; the minimum consistently detectable amount MCDA will be on the horizontal amount axis.


References


1. Currie, L.A. Limits for qualitative detection and quantitative determination: applica- tion to radiochemistry. Anal. Chem. 1968, 40(3), 586–593.


2. Lindstrom, R.M. In Lide, D.R., Ed. A Century of Excellence in Measurements, Standards, and Technology: A Chronicle of Selected NBS/NIST Publications, 1901-2000; NIST Special Publi- cation 958, 2001.


3. Hubaux, A. and Vos, G. Decision and detec- tion limits for linear calibration curves. Anal. Chem. 1970, 42(8), 849–855.


4. Moore, D.M. Statistics: Concepts and Contro- versies; Freeman: San Francisco, CA, 1979.


Stanley N. Deming, Ph.D., is an analytical chemist masquerading as a statistician at Statistical Designs, El Paso, TX, U.S.A.; e mail: statisticaldesigns@gmail.com; www. statisticaldesigns.com


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