M5–M7 Rules and Implementation Differences

Why the M5–M7 Labels Need an Exact Rule

A below-quantification-limit (BQL) result is not an exact concentration: it indicates that the concentration is below the applicable lower limit of quantification (LLOQ). The M5, M6, and M7 conventions turn such records into analysis values by substitution, and, for M5, by also omitting some records. These are data-handling rules, not three different ways to observe the underlying concentration.

The distinctions matter most when BQL records occur consecutively. M5 retains only the first BQL record in a consecutive run, substituting one-half of its LLOQ, and omits the rest of that run from the analysis. M6 substitutes one-half of the LLOQ for the first record in each run and assigns zero to the remaining records in that run. M7 assigns zero to every BQL record. Quantified records are unchanged by these rules.

Definition: Let \(Y_{ij}\) denote the reported result for record \(j\) from subject \(i\), and let \(L_{ij}\) be the LLOQ applicable to that record. A record is BQL when the assay reports it as below \(L_{ij}\). A run is a maximal consecutive sequence of BQL records for one subject in the specified record order: a quantified record or a change of subject ends the run. Under the standard conventions used here, M5 assigns \(L_{ij}/2\) to the first BQL record in a run and omits later records in that run; M6 assigns \(L_{ij}/2\) to the first and zero to later records; M7 assigns zero to every BQL record.

The run definition makes the conventions operational. If a quantified observation separates two BQL groups, the second group begins a new run, so it has its own “first” BQL record under M5 and M6. The applicable LLOQ is record-specific: when assay limits differ, use the limit attached to the BQL record being handled rather than a single assumed limit for the entire data set.

Worked Rules for a Sequence of Records

Worked Example: One Run and Then a Second Run

Consider one hypothetical subject with records in time order. The LLOQ is \(0.10\ \mathrm{mg/L}\) at every time. The reported sequence is \(1.20\ \mathrm{mg/L}\), BQL, BQL, \(0.18\ \mathrm{mg/L}\), BQL, BQL, BQL. There are two quantified records and five BQL records. The quantified result of \(0.18\ \mathrm{mg/L}\) separates the first BQL run from the second.

For either run, one-half the LLOQ is \(0.10/2=0.050\ \mathrm{mg/L}\). Under M5, the first two-record run becomes \(0.050\ \mathrm{mg/L}\) followed by an omitted record. The second, three-record run also contributes \(0.050\ \mathrm{mg/L}\), followed by two omitted records. Thus M5 retains the two quantified records and two substituted records; three BQL records are omitted from the analysis.

Under M6, the first run becomes \(0.050, 0\ \mathrm{mg/L}\), and the second becomes \(0.050, 0, 0\ \mathrm{mg/L}\). All seven records remain represented in the analysis. Under M7, all five BQL records become \(0\ \mathrm{mg/L}\); the two quantified results remain \(1.20\) and \(0.18\ \mathrm{mg/L}\). These transformations differ even though they start from the same assay reports.

The rules can be summarized for a run of \(k\) BQL records with record-specific limits \(L_1,\ldots,L_k\). M5 contributes one substituted value, \(L_1/2\), and omits \(k-1\) records. M6 contributes \(L_1/2\) followed by \(k-1\) zeros. M7 contributes \(k\) zeros. The first-run value uses the LLOQ of the first record in that run; each later BQL record is treated according to the rule for later records, not substituted using the first record’s limit.

\begin{aligned} \text{M5: } &(L_1/2,\ \text{omit},\ldots,\text{omit}),\\ \text{M6: } &(L_1/2,\ 0,\ldots,0),\\ \text{M7: } &(0,\ldots,0). \end{aligned}

A Short Justification: Count What Each Rule Represents

The operational difference follows directly by counting the BQL records in each run. If a run contains \(k\) records, M5 has one first record to substitute and \(k-1\) later records to omit. M6 has one first record to substitute and \(k-1\) later records to assign zero. M7 has no first-record exception, so all \(k\) records are assigned zero. For example, with \(k=3\), M5 contributes one analysis value from the run, whereas M6 and M7 each contribute three. This count is also a simple implementation check.

These are not likelihood statements about the probability of the unobserved concentration. In particular, replacing a BQL result by zero does not mean the assay measured an exact zero, and replacing one by half the LLOQ does not recover its unknown value. The substitutions create analysis values under a chosen convention.

Worked Example: Different LLOQs Across Records

Suppose a hypothetical subject has one BQL result with an LLOQ of \(0.08\ \mathrm{mg/L}\), followed by a quantified result, followed later by one BQL result with an LLOQ of \(0.12\ \mathrm{mg/L}\). Because the quantified result separates the BQL records, they are separate runs.

For the first BQL record, half the applicable LLOQ is \(0.08/2=0.040\ \mathrm{mg/L}\). For the second, it is \(0.12/2=0.060\ \mathrm{mg/L}\). M5 and M6 therefore substitute \(0.040\) and \(0.060\ \mathrm{mg/L}\), respectively; neither record is a later BQL in its run. M7 assigns zero to both. The quantified result is kept unchanged under all three rules.

Using one common limit, such as \(0.08\ \mathrm{mg/L}\), for both records would give the second BQL record the wrong M5 or M6 substitution. Applying \(0.12\ \mathrm{mg/L}\) to both would give the first record the wrong substitution. The record-to-limit mapping is part of the rule.

Documenting and Verifying an Implementation

An M5, M6, or M7 label alone may not fully specify the data transformation in a particular analysis. Reproducibility requires stating how records were ordered, how runs were defined, which LLOQ was used for each record, what happened to omitted observations, and how the transformed values were used in estimation. These details should describe the executed analysis, not merely the intended convention.

Specify the input and ordering.
Identify the subject, time or record order, BQL status, reported value, and applicable LLOQ. State how ties or duplicated records are ordered if they can occur.

Define a run.
State that consecutive BQL records within a subject form a run, and specify which record types end it. Do not let a run continue across subjects.

Apply the named transformation.
For each BQL run, apply the stated M5, M6, or M7 rule. Preserve quantified values and record whether M5 exclusions are omitted from the analysis.

Check and report the result.
Compare counts of input BQL records, substituted records, zeros, and omitted records with the rule’s expected counts. Record the software-specific implementation and any deviations.

For a hypothetical run of four BQL records, a reproducible report could say: “Within each subject, records were ordered by nominal time. Consecutive BQL records formed a run, interrupted by any quantified record. The first BQL in each run was set to half that record’s LLOQ; later BQL records in that run were set to zero. Quantified records were unchanged.” That description identifies M6 under the conventions in this tutorial and makes the handling testable. A report using M5 must additionally say that later records in each run were omitted from the estimation data.

A useful verification is to hand-check a small input sequence containing an isolated BQL, a run of multiple BQL records, a quantified result between runs, and more than one LLOQ. Compare every transformed value and the final record count against the stated rule. This catches errors that can be hidden in a large processed data set.

Assumptions, Diagnostics, and Limitations

The conventions require reliable BQL flags and a correctly matched LLOQ for each record. They also require a defined within-subject ordering so that “first” and “subsequent” have unambiguous meanings. If these inputs are wrong, the transformation can be wrong even when the code follows its written instructions. Before analysis, check for missing or inconsistent BQL status, impossible or missing limits, duplicate times whose order is unspecified, and records accidentally grouped across subjects.

Check the transformed data as well as the source data. Under M5, each BQL run should yield exactly one substituted value and \(k-1\) omitted records. Under M6, each run should yield one half-LLOQ value and \(k-1\) zeros. Under M7, every BQL record should become zero. Confirm that quantified observations remain unchanged, and tabulate the number of BQL records handled by each rule. A mismatch indicates either a coding error or a difference between the stated and implemented convention.

Zero substitutions deserve special attention. A zero is the assigned value under M6 or M7, not an assay measurement proving that the concentration was zero. Some observation models or transformations may not accommodate zero values directly; therefore the analyst must check that the chosen estimation setup can process the values as implemented. Do not silently alter the zeros or delete them: that would create a different rule and must be documented.

The main alternative is to model a BQL record as a left-censored observation, using the information that its value is below the LLOQ rather than assigning it a point value. The earlier material on censored likelihoods establishes that distinction. M5–M7 are simpler deterministic substitutions, but they discard or invent point values in ways that can affect the fitted model; they do not preserve all information in the censoring statement. Their results can also depend on the pattern of BQL runs and the chosen convention. A method label is therefore not a substitute for examining the data handling and its consequences.

Common Pitfalls

Treating M5, M6, and M7 as synonyms. They differ in whether later BQL records in a run are omitted or assigned zero, and whether the first record is set to half the LLOQ.
Starting a new run for every BQL record. Consecutive BQL records belong to one run; a quantified record separates runs.
Using the wrong LLOQ. M5 and M6 use half the LLOQ attached to the first record in each run, not an arbitrary study-wide value when limits differ.
Calling a substituted zero an observed zero. It is a data-handling assignment, not a quantified assay result.
Reporting only “M6 was used.” Without run ordering, limit selection, and handling of omitted or transformed records, that statement may not reproduce the analysis.

Key takeaway: Under the conventions stated here, M5 substitutes half the LLOQ for the first BQL in each consecutive run and omits the rest; M6 substitutes half the LLOQ first and assigns zero to the rest; M7 assigns zero to every BQL record. Define the run, use the correct record-specific limit, and document the implemented rule.

Check Your Understanding

Answer each question using the conventions defined above. Solutions follow the questions.

A subject has three consecutive BQL records at an LLOQ of \(0.20\ \mathrm{mg/L}\). What values or exclusions does each of M5, M6, and M7 produce?
A quantified record separates two BQL records. Are the BQL records in one run or two? What value does M6 assign to each if their LLOQs are \(0.06\) and \(0.10\ \mathrm{mg/L}\)?
For a run of four BQL records, how many records from that run are omitted under M5? How many zeros are assigned under M6 and M7?
Why is it incorrect to describe an M7 zero as the concentration measured by the assay?

Solutions. 1. Half the LLOQ is \(0.20/2=0.10\ \mathrm{mg/L}\). M5 yields \(0.10\ \mathrm{mg/L}\) and omits the other two records; M6 yields \(0.10,0,0\ \mathrm{mg/L}\); M7 yields \(0,0,0\ \mathrm{mg/L}\). 2. They are two runs because the quantified record separates them. Each is the first in its run, so M6 assigns \(0.06/2=0.030\ \mathrm{mg/L}\) and \(0.10/2=0.050\ \mathrm{mg/L}\), respectively. 3. M5 omits three records. M6 assigns three zeros, and M7 assigns four zeros. 4. BQL means the assay result is below its LLOQ; M7’s zero is a conventionally assigned analysis value, not an exact measurement.