Why Flow Reporting Matters
A population pharmacokinetic analysis may begin with many participants and concentration records, but only a defined subset may contribute to a particular analysis. A report should make that change visible. Readers need to know not only how many participants were analysed, but also how many concentration observations were available and why either count changed.
These counts describe different units. Excluding a participant can remove all of that participant’s records from an analysis, whereas excluding one concentration record may leave the participant in the analysis with other records. A clear flow account keeps these changes separate and makes the reported analysis population auditable.
The account begins with defined starting sets and stages. A stage might be the set of participants meeting the analysis eligibility criteria, or the set of concentration records retained for a specified analysis. State which analysis the flow describes; counts for different analyses may differ. This tutorial focuses on presenting those counts, not on defining the clinical sampling design or adjudicating data-handling decisions.
Set Up a Reconciled Flow
A simple sequential flow is appropriate when each later stage is a subset of the stage before it. At each transition, give the number removed and the reason. For a clean reconciliation, categories at that transition should be mutually exclusive: each removed unit is counted once. If a unit could meet more than one reason, state a priority rule for assigning one primary reason, or report overlapping reasons explicitly and do not add them as if they were disjoint.
The accounting is a direct consequence of partitioning the units at each stage. Suppose stage \(k-1\) contains \(N_{k-1}\) units. If \(E_{k,1}, \ldots, E_{k,J}\) are disjoint exclusion groups at the transition, and \(N_{k,j}\) is the count in group \(j\), then the retained count is:
Applying this equation at each transition gives a telescoping accounting: the starting count minus all disjoint removals equals the final count. This justifies the central check on a sequential flow. If the flow branches, reconcile each branch separately and state whether branches are mutually exclusive; do not force a single subtraction equation onto overlapping sets.
Use participant counts for participant transitions and record counts for concentration transitions. When an exclusion applies to a participant, show the participant count removed; when it applies to concentration records, show the number of records removed. If the same event affects both units, report both consequences only when both are known, and label them clearly.
Worked Example: Reconciling Participant Counts
Suppose a hypothetical analysis identifies 120 distinct participants for screening. Six do not meet the stated eligibility criteria. Of the 114 remaining, five have no usable dosing history for the planned analysis, and three lack a required covariate value. Assume each reason is assigned once, in this order, so the categories do not overlap. How many participants remain?
After eligibility review, \(120-6=114\) participants remain. After the dosing-history exclusion, \(114-5=109\) remain. After the covariate exclusion, \(109-3=106\) remain. Equivalently:
The flow should therefore show 120 screened, 6 excluded for eligibility, 114 continuing; then 5 excluded for unusable dosing history, 109 continuing; then 3 excluded for missing required covariate data, leaving 106 participants in the analysis population. The arithmetic is valid because the reasons were assigned as disjoint categories. The 106 is a participant count, not a count of concentration observations.
Account Separately for Concentration Records
Participant flow alone cannot show how much concentration data were retained. A participant may contribute multiple records, so the number of records is not interchangeable with the number of people. For each observation stage, identify the record unit and the rules defining that stage. A record may represent one concentration result linked to a participant and a collection time; a report should use its actual data structure and terminology consistently.
Missingness also requires careful wording. A participant with no concentration records is not the same as a participant with one or more concentration records whose value or collection time is missing. Likewise, a record excluded because a required field is absent is not necessarily a participant excluded from the analysis. State which unit is counted and avoid an ambiguous label such as “missing data” without a count and denominator.
Worked Example: Reconciling Concentration-Record Counts
In a hypothetical analysis, 106 participants contribute a total of 440 concentration records received for review. The records are screened in a fixed order: 12 have no concentration value, 11 have no usable collection time, 9 are duplicate entries, and 14 are outside the defined analysis window. Assume each record is assigned to its first applicable reason, so no record is counted twice. How many records remain?
The total removed is \(12+11+9+14=46\) records. The retained count is:
The accounting table should identify the 440 as records received, list each of the four disjoint removal categories with its count, and show 394 retained. These are observation counts, not participant counts. The information given does not establish how many of the 106 participants contributed at least one retained record, so the table should not imply that all 106 did.
Choose a Diagram and Table That Fit the Data
A participant-flow diagram is useful for a short, ordered sequence of participant stages. A concise table is often clearer for detailed observation accounting, particularly when there are several reasons for removing records. They are complementary: the diagram gives the broad progression, while a table can preserve the detail needed to audit counts. Neither should make a decision appear more defensible merely by displaying it.
If analysis sets are not nested—for example, if different model fits use different subsets—show separate branches from the shared starting set. Label each branch with its analysis and its participant or record count. A single line of boxes would incorrectly suggest that every later set is simply the preceding set after exclusions.
Worked Example: Presenting Participant and Observation Flow Together
Suppose a hypothetical report has the participant flow from the first example and the concentration-record flow from the second. One compact participant diagram could read:
Six did not meet eligibility criteria; 114 continued.
Five lacked usable dosing histories and three lacked a required covariate value; 106 participants remained.
This is a participant count. It does not state how many concentration records were retained.
The accompanying observation table can preserve the separate record accounting:
| Observation-record stage or reason | Records |
|---|---|
| Records received for review | 440 |
| No concentration value | 12 |
| No usable collection time | 11 |
| Duplicate entry | 9 |
| Outside the defined analysis window | 14 |
| Records retained | 394 |
The table’s categories sum to 46 removed records, and \(440-46=394\) records remain. The diagram accounts for participants; the table accounts for records. Presenting both avoids the misleading shortcut of reporting “106 participants and 394 observations” without showing how either number was obtained.
Denominators, Checks, and Failure Modes
Percentages can help readers compare the size of a change, but only when the denominator is explicit. For example, in the record example, 12 of 440 records had no concentration value, or \(12/440 \times 100=2.73\%\) of records received. That percentage does not mean 2.73% of participants had missing concentrations. A participant-level percentage would require the number of distinct participants affected and a participant denominator.
Before finalizing a flow, check each transition from its stated starting count. Confirm that all disjoint exclusion categories sum to the reported total removed, and that the retained count equals the preceding count minus removals. Check participant and record totals independently. If a reason count overlaps another reason, either recode the categories using a stated primary-reason rule or label the reasons as overlapping and provide the appropriate distinct total separately.
- Mixing units. A box that says “12 missing” is not interpretable. Say whether this means 12 participants or 12 concentration records.
- Using vague stages. Labels such as “analysed” can conceal the actual transition. Name the stage and the analysis population it represents.
- Double-counting reasons. If a record has both a missing value and a missing time, adding both counts as unique exclusions overstates removals unless a primary-reason rule is used.
- Leaving unexplained gaps. If 100 units enter a stage and 92 continue, account for the other 8 or clearly state that their status is unknown.
- Implying all participants contributed data. A final participant count does not establish the number of participants with at least one retained concentration record.
A prose-only description may be adequate for a very small, simple flow, but readers can find sequential counts difficult to verify in a long paragraph. A diagram makes stages and branches visible; a table makes reason counts easy to compare. For a complex flow, use both, with matching labels and counts. The limitation is that a flow account documents counts and categories; it does not by itself explain the clinical or statistical implications of the data that remain.
Check Your Understanding
Try each question, then compare your reasoning with the solutions below.
- A flow begins with 80 participants. Four are excluded for eligibility and six more lack a required dosing history. How many remain if the categories are sequential and disjoint?
- There are 300 concentration records. Eight lack a value and five lack a collection time. Can the retained count be calculated by subtracting 13? What assumption is needed?
- A report says “10 records were missing” but gives no denominator or missing field. What information should be added?
- Two separate analyses use different subsets, and some participants are included in both. Should their counts be shown as one sequential flow or separate branches?
Solutions. (1) \(80-4-6=70\) participants remain. (2) Yes, if the two reason categories are disjoint and account for all removals; then \(300-(8+5)=287\) records remain. If records can appear in both categories, the unique number removed is not known from these counts alone. (3) State that the count is 10 concentration records, identify which field is missing, give the relevant stage and denominator, and clarify whether those records were excluded or otherwise handled. (4) Use separate branches labelled by analysis. A single sequential flow would imply nested sets and could misrepresent participants included in both analyses.