Attrition, Missing Samples, and Replacement Rules

Why Attrition Is a Design Problem

A planned single-cell study can lose donors before collection, specimens during transport, or libraries during processing and sequencing. These losses are not interchangeable. A donor who withdraws before any sample is collected removes an independent biological replicate. A failed library removes a measurement from a specimen, while a low cell yield may leave a donor represented but with less information. Counting cells cannot generally replace a missing donor: cells from one donor share biological context and do not provide independent donor-level replication.

Attrition matters for both the number of usable replicates and the meaning of the comparison. If a condition has fewer observed donors, precision can decline. If the chance of a usable sample is related to the condition or to the outcome being studied, the observed donors may also differ systematically from the intended donor population. A defensible plan therefore sets recruitment, replacement, and exclusion rules before outcomes are examined.

Definition: Let \(A_i\) denote the assigned condition for donor \(i\), \(Y_i\) the donor-level outcome of interest, and \(R_i=1\) if that outcome is observed and eligible for the planned analysis, with \(R_i=0\) otherwise. Attrition is loss of an intended donor or measurement before the analysis endpoint. A replacement is a new eligible donor recruited under a rule specified in advance to restore a planned sample count; it does not restore the missing outcome of the original donor.

For a multi-condition study, the outcome may be a donor-level pseudobulk expression summary, a cell-type proportion, or another prespecified donor-level quantity. The exact outcome is not needed to write an attrition rule, but the rule should say what makes a donor-condition specimen usable and at what stage that decision is made. A cell-yield threshold, for example, should be applied consistently and without reference to the observed condition effect.

State the Missingness Assumptions

Missingness is described relative to the outcome and other recorded information, not merely by whether a file is present. These distinctions help separate a loss that mainly reduces precision from one that may change the comparison.

Definition: Missingness is completely at random (MCAR) when observation status is independent of the outcome and all relevant study variables. It is at random (MAR) when, conditional on recorded variables such as condition, site, and baseline donor characteristics, observation status no longer depends on the unobserved outcome. It is not at random (MNAR) when observation still depends on the unobserved outcome after conditioning on recorded information.

These are assumptions about the mechanism, not labels that can usually be established from the observed data alone. For example, if a processing protocol causes more failures in one condition but, within each condition, failure is unrelated to the donor’s outcome, missingness is not MCAR across the whole study. It may be MAR conditional on condition. If high values of a donor-level outcome make a specimen more likely to fail, then missingness may be MNAR unless the relevant cause is recorded and adequately accounted for.

Condition-dependent missingness does not automatically bias a condition-specific mean. If donors are missing at random within each condition, the observed mean in each condition can still estimate its intended mean, though fewer observed donors increase uncertainty. Bias arises when the observed donors are systematically different in outcome from the intended donors, or when the missingness process otherwise changes who is represented in the comparison.

Recruitment and Replacement Rules

Suppose each recruited donor independently has probability \(r\) of producing a usable outcome, and the planned analysis requires at least \(m\) usable donors in a group. If \(N\) donors are recruited, the usable count \(U\) follows a binomial distribution with parameters \(N\) and \(r\). Its expected value is \(Nr\), but planning only for \(E(U)\geq m\) does not ensure a high probability of reaching \(m\). When the target is important, choose \(N\) using \(P(U\geq m)\), with a stated retention assumption and desired assurance probability.

U\sim\operatorname{Binomial}(N,r),\qquad P(U\geq m)=\sum_{k=m}^{N}\binom{N}{k}r^k(1-r)^{N-k}

The binomial calculation assumes independent donors with a common usable-sample probability. It is an approximation if risks differ substantially by site, condition, or donor characteristics. In that situation, planning should use stratum-specific retention estimates or a design simulation rather than treating all donors as exchangeable.

Worked Example: Recruiting for a Usable-Donor Target

A hypothetical study needs at least 12 usable donors in each condition. Based on pilot feasibility information, the team assumes each recruited donor has an 85% chance of yielding an eligible specimen. The team wants at least a 90% probability of obtaining 12 usable donors in a condition.

With \(N=15\), the chance of reaching 12 is the chance of at most 3 failures. Using failure probability \(0.15\),

P(U\geq12)=\sum_{k=0}^{3}\binom{15}{k}(0.15)^k(0.85)^{15-k} \approx 0.8227.

Thus, recruiting 15 gives only about an 82.3% chance of meeting the target, even though the expected usable count is \(15(0.85)=12.75\). With \(N=16\), the chance of at most 4 failures is

P(U\geq12)=\sum_{k=0}^{4}\binom{16}{k}(0.15)^k(0.85)^{16-k} \approx 0.9209.

Under the stated assumptions, 16 recruits per condition meet the 90% assurance goal. This is a planning calculation, not a guarantee: the actual retention probability may differ, and the calculation does not address bias if failure is related to outcomes.

A replacement rule should specify eligibility, timing, and allocation. It should also distinguish ordinary attrition from exclusion. For example, an exclusion can be triggered by a predeclared sample-identity mismatch or a minimum library-quality criterion; it should not be triggered because a donor’s expression value weakens a preferred result. Replacement is most straightforward when a failure is identified before condition exposure or before the relevant specimen is processed, and when the reserve donor can enter the same allocation process as the original recruits.

Define the analysis unit.
State whether the target count is donors, donor-condition specimens, or another independent unit. Do not treat extra cells from one donor as additional donors.

Set eligibility and failure criteria.
Specify objective thresholds, who applies them, and when, preferably without access to condition-effect estimates.

Predefine replacement.
Record whether replacements come from a reserve list, how they are allocated, and whether a replacement must provide every specimen needed for a paired comparison.

Log every disposition.
For each donor and specimen, retain condition, stage, reason for loss, quality metrics, and whether a replacement was recruited.

When Missingness Can Distort a Comparison

Let \(Y_i\) be the outcome that would be available for donor \(i\) under the relevant condition. The intended condition mean is \(E(Y_i\mid A_i=a)\), whereas the observed mean is \(E(Y_i\mid A_i=a,R_i=1)\). Their difference is the selection shift in condition \(a\). If those means differ, the observed condition mean does not represent the intended condition mean. Consequently, a contrast of observed means can combine the condition effect with different selection shifts in the two conditions.

\begin{aligned} \Delta_{\mathrm{target}}&=E(Y\mid A=1)-E(Y\mid A=0),\\ \Delta_{\mathrm{observed}}&=E(Y\mid A=1,R=1)-E(Y\mid A=0,R=1). \end{aligned}

This distinction is especially important when a condition changes cell viability or specimen quality and those changes are related to the biological outcome. A difference in failure rates is a warning to investigate, but it does not by itself prove bias. Conversely, similar failure rates do not prove that the missing outcomes are comparable: different donors can be lost in each condition while the overall percentages happen to match.

Worked Example: Outcome-Related Dropout Can Erase a Contrast

Consider a hypothetical four-donor group in each of two conditions, with a donor-level expression summary in arbitrary units. The intended control outcomes are \(0,2,4,6\), with mean \((0+2+4+6)/4=3\). The intended treated outcomes are \(2,4,6,8\), with mean \((2+4+6+8)/4=5\). The intended contrast, treated minus control, is \(5-3=2\) units.

Suppose all four control specimens are observed, but in the treated group the two largest outcomes fail quality control. The observed treated outcomes are \(2\) and \(4\), so their mean is \((2+4)/2=3\). The observed contrast is now \(3-3=0\) units, rather than the intended 2-unit contrast. Here half the treated specimens are missing, and their loss is directly related to the outcome; the observed comparison is biased for the stated target.

By contrast, if two treated specimens had been lost by a mechanism independent of outcome within the treated group, the observed treated mean would not be forced downward in this way. It would remain an estimate with greater sampling uncertainty. The example shows why recording only a group-level failure percentage is insufficient: the reason and relationship of failure to outcome matter.

Paired Conditions and Alternative Responses

In a paired design, each donor is intended to contribute measurements under multiple conditions. The primary paired contrast uses donors with the required measurements in both conditions. If one condition is missing for a donor, that donor cannot supply a complete within-donor contrast. Replacing only the missing condition with a specimen from a different donor does not recreate the original pair; it mixes a between-donor difference into a within-donor comparison.

Worked Example: Replacing a Failed Specimen in a Paired Design

A hypothetical study plans 10 donors, each measured under conditions A and B. After processing, donor D4 has a usable A specimen but a failed B library. The primary paired analysis therefore has nine complete donor pairs, not 10. If a reserve donor is recruited and both A and B are collected and processed for that donor, the study can have 10 complete pairs, provided the replacement was eligible and allocated under the predeclared rule.

If only a B specimen from a new donor is added and matched to D4’s A specimen, it is not a valid within-donor pair: the measurements come from different people. A prespecified secondary analysis might use available unpaired observations under an appropriate model, but it answers a different analysis question and does not make the missing paired outcome observed.

The main responses to missing specimens serve different purposes. Recruiting reserve donors before data collection can protect the planned usable count, but it cannot correct selective loss. Complete-case analysis is transparent and may be suitable when the required missingness assumptions are defensible; it can be biased under outcome-related selection. Weighting or model-based approaches can use recorded predictors of missingness, but their validity depends on assumptions such as adequate measurement of those predictors and a correctly specified model. None can recover information with certainty when dropout depends on unobserved outcomes.

Approach	Useful when	Main limitation
Reserve recruitment	Expected technical or logistical loss threatens the target count.	Does not remove selection bias from outcome-related attrition.
Complete-case analysis	Missingness assumptions make observed cases representative for the target contrast.	Can lose precision and can be biased if observed cases differ systematically.
Weighting or model-based adjustment	Important predictors of missingness are recorded and assumptions are credible.	Results depend on the model and cannot generally identify effects of unmeasured causes of dropout.

Diagnostics, Limitations, and a Common Mistake

Before interpreting a comparison, summarize intended, recruited, usable, and analyzed donors by condition and relevant design strata. Tabulate failure stage and reasons, and compare recorded baseline characteristics and quality metrics between retained and lost donors. Examine whether losses cluster by site, operator, processing day, or condition. These checks can reveal allocation or workflow problems, but they cannot establish MCAR or rule out dependence on an unobserved outcome.

A common mistake is to say, “We recruited extra donors, so attrition cannot bias the result.” Extra recruitment can restore a count, but it does not make a selective replacement representative or recover the outcome of a selectively lost donor. The correction is to separate two questions: whether enough independent donors remain, and whether the observed donors represent the target population for the comparison. Plan for both, report both, and state the assumptions behind any adjustment.

Replacement also has limits. Donors recruited later may differ by calendar time or site, and processing conditions may change; if so, replacement status can become associated with technical factors. Preserve the allocation structure where possible, record recruitment timing, and avoid replacing a donor based on observed outcome values. In studies where specimens are tied to a particular donor or paired condition, replacements must respect that structure rather than simply filling a missing cell in a table.

Key takeaway: Attrition planning protects the number and structure of independent replicates; missingness assessment protects the interpretation of the comparison. A sound protocol states both the replacement rule and the assumptions under which the observed data represent the intended donor population.

Check Your Understanding

Answer each question, then compare with the solutions.

A study needs 20 usable donors and expects 10% attrition. What is the smallest recruitment number whose expected usable count is at least 20, and why is this not a 90% assurance calculation?
If specimen failure is more common in condition B but unrelated to outcomes within each condition, what is the main immediate concern for the comparison?
In a paired study, one donor’s B specimen fails. Can a B specimen from a new donor complete that donor’s pair? Explain.
Why can matching failure percentages across conditions fail to rule out bias?

Solutions. (1) Solve \(0.90N\geq20\), giving \(N\geq22.22\), so 23 donors are needed for an expected count of at least 20. This uses the expectation only; it does not calculate the probability of reaching 20, which requires the binomial tail probability. (2) The observed group sizes and hence precision may differ; if failure is genuinely unrelated to outcomes within condition, the condition means need not be biased solely because failure rates differ. (3) No. The new donor’s B measurement is not paired with the original donor’s A measurement; a replacement pair requires measurements from the same replacement donor. (4) Equal rates do not imply that the same types of donors are lost in each condition. Outcome-related or otherwise selective loss can differ between groups even when their percentages missing are equal.