Biological Zeros, Detection Limits, and Censored Signals

Why the Meaning of Zero Matters

An entry of zero in an omics table can describe different things. A biological target may truly be absent from a specimen; a sequencing assay may have sampled no molecules from a feature that is present; or an instrument may have registered a signal too weak to report quantitatively. These states are not interchangeable. If they are collapsed into the same numeric zero, downstream summaries and comparisons can give a misleading account of the data.

The measurement target and scale matter. A zero read count is an observed count. A value reported as “below detection limit” is generally an observation that the underlying signal lies in a range the assay cannot quantify. A feature not called by a processing pipeline may instead mean that the available evidence did not meet a detection or identification rule. These distinctions build on the measurement targets and scales used in sequencing, proteomic, and metabolomic tables.

Three Measurement States

A biological zero is about the specimen. For example, a particular transcript may be absent from the biological material being measured. Whether “absent” is meaningful depends on the target definition: absence from a sampled cell type is not necessarily absence from every cell in a tissue. Even when a target is truly absent, the assay may still produce background signal, contamination, or an erroneous call.

An observed zero is about the data produced by the assay. In sequencing, the recorded count for a feature can be zero because no molecules from that feature were sampled and sequenced. The feature may nevertheless be present. A count table records what was observed, not a direct census of every molecule in the specimen.

A below-limit result is about the assay’s resolution. If a quantitative assay reports values only at or above a limit \(L\), a report of “less than \(L\)” says the signal is in the range below \(L\). It does not distinguish a true zero from a small positive value in that range. This is called left censoring when the observation tells us a value is below a threshold but not its exact value.

The assumptions behind any interpretation should therefore be stated. A reported zero should be understood relative to the table’s data-generation rules: whether zero is an actual measured count, a thresholded signal, or a placeholder for a feature not reported. Interpreting “below \(L\)” as censoring assumes the assay documentation identifies \(L\) on the relevant scale and that the entry really means the unreported value fell below that threshold. It should not be assumed that every unreported feature has that meaning.

Why a Sequencing Zero Can Occur by Chance

A simple model illustrates why a zero read count does not prove a feature is biologically absent. Suppose, as a simplifying assumption, that the count \(X\) for a feature follows a Poisson distribution with mean \(\lambda\). This model treats reads as independent sampling events with a fixed expected count for that feature in the library. It is an illustration of sampling, not a complete model for all sources of variation in sequencing data.

For a zero count, substitute \(k=0\). Since \(0!=1\) and \(\lambda^0=1\), the probability is

Thus, whenever the expected count is positive, the probability of seeing zero is also positive under this model. A small expected count makes a zero especially plausible. This derivation justifies treating zero reads as evidence about sampling and abundance, not as proof that abundance equals zero.

Below-Limit Signals Are Censored, Not Exact Zeros

For a quantitative assay, let \(Y\) be the signal on the assay’s reported scale and \(L\) its detection limit. If the instrument reports only “\(Y<L\),” the data tell us that the value lies below \(L\), without specifying where. For a nonnegative signal, the compatible interval is \(0\leq Y<L\). If the measurement model has noise, this interval describes the reported observation; it does not necessarily mean the underlying biological amount itself is known to lie exactly in that interval.

When a probability model for the signal is appropriate, a censored observation contributes the probability of falling below the threshold, rather than the probability of taking the exact value zero. For example, if a noisy instrument response \(Y\) is normally distributed with mean \(\theta\) and standard deviation \(\sigma\), then the probability of a report below \(L\) is \(P(Y<L\mid\theta)=\Phi((L-\theta)/\sigma)\), where \(\Phi\) is the standard normal cumulative distribution function. This statement assumes the stated normal measurement model and a known threshold; it does not imply that every assay signal is normally distributed.

Metabolomics: Keep the Limit and the Value Distinct

Metabolomics tables can include peaks that are detected but not confidently assigned to a compound, compounds with a signal below a quantitative limit, and features not reported under the processing rules. These statements answer different questions: whether a signal was observed, whether it could be assigned to a compound, and whether its amount could be quantified. A missing compound label is not, by itself, proof of biological absence.

Practical Checks and Common Pitfalls

Before analysing a column containing zeros or non-detections, check what the assay and data-processing documentation say about its entries. Useful questions include: Is zero an actual count or an assigned code? What is the feature-specific or assay-wide reporting limit, and in what units? Does “not detected” mean no signal, a signal below a threshold, or failure to meet an identification rule? Are detection and quantification thresholds different? These checks clarify the meaning of the values; they do not recover an exact amount that the assay did not measure.

A common mistake is to reason, “The table contains zero, so the biological target is absent.” The correction is to identify what the zero encodes. A zero sequencing count is an observed count and can arise from finite sampling; a below-limit report is an interval; a biological zero is a claim about the specimen. If the table has replaced a threshold report with zero, that encoding should be verified rather than assumed.

Another tempting shortcut is to substitute half the detection limit for every below-limit observation. This may be a convenient approximation in some analyses, but it asserts a specific value that was not observed and can affect summaries, especially when many observations are censored. Likewise, replacing every zero count with a small positive number changes the recorded count and can alter subsequent calculations. Neither substitution makes the underlying state known.

No single approach resolves every case. Retaining the observed count preserves the sequencing result but does not classify biological absence. Recording “below \(L\)” as an interval preserves the known threshold information but not the unknown exact value. A zero or half-limit substitution yields a complete numeric table, but adds assumptions and can distort summaries. A probability model for censored values can use the threshold information, but relies on a suitable model for the signal and measurement process. The appropriate choice depends on the data’s meaning and the question being asked.

These distinctions have limits. Assay documentation may not record the relevant threshold, and a threshold can vary by feature or run. A value above a detection threshold may still have uncertain identification or quantification. Conversely, a biological target that is absent may produce background signal. The table alone may therefore be insufficient to classify a recorded zero or non-detection with certainty.