Detection limits

Why does a laboratory report a contaminant as \"below the limit of detection\" rather than simply saying none was there?

A water sample goes to a laboratory. The report comes back: the contaminant was "below the limit of detection". It reads like a lawyer's sentence — a refusal to say the plain thing, which is that there was none.

But suppose the laboratory is entirely honest and simply cannot say what we want it to. What would have to be true of measurement itself for "none was there" to be a claim no instrument can make?

Start at the simplest experiment. Take a sample prepared to contain none of the substance — a blank — and measure it. Does the instrument read zero?

It does not. It reads something: electronic noise, stray light, a trace of the analyte in the reagents themselves, material bleeding off a chromatography column, a baseline drifting with the room. Run that blank twenty times and you get twenty different numbers scattered around an average. The instrument's answer to "nothing" is not a value; it is a distribution with a centre and a width.

Everything follows from that. When a sample reads slightly above the blank average, two explanations are available: a little of the substance was present, or this was a blank on the high side of its scatter. Nothing in the number distinguishes them. Detection is therefore not a measurement but a decision, and a decision needs a threshold.

Put the threshold low and you report contamination that is not there; put it high and you miss contamination that is. No placement avoids both, and the discipline fixes the trade by convention: a stated number of blank standard deviations above the blank average.

Take the common choice of three. Under a normal distribution a one-sided threshold three standard deviations up is exceeded by roughly one blank in seven hundred — read off a table, not taken on faith. Two things spoil that comfort: real blanks drift and occasionally spike, and their standard deviation is usually estimated from a handful of replicates, itself wildly uncertain when it comes from five numbers. The true false-positive rate is worse than the table says.

The second point is deeper, and most often missed. Ask what happens if a sample truly contains the substance at a concentration whose average signal sits exactly at the threshold. Half its measurements land above the line and half below: a sample sitting precisely at the "limit of detection" is missed half the time. So one threshold cannot serve as both a decision rule and a capability statement, and the careful treatment — Currie's — separates them. The critical value is the line a single result is compared against, chosen for an acceptable false-positive rate; the detection limit is the higher concentration at which you will reliably exceed that line. For a five percent false-positive rate, set the line about 1.65 blank standard deviations up; to have no more than a five percent chance of missing a genuine sample, that sample must average another 1.65 above the line. Add them: the detection limit sits near 3.3 standard deviations, roughly twice the critical value. The factor of two is not padding — it is the price of controlling both errors instead of one.

The same reasoning gives the next tier: a limit of quantitation at ten standard deviations, where the scatter is about a tenth of the value, so a number quoted there carries roughly ten percent relative uncertainty. Below it you may say something is present but not how much.

Now the report makes sense. "Below the limit of detection" says: this result did not clear a threshold set on our own noise, using our chosen error rates. It is a statement about the measurement system and the decision rule, not about the sample, and properly an upper bound. Absence is not something a measurement establishes; only presence can be, and absence only bounded.

A second, chemical trap sits beside the statistical one: what the instrument achieves on a clean standard is not what the method achieves on real material, since poor recovery, a suppressing matrix or a smaller aliquot all make it worse.

Three clarifications, two of them about neighbouring ideas that look identical and are not.

This is not the base-rate problem behind screening false positives. That question begins after a test has produced a yes or no, and asks what a yes is worth when the condition is rare; this one asks where the yes came from — a continuous signal, and a line drawn on it by someone choosing an error rate. Nor is it censoring, as in a study whose patients ran out of follow-up time: a censored observation was never made, while here it was made in full and simply cannot be told apart from nothing.

The threshold is empirical, not a property of the substance. Change the reagents, let the lamp age, analyse a dirtier matrix, and the blank's scatter changes with it. Two laboratories can honestly report "not detected" and a definite number for the same sample without either being wrong, so a limit quoted with no conditions attached is not information.

And the account is testable in an afternoon. Prepare twenty blanks and twenty samples spiked at the claimed detection limit and run them blind. The predictions are specific: blanks should exceed the threshold only rarely; samples spiked at the critical value missed about half the time; samples spiked at the detection limit caught about nineteen times in twenty. If the spiked-at-limit samples came back positive essentially always, the stated limit is more conservative than its arithmetic claims. If blanks came back positive routinely, their spread was underestimated — almost always because too few were used.

• Why regulatory limits are sometimes set below what routine methods can detect.

Every term the collection defines is gathered in the glossary.