Report missing observations and changing denominators
Trace who contributes to each result, label every percentage denominator, and report why analysis sample sizes change without hiding missing observations.
About 12 min read · Allow extra time for the practical checks
A study may begin with 520 eligible people, describe 468 baseline respondents, calculate a percentage from 441 observed outcomes and fit a model with 412 complete records. None of those numbers is automatically wrong. The reporting problem begins when the reader cannot reconstruct why they differ.
A single “sample size” in the methods section is therefore rarely enough. Every important result needs an analysis-sample trace: the unit counted, the records required, the reasons records were unavailable or excluded, the missing-data method and the final number contributing to the result.
For the meanings of dataset fields and missing-value codes that sit behind this trace, see the separate guide to building a data dictionary.
This guide shows how to build that trace. It is a reporting framework, not a rule that every analysis should use complete records, imputation or any other universal missing-data method.
Name the unit before the number
The symbol n is ambiguous unless you say what it counts. Depending on the design, it may refer to people, visits, documents, measurements, clusters, events or units of person-time.
On a narrow screen, scroll the table sideways to read every column.
| Quantity | What it counts | Reporting question |
|---|---|---|
| Participants | Distinct people contributing to an analysis | Can one person appear more than once? |
| Records or observations | Rows, visits, responses or measurements | How are repeated observations linked to participants? |
| Outcome events | Occurrences such as completions, admissions or publications | Can a participant experience more than one event? |
| Clusters | Classes, clinics, institutions or other sampled groups | How many clusters and units within clusters were analysed? |
| Person-time | Observed time at risk | What starts, stops or censors contribution time? |
Write “412 participants,” “1,846 visits from 412 participants” or “286 completion events among 441 participants,” not merely “n = 412.” If rows rather than people are analysed, report both when both matter.
Missing also does not mean “no.” A missing outcome says the value was not observed or available under the study’s data rules. Coding it as a negative outcome changes the analysis unless that interpretation is justified by the data-generating process.
Keep three records connected
A transparent report normally needs three related records.
- Participant or record flow traces movement from the target or invited population through eligibility, enrolment, follow-up and broad analysis stages.
- Variable-level missingness shows how many values are observed and missing for important exposures, outcomes and covariates.
- Analysis-specific denominators show exactly which records contributed to each percentage, table, model or sensitivity analysis.
The STROBE Explanation and Elaboration asks observational-study authors to trace individuals through eligibility, inclusion, follow-up and analysis, give reasons for nonparticipation or exclusion, and report missing values for important variables. A flow diagram can help, but it does not replace the analysis-specific record.
Likewise, the SAMPL Guidelines recommend giving sample sizes for each analysis and the numerators and denominators behind percentages. One headline sample size cannot perform all of those jobs.
A flow record answers “How did the study population narrow?” A missingness table answers “Where are values absent?” A denominator ledger answers “Which records produced this result?” Keep them aligned, but do not collapse them into one overloaded table.
Build a denominator ledger
Create the ledger while you analyse, not after the manuscript has been written. Give each output a stable name that matches the table, figure or model label used in the draft.
On a narrow screen, scroll the table sideways to read every column.
| Field | What to record | Why it matters |
|---|---|---|
| Output | Table, figure, percentage, model or sensitivity analysis | Connects the ledger to a visible result |
| Counting unit | Participants, records, events, clusters or person-time | Prevents different units from being compared as if they were identical |
| Starting set | The eligible dataset before analysis-specific rules | Makes the reference population explicit |
| Required fields | Outcome, exposure and covariates needed for this output | Explains why different models can have different samples |
| Unavailable or excluded | Counts and non-overlapping reasons, preferably in the order applied | Lets the reader reconstruct the final count |
| Missing-data method | Complete-record, weighting, imputation, model-based handling or another justified method | Links the denominator to the actual analysis |
| Final denominator | Number of units contributing to the result | Identifies the analysis sample |
| Numerator | Event or category count when the result is a proportion | Makes a reported percentage reproducible |
| Notes | Assumptions, repeated measures, weighting, sensitivity role or confidentiality rule | Preserves qualifications that the count alone cannot show |
Sequential exclusion reasons should add to the difference between the starting set and the final complete-record sample. If reasons overlap, label them as overlapping rather than adding them. A participant can lack both an exposure and an outcome; two marginal missing counts cannot automatically be summed.
Worked example: trace every denominator
The institution, participants, programme and all counts below are wholly fictional. They were constructed to demonstrate reporting and are not participant data.
A fictional college examines whether use of an optional study-planning service is associated with completion of a follow-up learning task. The study identifies 520 eligible students, enrols 488 and receives a baseline questionnaire from 468. The follow-up completion outcome is observed for 441 baseline respondents.
Among the 468 baseline respondents, year level is available for all 468, service-use exposure for 455, commuting time for 448 and programme mode for 453. These variable-level counts overlap; they should not be subtracted from 468 independently to invent an analysis sample.
For the primary complete-record model, records are filtered sequentially: 27 lack the follow-up outcome, 11 of the remaining records lack the exposure and 18 of the remainder lack commuting time. The primary analysis therefore contains 468 − 27 − 11 − 18 = 412 participants.
The broad flow is:
On a narrow screen, scroll the table sideways to read every column.
| Stage | Participants | Change from previous stage |
|---|---|---|
| Eligible and invited | 520 | Starting set |
| Enrolled | 488 | 32 did not enrol |
| Baseline questionnaire | 468 | 20 enrolled students did not provide baseline data |
| Observed follow-up outcome | 441 | 27 baseline respondents had no recorded follow-up outcome |
That flow still does not show every analysis. The ledger does.
On a narrow screen, scroll the table sideways to read every column.
| Output | Required information | Method and exclusions | Contributing participants | Numerator and denominator |
|---|---|---|---|---|
| Baseline description | Each listed variable separately | Available values shown row by row | Valid n ranges from 448 to 468 | Shown for each categorical row |
| Follow-up completion percentage | Observed completion outcome | 27 outcomes unavailable | 441 | 286 / 441 |
| Primary adjusted model | Outcome, exposure, year level and commuting time | Complete-record; sequentially excludes 27, 11 and 18 | 412 | Not a percentage |
| Reduced-covariate sensitivity model | Outcome, exposure and year level | Complete-record; excludes 27 and then 11 | 430 | Not a percentage |
| Programme-mode subgroup model | All primary-model fields plus programme mode | Excludes 15 programme-mode values from the primary sample | 397 | Report subgroup counts separately |
| Multiple-imputation sensitivity analysis | Prespecified baseline variables and follow-up outcome | Fictional plan uses 40 imputed datasets; assumptions and model specification reported in Methods | 468 baseline participants | Observed counts remain reported separately |
The ledger shows why the model counts differ. It also prevents a subtle error: saying “the study included 468 participants” beside a result that actually used 397, without telling the reader.
The final row is an example, not a recommendation to impute. The appropriate strategy depends on the analysis objective, missingness process, available auxiliary information and defensible assumptions. The observed-value counts remain facts even when a model uses imputation.
Label the denominator behind every percentage
In the fictional example, 286 of the 441 participants with an observed follow-up outcome completed the task:
286 ÷ 441 × 100 = 64.9%
Using the baseline count instead would give:
286 ÷ 468 × 100 = 61.1%
Those percentages answer different questions. The first is the observed completion proportion among participants with recorded outcomes. The second implicitly places the 27 missing outcomes in the denominator; unless the analysis defines what those missing outcomes mean, it may be misleading.
A transparent sentence is:
Among the 441 baseline respondents with an observed follow-up outcome, 286 completed the task (64.9%); outcomes were unavailable for 27 of 468 baseline respondents.
If a table includes categories, state whether percentages are calculated from all eligible records, all respondents, valid responses within each variable, a subgroup total or a weighted denominator. Column percentages, row percentages and percentages of the full sample are not interchangeable.
Weighted analyses require extra care. A weighted percentage may not equal the raw numerator divided by the raw denominator. Report the unweighted contributing count, identify the weighting method and label the estimate as weighted.
Write the sample trace into the manuscript
Do not make readers reconstruct the sample by comparing footnotes scattered across the paper. Use short, linked statements in the methods, results and tables.
Methods
Name the planned analysis set and the information each analysis requires.
The primary model included baseline respondents with observed follow-up completion, service-use exposure, year level and commuting time. We also fitted a reduced-covariate complete-record model and a prespecified multiple-imputation sensitivity analysis. The imputation model, number of datasets and assumptions are reported below.
The exact wording must match what was actually planned and done. Do not copy this example into a study that used different rules.
Results
Move from broad flow to variable-level missingness and then to the analysis sample.
Of 520 eligible students, 488 enrolled and 468 completed the baseline questionnaire. Follow-up completion was observed for 441 participants. Exposure was available for 455 participants and commuting time for 448. After sequential exclusion of 27 records without the outcome, 11 further records without exposure and 18 further records without commuting time, 412 participants contributed to the primary complete-record model.
Tables and figures
Place the relevant denominator where the result appears.
- Put valid n and missing n beside descriptive variables when they vary.
- Put group counts in table headings only when every row genuinely uses those counts.
- Add a footnote when denominators vary by row, model or outcome.
- Show numerator/denominator with a percentage when the denominator is not obvious.
- Identify whether counts are participants, repeated observations or events.
- Label imputed, weighted and complete-record results rather than presenting them as one undifferentiated sample.
The descriptive-statistics table guide focuses on choosing summaries for variable type and showing valid counts. The denominator ledger extends that work across figures, percentages and models.
Report the missing-data method without prescribing one
The TARMOS framework described by Lee and colleagues separates planning, examination of observed missingness, analysis, reporting and interpretation. Its central reporting lesson is that authors should show the extent and reasons for missing data, the methods and assumptions used, and how conclusions behave under relevant sensitivity analyses.
That transparency does not make one method universally best. Complete-record analysis can discard information and can be biased under some missingness processes. Multiple imputation can also be inappropriate when its model or assumptions are poorly aligned with the data and estimand. Weighting and model-based approaches have their own requirements.
For each primary or sensitivity analysis, report:
- which variables had missing values and how many;
- what information was required for the analysis;
- whether exclusions were automatic, prespecified or decided after examining the data;
- the method used to handle missingness;
- the assumptions that make that method relevant;
- important auxiliary variables, models or weighting steps;
- software and settings needed for reproducibility;
- how the result changed across justified sensitivity analyses; and
- how remaining uncertainty or possible bias affects interpretation.
Do not claim that similar estimates prove missingness was harmless. Agreement can be reassuring within the examined assumptions, but unexamined mechanisms may remain.
Protect confidentiality while remaining transparent
Transparent reporting does not require publishing participant-level records. The American Statistical Association’s Ethical Guidelines join openness about data processing, assumptions, bias and limitations with duties to protect confidentiality.
Use aggregate flow counts and missingness summaries. Follow the study’s disclosure-control rules for small cells, rare combinations and sensitive subgroups. If a count must be suppressed, say that it was suppressed and why; do not silently remove the row or replace a confidential value with an invented number.
A denominator ledger can remain an internal analysis-control document when its finest detail would disclose identities. The manuscript can then report the aggregate information readers need to evaluate the result.
A 12-step denominator workflow
- Name the target population, invited set and eligible set.
- Record enrolment, participation and follow-up counts with non-overlapping reasons where possible.
- Define the unit for every dataset and output.
- List the variables required for each table, percentage and model.
- Record observed and missing counts for important variables.
- Apply exclusions in a recorded order, or mark reasons as overlapping.
- Recalculate the final denominator independently from the analysis dataset.
- Record the missing-data method and assumptions for each analysis.
- Pair every percentage with its numerator and denominator.
- Compare denominators across results and explain every change.
- Transfer the trace into methods, results, tables, figures and footnotes.
- Check confidentiality, arithmetic and agreement between manuscript text and output.
Before release, verify three equalities whenever sequential exclusions are claimed:
- starting count minus exclusions equals the final analysis count;
- category numerators sum to the stated valid denominator when categories are exhaustive; and
- reported percentages reproduce from the labelled numerator and denominator, allowing only stated rounding.
Final denominator check
Ask these questions for every important result:
- What exactly is being counted?
- What was the starting set?
- Which variables or observations were required?
- Why were records unavailable or excluded?
- Do the reasons overlap?
- What method handled missing values?
- What assumptions does that method require?
- What is the final contributing count?
- If a percentage is shown, what are its numerator and denominator?
- Does the table, figure, text and analysis output use the same count?
- Could a reader mistake missing for “no” or “none”?
- Can the result be reported without exposing confidential small cells?
The goal is not to force every analysis to have the same n. It is to make each denominator auditable. When counts change for a defensible reason, say so. When missingness limits the inference, keep that limitation attached to the result it affects.
Statistics and data-analysis support can help you reconcile analysis samples, missing-data decisions and reporting tables using your own research plan and data while you retain responsibility for the analysis and conclusions.
