Below-Detection Measurements As MNAR

Why a Missing Measurement May Still Contain Information

Laboratory instruments do not always return a numerical value. An assay may report a concentration only when it reaches a specified detection limit. A sample below that limit may be recorded as “below detection,” while a different sample may have no result because a tube was lost or the instrument failed. Both records lack an exact numerical measurement, but they do not mean the same thing. In the first case, the absence of a number is tied to the value itself; in the second, it may be unrelated to the value.

This distinction matters because missingness tied to the unobserved measurement is missing not at random (MNAR). It also matters because “below detection” is not necessarily a blank with no information: it can tell us that the value lies below a known boundary. The classification should follow the measurement and recording process, not merely the appearance of an empty cell.

Definition: Let \(Z\) be the measurement that would be obtained without a detection limit, let \(c\) be the detection threshold on the scale of \(Z\), and let \(R=1\) indicate that an exact numerical measurement is recorded and \(R=0\) indicate that it is not. Under a pure threshold rule, \(R=1\) when \(Z\geq c\) and \(R=0\) when \(Z<c\). If \(P(Z<c)>0\) and \(P(Z\geq c)>0\), this rule makes response depend on the unobserved value and produces MNAR nonobservation.

The condition that there is positive probability on both sides of the threshold is important. A rule cannot create observed nonresponse in a population where every possible value is above the threshold. Nor does a threshold divide the population if all possible values are below it. In either case, response does not vary over the measurement’s support in the relevant way; the ordinary situation of some detectable and some undetectable measurements is the one described by the MNAR classification.

Threshold Censoring and the Information in a Limit

A below-limit result and a failed assay can both have \(R=0\), but they differ in what is known about \(Z\). If the laboratory records a specific “below detection” code and the threshold is known, the record implies \(Z<c\). This is often described as left-censoring: the exact value is unavailable, but it is known to lie in a specified interval. If a sample is lost for an unrelated operational reason, the record alone may give no such bound.

A simple distributional model makes the distinction concrete. Suppose, for illustration, that \(Z\) follows a normal distribution with mean \(\mu\) and standard deviation \(\sigma\), and that \(c\) is a known, fixed threshold. Then the probability that a measurement falls below the limit is the lower-tail probability \(P(Z<c)\). For a detected result, the observed values come from the part of the distribution at or above \(c\); they are not a random sample of all values from the original distribution.

P(Z<c)=\Phi\!\left(\frac{c-\mu}{\sigma}\right), \qquad f(z\mid Z\geq c)=\frac{f(z)}{P(Z\geq c)} \quad (z\geq c).

Here, \(f\) is the density of \(Z\), and \(\Phi\) is the standard normal cumulative distribution function in the normal example. The second expression is the conditional density among detected measurements: it is the original density restricted to values above the limit and divided by the probability of being above the limit. A below-limit code contributes information about the event \(Z<c\), rather than identifying the exact value.

Worked Example: Calculating the Fraction Below a Limit

Suppose a hypothetical assay measures the logarithm of a concentration, \(Z\), and returns an exact value only when \(Z\geq 2\). For illustration, assume \(Z\sim N(3,1)\), so the mean is 3 and the standard deviation is 1. This distributional assumption is a simplifying model for this example, not a conclusion established by the missingness rule.

The standardized threshold is \( (2-3)/1=-1 \). Therefore, \(P(Z<2)=\Phi(-1)\approx 0.1587\), and \(P(Z\geq 2)=1-0.1587=0.8413\).

In a hypothetical set of 1,000 measurements generated under this model, about \(1{,}000(0.1587)=159\) would be expected below the threshold and about \(1{,}000(0.8413)=841\) would be expected to yield exact numerical values. These are expected counts under the assumed model, not guaranteed counts in a particular set of 1,000 samples. Because both probabilities are positive, the threshold rule makes the chance of an exact result depend on \(Z\): values below 2 are not returned numerically, whereas values at or above 2 are.

The threshold model also explains why detected measurements can look systematically high relative to the full distribution: the smaller values have been excluded from the exact numerical results by design. This is not evidence that the assay malfunctioned. It is a consequence of the reporting rule. Where the below-limit status is retained, that status should not be discarded as if it conveyed nothing.

Independent Instrument Failure Is a Different Process

Now consider an instrument that sometimes fails for operational reasons. Let \(F=1\) mean that the instrument is functioning for a sample and \(F=0\) mean that it fails. If failure is independent of \(Z\), it does not preferentially remove low or high values. If the instrument returns an exact value whenever it functions and \(Z\geq c\), then both the threshold and instrument operation determine whether an exact number is recorded.

Definition: In a threshold-plus-failure model, suppose \(P(F=1)=q\), with \(F\) independent of \(Z\). The exact-result indicator is \(R=F\,I(Z\geq c)\), where \(I(\cdot)\) equals 1 when its condition is true and 0 otherwise. Thus an exact result requires both an operating instrument and a value at or above the limit.

The independence assumption concerns the failure process, not the threshold process. Even if instrument failure is independent of the true measurement, the combined missingness is still value-dependent when the threshold rule excludes some values and allows others. In a real data set, a failure code or laboratory log may help distinguish these causes. If the reason for absence is not recorded, the same blank cell could represent either process, and the classification may be uncertain.

Worked Example: Separating Failures From Below-Limit Results

In a hypothetical batch of 100 samples, the laboratory records 70 exact numerical results, 22 “below detection” results, and 8 instrument failures. Assume the failure codes correctly identify operational failures and that the detection limit is known.

The 22 below-detection records tell us that those samples’ measurements were below the threshold, even though their exact values are unavailable. Their lack of an exact number is tied to the measurement value by the reporting rule. The 8 failure records indicate a separate operational reason for no numerical result; whether those failures are independent of the sample values must be justified by how the failure process works, not inferred just from the fact that they are labelled “failure.”

There are 30 records without exact numerical values, but they should not automatically be treated as 30 equivalent missing measurements. The 22 below-limit records carry interval information, while the 8 failure records do not establish that their values were below the limit. The observed proportion without an exact result is \(30/100=0.300\), or 30.0%, but that single proportion does not identify how much nonobservation came from thresholding versus failure.

A Short Derivation for a Mixed Process

Under the threshold-plus-failure model, conditioning on the true measurement gives a direct way to see where the value dependence enters. If \(z<c\), the threshold condition is not met, so an exact result cannot be recorded even if the instrument works. If \(z\geq c\), an exact result is recorded precisely when the instrument functions.

P(R=1\mid Z=z)= \begin{cases} 0, & z<c,\\ q, & z\geq c. \end{cases}

When both \(P(Z<c)>0\) and \(P(Z\geq c)>0\), these conditional response probabilities differ across possible measurement values. Therefore response depends on \(Z\), even though the additional failure indicator \(F\) is independent of \(Z\). Averaging over the distribution of \(Z\) gives \(P(R=1)=qP(Z\geq c)\). The overall probability of no exact result is consequently \(P(Z<c)+(1-q)P(Z\geq c)\): a below-limit result or an instrument failure for a value above the limit can each prevent an exact result.

Worked Example: Quantifying Threshold and Failure Contributions

Use the hypothetical normal model \(Z\sim N(3,1)\) with threshold \(c=2\), and suppose an operating instrument returns 95% of otherwise detectable results, so \(q=0.95\). Assume instrument failure is independent of \(Z\), and that the normal model and threshold are correct for this calculation.

From the earlier calculation, \(P(Z<2)=0.1587\) and \(P(Z\geq 2)=0.8413\). The probability of an exact result is \(0.95(0.8413)=0.7992\), to four significant figures. Thus the probability of no exact result is \(1-0.7992=0.2008\), or 20.08%.

The below-threshold contribution is 0.1587. The additional failure contribution among detectable values is \((1-0.95)(0.8413)=0.05(0.8413)=0.04207\). Adding them gives \(0.1587+0.04207=0.20077\), which rounds to 0.2008. The response probability is 0 below the threshold and 0.95 above it, so the combined nonobservation remains value-dependent. Independent instrument failure does not remove the MNAR feature introduced by the threshold.

How to Classify a Real Measurement Process

A useful classification begins with the reason a result is absent and the information retained about it. The substantive question is not simply whether the data table contains a blank. Ask whether the laboratory’s rule deterministically or probabilistically relates the chance of an exact numerical result to the value that would have been measured.

Identify the measurement scale and threshold.
Check the assay documentation for the limit and whether it applies to the scale in the analysis. If the measurement is transformed, express the threshold on that same scale.

Inspect the recorded status.
Separate explicit below-limit codes from instrument failure, lost samples, and other reasons whenever the records permit. A below-limit code usually conveys a bound; a failure code need not.

State the response rule and its conditions.
Specify what values yield an exact result. For a threshold rule, check whether there is positive probability on both sides of the threshold before describing the observed nonresponse as value-dependent.

Assess other routes to nonobservation.
Determine whether failures depend on sample characteristics, batch, concentration, or other variables. Do not assume failures are independent merely because they are not called “below detection.”

Useful diagnostics include assay manuals, batch-level logs, status codes, and plots of exact results relative to the limit. A pile-up of recorded values exactly at the limit can indicate that below-limit results were replaced by the limit rather than retained as censored statuses. That practice changes what the recorded number means: a substituted threshold is not an observed exact measurement. Conversely, an absence of such a pile-up does not prove that no thresholding occurred; a separate code may have been used.

Observed data alone generally cannot establish that failures are independent of the unobserved measurements. Process information is therefore important. If failures occur more often in particular batches or under operating conditions associated with sample values, an independence claim may be inappropriate. A relationship between failure and observed predictors can also change the missingness classification conditional on those predictors. The standard names MCAR, MAR, and MNAR describe assumptions about the data-generating and missingness processes; labels in a file are not, by themselves, proof of any one assumption.

What the Classification Does and Does Not Establish

Calling below-threshold nonobservation MNAR describes how the response indicator depends on the unavailable value. It does not mean that every analysis of below-limit data is impossible. When the censoring threshold is known, the below-limit status provides an inequality about the value. A suitable analysis can use that information and make explicit assumptions about the measurement distribution. Those assumptions matter: the threshold rule alone does not specify the distribution of values below the limit or determine their exact values.

It is also important not to conflate classifying a mechanism with choosing a method to fill blank cells. Replacing every below-limit value by zero, by the limit, or by a fixed fraction of the limit treats an interval statement as though it were an exact observation. Such substitutions can distort summaries and associations. The later material on single imputation and missing-value uncertainty addresses why a single filled-in value does not represent uncertainty about the missing measurement.

Reason no exact value is recorded	Information available	Key classification point
Below a known detection limit	The value is known to be below the threshold	Nonobservation depends on the value when values occur on both sides of the limit
Instrument failure independent of the measurement	Usually no bound on the value from the failure alone	The failure component can be independent, but check the operational process
Thresholding plus independent instrument failure	Below-limit statuses may give a bound; failures may not	The overall exact-result indicator remains value-dependent if the threshold excludes some values

Common Mistakes

Calling every blank a below-limit result. A lost sample or instrument failure does not imply that its measurement was small. Use recorded reasons and process documentation.
Calling every instrument failure MCAR without checking. Independence is an assumption about the failure process. Failures can be related to batch, sample type, or other features.
Ignoring the support condition. A threshold rule is value-dependent in the relevant sense when both sides of the threshold have positive probability. Merely writing a threshold does not prove that any observations are omitted.
Treating a below-limit code as an exact zero or an exact threshold value. A code that means \(Z<c\) supplies an interval constraint, not an exact measurement.
Thinking independent failures cancel the threshold mechanism. They do not. In a combined process, response still varies with \(Z\) when the threshold has positive probability on both sides.

Key takeaway: A detection-limit rule makes exact-result nonresponse depend on the unobserved measurement when there is positive probability both below and at or above the threshold. Independent instrument failure is a distinct process, and a below-limit status may retain useful interval information that a generic failure code does not.

Check Your Understanding

For each question, first decide what the recording process says about the unobserved value. Solutions follow each question.

An assay reports values only when \(Z\geq 4\). Suppose \(P(Z<4)=0.12\) and \(P(Z\geq4)=0.88\). Is the threshold-based nonobservation MNAR, and what condition supports the answer?
Solution: Yes. The response rule depends on \(Z\), with nonresponse below 4 and response at or above 4. Both sides have positive probability, as shown by 0.12 and 0.88.
Every possible value in a population is above the assay’s detection limit. Does the threshold rule produce below-limit nonresponse in this population?
Solution: No. Here \(P(Z<c)=0\), so no values are excluded by the threshold. A threshold exists, but it does not produce nonresponse for values in this population’s support.
A record is coded “below detection” for a known threshold \(c\). What can be concluded about its exact value?
Solution: The code indicates \(Z<c\), subject to the accuracy of the assay coding and threshold. It does not reveal the exact value; it supplies an interval constraint.
In the mixed example, suppose \(P(Z\geq c)=0.80\) and \(q=0.90\). Calculate the probability of an exact result.
Solution: An exact result requires both a detectable value and a functioning instrument, so \(P(R=1)=qP(Z\geq c)=0.90(0.80)=0.72\). The probability is 72% under the stated model.
A data file has 15% blank values, but it contains no reason codes or assay documentation. Can the blanks be classified as independent failure or below-limit nonresponse from that percentage alone?
Solution: No. The overall missing proportion does not reveal the cause. Status codes, laboratory records, or additional process information are needed to distinguish thresholding from failure and assess assumptions about failure independence.