Rare Cell Types and Unstable Proportion Estimates

Why Rare Proportions Are Especially Uncertain

A cell type can be genuinely rare and still have a highly variable observed proportion. When only a few cells of that type are captured, adding or losing one cell can substantially change the estimate. This matters when describing a cell population and when comparing conditions: a striking relative change may rest on only a handful of observations.

The key is to keep the count, the denominator, and the target of inference visible. A proportion based on 2 cells among 800 captured cells is not as precisely estimated as the same proportion based on 200 among 80,000, even though both observed proportions equal 0.25%. A count threshold can flag limited precision, but it should not turn “rare and uncertain” into “not present.”

Definition: In one sample, let \(n\) be the number of eligible captured cells in the chosen denominator, \(k\) the number assigned to the cell type of interest, and \(\hat{p}=k/n\) its observed proportion. Under a binomial sampling model, each of the \(n\) cells is independently classified as belonging or not belonging to that type, with the same probability \(p\). Then \(K\sim\mathrm{Binomial}(n,p)\), \(E(K)=np\), and \(\hat{p}\) estimates \(p\). For a cell composition sampled multinomially, the count for one cell type has this binomial marginal model; the other cell-type counts are not independent of it.

This model describes sampling uncertainty conditional on the sample and its denominator. It does not, by itself, account for biological differences between donors, unequal capture or classification, or dependence among cells. As emphasized in the earlier material on donors as independent units, many cells from one donor do not create the same evidence about a condition as many independent donors.

Small Counts, Large Relative Uncertainty

Under the binomial model, the variance of \(\hat{p}\) is \(p(1-p)/n\). Replacing the unknown \(p\) by \(\hat{p}\) gives the familiar estimated standard error. For a rare type, \(p\) is small, so the standard error as a fraction of the estimated proportion can be large.

\operatorname{Var}(\hat{p})=\frac{p(1-p)}{n}, \qquad SE(\hat{p})\approx\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \qquad \frac{SE(\hat{p})}{\hat{p}}\approx\frac{1}{\sqrt{k}} \quad\text{when }p\text{ is small and }k=np.

The last approximation follows by replacing \(1-p\) with 1 and \(\hat{p}\) with \(k/n\): \(SE(\hat{p})/\hat{p}\approx\sqrt{(k/n)/n}/(k/n)=1/\sqrt{k}\). It is a useful scale check, not a confidence interval. With about 4 observed cells, the relative standard error is roughly 50%; with about 100, it is roughly 10%.

Worked Example: Expected Counts and the Chance of Seeing No Cells

Suppose a hypothetical sample has \(n=500\) eligible cells and the true proportion of a rare type is \(p=0.002\), or 0.2%. Find the expected count, the standard deviation of the estimated proportion, and the probability of observing no cells of that type.

The expected count is \(np=500(0.002)=1\) cell. The standard deviation of \(\hat{p}\) under the model is

\sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.002(0.998)}{500}} = \sqrt{0.000003992} \approx 0.001998.

That is about 0.200 percentage points, as large as the true proportion itself. The probability of zero observed cells is

P(K=0)=(1-p)^n=(0.998)^{500}\approx 0.3675.

Thus, even if the true proportion were 0.2%, a sample of this size would show no cells of the type about 36.8% of the time under the assumptions. A zero count is compatible with a nonzero population proportion; it is not proof of biological absence.

Intervals for a Single Rare Proportion

A standard symmetric interval \(\hat{p}\pm1.96SE(\hat{p})\) can behave poorly when counts are sparse: it may extend below zero, and its actual coverage can be inaccurate. The Wilson score interval is a useful alternative for a binomial proportion. It stays within 0 and 1 and generally behaves better near the boundaries than the simple Wald interval.

Definition: For a two-sided confidence level based on critical value \(z\), the Wilson score interval for \(p\) is

\frac{ \hat{p}+\frac{z^2}{2n} \ \pm\ z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}+\frac{z^2}{4n^2}} }{ 1+\frac{z^2}{n} }.

For a 95% interval, use \(z=1.96\). The formula comes from inverting the score-test inequality \((\hat{p}-p)^2\le z^2p(1-p)/n\): collecting terms gives a quadratic inequality in \(p\), and its two roots are the displayed lower and upper limits. The interval is an approximate frequentist interval under the binomial model. Its 95% interpretation concerns the long-run coverage of the procedure, not a 95% probability that a fixed \(p\) lies in a particular computed interval.

Worked Example: A Wilson Interval for a Sparse Cell Type

In a hypothetical sample, 4 of \(n=1{,}000\) eligible cells belong to a rare type, so \(\hat{p}=4/1{,}000=0.004\), or 0.4%. Calculate a 95% Wilson interval using \(z=1.96\) and \(z^2=3.8416\).

The denominator is \(1+3.8416/1{,}000=1.003842\). The center of the interval is

\frac{0.004+3.8416/(2{,}000)}{1.003842} = \frac{0.0059208}{1.003842} \approx 0.005898.

The half-width is

\frac{1.96\sqrt{0.004(0.996)/1{,}000+3.8416/(4{,}000{,}000)}}{1.003842} \approx 0.004341.

The resulting interval is approximately \(0.005898\pm0.004341\), or 0.001557 to 0.01024. In percentage terms, the observed 0.4% is compatible with proportions from about 0.156% to 1.024% under this interval procedure and model. The width is substantial relative to the estimate, reflecting the small count.

When Condition Contrasts Are Unstable

A contrast can be unstable for two related reasons. First, the difference between two estimated proportions has sampling uncertainty from both samples. Second, when proportions are very small, a relative comparison such as a ratio can change sharply with a one-cell difference. A fourfold ratio from 1 versus 4 observed cells may sound decisive, but the absolute difference could be only a few tenths of a percentage point and both estimates may be imprecise.

For independent samples with denominators \(n_A,n_B\), a large-sample standard error for the difference \(\hat{p}_B-\hat{p}_A\) is

SE(\hat{p}_B-\hat{p}_A) \approx \sqrt{ \frac{\hat{p}_A(1-\hat{p}_A)}{n_A} + \frac{\hat{p}_B(1-\hat{p}_B)}{n_B} }.

This approximation is most defensible when counts and non-counts in each sample are not sparse and observations are independent under the model. With very small counts, a normal interval for the difference should not be treated as reliable simply because software prints one. Wilson intervals for each proportion are useful for displaying individual uncertainty; overlap or non-overlap of two separate intervals is not, by itself, a formal test or confidence interval for their difference.

Worked Example: A Contrast With Enough Counts for a Normal Approximation

Consider two hypothetical independent cell samples, each with \(n=10{,}000\). Condition A has \(k_A=40\) cells of the type; condition B has \(k_B=100\). Assume for this calculation that cell classifications are independent within each sample, the samples are independent, and the counts are not so sparse that a normal approximation is unreasonable.

The observed proportions are \(\hat{p}_A=40/10{,}000=0.004\) and \(\hat{p}_B=100/10{,}000=0.010\). The estimated difference, B minus A, is 0.006, or 0.6 percentage points. Its estimated standard error is

\sqrt{ \frac{0.004(0.996)}{10{,}000} + \frac{0.010(0.990)}{10{,}000} } = \sqrt{0.0000013884} \approx 0.001178.

A 95% normal-approximation interval is \(0.006\pm1.96(0.001178)\), which is approximately 0.003691 to 0.008309, or 0.369 to 0.831 percentage points. This interval describes a positive difference in the hypothetical cell-sampling model. It does not account for donor-to-donor variation, so it is not by itself a population-level condition comparison when cells come from a limited set of donors.

Now compare that with a sparse contrast: 1 of 1,000 cells in one sample and 4 of 1,000 in another. The estimates are 0.1% and 0.4%, a fourfold ratio but an absolute difference of only 0.3 percentage points. One extra cell changes either estimate by 0.1 percentage points, a large change relative to the observed values. A ratio can therefore look dramatic while being poorly determined. Report the counts and denominators alongside the proportions, and be cautious about both the absolute contrast and any relative summary.

Count Thresholds Should Describe Precision

There is no universal minimum count that makes a rare cell type reliable. A threshold is most informative when tied to a stated purpose: screening for estimates too unstable to interpret, deciding which results need a caution flag, or setting a desired resolution. It should not silently redefine the population of interest by dropping every type below the threshold.

Worked Example: Choosing a Threshold From One-Cell Resolution

A hypothetical analysis team wants each captured cell to change a reported sample proportion by no more than 0.05 percentage points. What denominator is required for that resolution, and what does the rule imply for a sample with 2 cells of a type among 800?

One cell changes a proportion by \(1/n\). Since 0.05 percentage points is 0.0005 as a proportion, require \(1/n\leq0.0005\). Rearranging gives \(n\geq2{,}000\). With \(n=800\), one cell changes the proportion by \(1/800=0.00125\), or 0.125 percentage points. The observed proportion for 2 cells is \(2/800=0.0025\), or 0.25%. Thus, one cell represents half of this observed count, and the sample does not meet the chosen resolution goal. The appropriate conclusion is that the estimate is coarse under this rule—not that the cell type should be erased or is absent.

A practical report can show the count, denominator, proportion, and interval, then flag estimates below a prespecified count or precision threshold. Keep flagged values visible when biological rarity is meaningful. If a threshold is used to exclude observations from a later analysis, explain the exclusion and assess how it changes the target population; do not present a technical stability rule as a biological definition.

Diagnostics, Alternatives, and Limits

Before interpreting an interval or contrast, check what is driving it. A narrow interval based on a very large cell count may still understate uncertainty if cells from the same donor are dependent or if donors differ substantially. Conversely, a wide interval is not evidence that the type is unimportant; it indicates that the observed sample gives limited precision for that proportion.

Check counts and denominators together. The same percentage can be supported by very different numbers of observed cells.
Check the one-cell increment. Compute \(1/n\) in percentage points to see how sensitive the estimate is to one classification.
Check model assumptions. If classification probabilities differ across cells or cells are dependent, the binomial standard error may be too small.
Keep cell-sampling and biological uncertainty distinct. A cell-level interval conditional on one sample does not substitute for replication across independent donors.
Inspect the absolute scale. A large relative ratio for two tiny proportions may correspond to a small and uncertain absolute difference.

The Wald interval is simple but can extend beyond the possible range and perform poorly for sparse proportions. The Wilson interval is generally preferable for a single binomial proportion near zero or one; an exact binomial interval is another option and can be conservative. For a contrast, a difference-based approximation may be reasonable with adequate counts and its assumptions stated, but sparse contrasts require methods designed for that setting. More elaborate count models can address additional sampling structure, but no model removes the need to identify the independent biological units and the intended estimand.

Key takeaway: A small count makes a rare proportion uncertain because one or a few cells carry much of the estimate. Report counts and denominators, use an interval suited to sparse proportions, and treat thresholds as transparent precision flags—not evidence that a genuinely rare population does not exist.

Check Your Understanding

Answer each question, then compare with the solutions below.

A type is observed 5 times among 2,000 cells. What is its observed proportion, and what is the approximate relative standard error using the rare-type approximation?
Why does a zero observed count not establish that the true proportion is zero?
Two samples have estimates of 0.1% and 0.4%. Give one reason a fourfold ratio may be unstable.
A precision rule requires a one-cell increment no greater than 0.1 percentage points. What minimum denominator meets the rule?

Solutions. (1) \(\hat{p}=5/2{,}000=0.0025\), or 0.25%. The approximate relative standard error is \(1/\sqrt{5}\approx0.447\), or 44.7%. (2) Under a binomial model with \(p>0\), the probability of zero cells is \((1-p)^n\), which can be appreciable when \(p\) is small or \(n\) is limited. (3) With rare counts, one additional cell can change an estimate substantially relative to its size, so the ratio may be driven by only a few observations; counts and denominators are needed to assess precision. (4) Since 0.1 percentage points is 0.001 as a proportion, \(1/n\leq0.001\) requires \(n\geq1{,}000\).