Balance Tables and Allocation Diagnostics

Why Allocation Summaries Matter

A study can have the intended number of donors overall and still provide weak support for a planned comparison. Donors may be concentrated in different batches by condition, some donors may lack a required condition, or the sample totals may conceal a small number of highly productive donors. Allocation summaries make these patterns visible before analysis. They complement design-matrix rank and alias checks: a table or plot describes how observations are distributed, while a rank check addresses whether specified model terms are algebraically distinguishable.

For single-cell studies, the unit counted must be explicit. For a donor-level comparison, count independent donors or donor-condition samples, not cells, as though every cell were an independent biological replicate. Cell yields can be useful technical summaries, but they do not replace the donor allocation summary.

Build Tables With the Right Unit and Denominator

Start from one row per experimental unit or sample, with unambiguous identifiers for donor, condition, and batch. Decide whether the table will count donors, donor-condition samples, or another unit; those counts answer different questions. If each donor contributes one sample to one condition, donor counts and sample counts coincide. In a repeated or paired design, one donor can contribute multiple samples, so they do not.

A condition-by-batch table is a useful first summary. Report the counts, the row or column totals, and percentages with their denominators stated. For a table with condition \(c\) and batch \(b\), let \(n_{cb}\) be the number of units in that combination. The condition composition within batch \(b\) is \(n_{cb}/n_{\cdot b}\), whereas the batch composition within condition \(c\) is \(n_{cb}/n_{c\cdot}\). These percentages answer different questions and should not be substituted for one another.

The conditional-proportion notation here describes the observed allocation table; it is not a model-based probability claim. Reporting both directions helps reveal whether a batch contains a mix of conditions and whether each condition is spread across batches. Also show overall condition and batch totals, since a table of percentages alone can make a group of two units look as substantial as a group of twenty.

Check Coverage at the Donor Level

For a within-donor comparison, the relevant summary is not only the number of samples per condition. It is also the number of donors with the required conditions observed. A donor-by-condition coverage table makes this explicit: rows represent donors, columns represent conditions, and an entry indicates whether a usable sample is present. A compact summary can then count donors with all required conditions, only a subset, or none.

The definition of “complete” depends on the planned comparison. For a two-condition paired comparison, a complete donor has an observation in both conditions. In a multi-condition design, a comparison may require only a particular pair of conditions, not every condition in the study. State the coverage rule before counting, and distinguish planned samples from samples actually available for analysis.

Plots That Reveal Allocation Structure

A table is usually the clearest source of exact counts; a plot can make a pattern easier to notice. Choose the display to match the question, and retain counts or denominators so visual area is not mistaken for sample size.

• Grouped bar plot: show counts for each condition within each batch. This makes empty or small combinations visible; use a count axis when group size matters.
• Stacked bar plot: show condition composition within each batch. A 100% stacked version emphasizes composition but hides differences in total batch size, so pair it with batch counts.
• Donor-by-condition tile plot: display presence or absence of each required sample for each donor. This is useful for paired and repeated-condition coverage, especially when donor order is grouped by a relevant design feature.
• Dot plot of donor-level yields: show cells recovered per donor, grouped by condition or batch. Keep each donor visible rather than showing only pooled cell totals.

Use consistent condition and batch labels across tables and plots. If the design includes several categorical factors, begin with the factors needed to assess the planned comparison rather than producing a large collection of unreadable cross-tabulations. A sparse table can be more informative than a complicated plot: zeros and small counts should remain visible rather than being smoothed away.

What Imbalance Does—and Does Not—Show

An imbalance is a warning about the information available for a comparison, not by itself proof that an estimate is biased or that analysis is impossible. An absent condition-batch combination may prevent a particular effect from being separated from batch, while partial representation may still leave some comparisons estimable. Whether this occurs depends on the full design and the target comparison, so use allocation summaries alongside the design-matrix checks already introduced.

Marginal balance can be misleading. Equal numbers of donors in two conditions say nothing about whether the conditions are distributed similarly across batches, sites, or other recorded design factors. Conversely, unequal condition totals do not automatically make a comparison unusable; the important questions include which units contribute to the target contrast and whether relevant design factors overlap across conditions. Avoid turning a descriptive table into a claim about causal effects.

For continuous donor characteristics, a balance table can report group-specific means, standard deviations, or medians and ranges, with units. Such summaries can reveal obvious differences in the recruited groups, but no single numerical balance threshold establishes that groups are exchangeable. Categorical characteristics should be shown with counts and denominators. These descriptions help assess coverage; they do not determine which variables belong in a model.

A Short Justification for Checking Both Margins

The need to inspect both batch composition and condition composition follows directly from the table denominators. In the first worked example, each batch contains 10 donors, so \(n_{\cdot b}=10\). The proportion control within Batch 1 is therefore \(8/10=0.80\). But among the 10 control donors, \(n_{c\cdot}=10\), so the proportion assigned to Batch 1 is also \(8/10=0.80\). In other tables, those denominators can differ substantially. For example, a large batch can contain most donors while still having a balanced condition mix. Consequently, a percentage is interpretable only with its denominator named, and reporting both conditional directions prevents a one-sided summary from hiding the allocation pattern.

Common Failure Modes and Alternatives

The first alternative to a table is a plot, not an automatic significance test. A test of association between condition and batch answers a different inferential question and can be unhelpful in a small or deliberately allocated study: a non-significant result does not demonstrate balance, and a significant result does not quantify whether the planned comparison is recoverable. For allocation diagnosis, exact counts, denominators, and plots are usually more directly interpretable.

• Counting cells as donors. A condition with more cells may simply have higher cell recovery. Report donor counts for biological replication and cell yields separately.
• Reporting percentages without denominators. “80% control” is incomplete: state whether that is 80% of a batch or 80% of a condition.
• Checking only overall totals. Equal condition totals can coexist with strong batch imbalance, as the first example shows. Inspect the cross-tabulation.
• Summarizing paired samples only by condition totals. Equal sample totals do not reveal how many donors have complete pairs. Report donor-level coverage.
• Dropping zero-count combinations from the display. Their absence is often the most important diagnostic. Show the full set of planned combinations.
• Using a plot without exact counts. Bars or tiles can exaggerate differences when totals are small. Include counts or a table alongside the visualization.

A balance table is descriptive and depends on accurate metadata. Check for duplicate sample identifiers, inconsistent condition labels, missing batch assignments, and donors assigned to impossible combinations. Confirm that the table totals reconcile with the analysis data after exclusions. Otherwise, a polished plot can faithfully display a data-processing error.