Differential-Abundance Testing With propeller

Why Propeller Starts With Samples

A differential-abundance analysis asks whether a cell type’s relative representation differs across conditions. The rows of evidence must represent independent biological replicates: thousands of cells from one donor do not provide thousands of independent replicates. Propeller addresses this by first calculating a cell-type proportion for each sample and then modeling those sample-level values. Its central statistical feature is that it uses a linear-model framework on transformed proportions and moderates variance estimates across cell types.

This approach makes the sample—not the individual cell—the unit that supplies replication. It can be useful when the study has several biological samples per condition and the analyst wants component-wise tests for cell types. The resulting tests concern differences in transformed sample-level proportions; they are not tests of absolute cell counts, and a component-wise result does not by itself describe the full joint change in composition.

The denominator must be defined consistently. For example, if the question concerns proportions among a specified set of immune cell types, \(N_i\) should count those cells, not an unrelated set of cells in some samples. The earlier material on aggregating cells into donor-level count tables establishes why counts should be summarized by biological sample before testing. The earlier material on multiplicity control applies because many cell types are usually tested.

Transformation, Model, and Assumptions

A common propeller transformation is \(y_{ik}=\arcsin(\sqrt{p_{ik}})\), where the arcsine is in radians. It maps proportions between 0 and 1 to values between 0 and \(\pi/2\). A second supported choice is the logit transformation, \(\log(p_{ik}/(1-p_{ik}))\), which requires special handling of proportions at 0 or 1. The transformation changes the scale on which differences are modeled; it does not change the fact that the observations are samples.

For each cell type \(k\), the transformed values are modeled with a design matrix \(X\). In a two-condition study, a simple model is \(y_{ik}=\beta_{0k}+\beta_{1k}z_i+\epsilon_{ik}\), where \(z_i=0\) for the reference condition and \(z_i=1\) for the comparison condition. The coefficient \(\beta_{1k}\) is the comparison-minus-reference difference in mean transformed proportion under this model. More complex designs can include covariates or blocking terms, provided the design can distinguish the condition effect from those terms.

The usual linear-model assumptions concern the residuals on the transformed scale: samples are independent after accounting for the design; the mean structure is correctly specified; residual variation is reasonably represented by the model; and residuals are sufficiently well behaved for the small-sample \(t\)-based inference to be useful. Exact normality is not guaranteed by transforming proportions. Empirical-Bayes moderation can stabilize noisy variance estimates, especially when the number of samples is modest, but it cannot repair dependence, confounding, a badly specified design, or a lack of biological replication.

In a two-group comparison with equal variance, the unmoderated residual variance for a cell type is the residual sum of squares divided by its residual degrees of freedom \(df\). The estimated variance of a coefficient is that residual variance multiplied by the appropriate design-matrix factor. Propeller’s empirical-Bayes step combines each cell type’s residual variance with information about variance across the tested cell types. A standard moderated variance has the form below. The prior variance \(s_0^2\) and prior degrees of freedom \(d_0\) are estimated across cell types by the procedure; they are not extra observations from the study.

Here \(s_k^2\) is the cell-type-specific residual variance estimate and \(v_k\) is the design factor for the tested coefficient or contrast. The moderated statistic is compared with a \(t\) distribution using moderated degrees of freedom, commonly \(d_0+df_k\) in this simple description. In an actual analysis, software estimates the moderation parameters from the collection of cell types and may apply implementation-specific refinements. Report the model, transformation, contrast, and multiplicity adjustment along with the resulting tests.

Worked Examples

Diagnostics and Failure Modes

A propeller result is only as credible as its sample-level data and design. Before interpreting a small \(p\)-value, inspect the following features:

• Biological replication: Count independent samples in each condition. A large number of cells does not compensate for one or two donors per condition, and variance moderation does not create replication.
• Design rank and confounding: Confirm that condition is estimable after including covariates or blocking terms. If all reference samples are from one batch and all comparison samples from another, the condition effect cannot be separated from batch using the design alone.
• Sample-level proportions: Plot each sample’s proportion, not just a group mean. Look for outliers, clusters by batch, heavy skew, or a result driven by one sample. A cell type absent from some samples can produce boundary values; arcsine-square-root transformation is defined at zero, but repeated zeros may still leave little information for estimating a condition effect.
• Residual variance and fit: Examine residuals and sample-level scatter on the modeled scale. Large residuals, unequal spread, or a mean pattern not captured by the design can make a small-sample \(t\)-test unreliable. Moderation stabilizes variance estimates; it does not eliminate the need to assess the fit.
• Group imbalance: Unequal sample sizes are allowed in a linear model, but a small group can make the estimate sensitive to individual samples and reduce precision. Report group sizes and inspect whether missingness or low cell recovery is concentrated in one condition.
• Multiplicity and dependence: Apply a suitable multiplicity adjustment across the family of cell types being tested. Because proportions share a denominator and sum to one, component-wise tests are related; a result for one type should not be presented as a complete account of composition change.

What the Test Does—and Does Not—Say

Propeller offers a practical way to compare sample-level cell-type proportions while borrowing information across cell types to stabilize residual variances. Compared with a separate unmoderated \(t\)-test for every cell type, moderation can reduce the instability of variance estimates when each test has few samples. The linear-model framework also accommodates designs that include suitable covariates or contrasts. The arcsine-square-root transformation accepts zero proportions without a continuity correction, though zero-heavy data may still be uninformative.

Other approaches answer related but not identical modeling questions. Binomial or beta-binomial regression models a cell type against the remaining cells and can represent sampling variation in counts; joint multinomial, Dirichlet-multinomial, logistic-normal, or Dirichlet models address the composition as a whole in different ways. Bayesian methods such as scCODA use a joint compositional framework and different assumptions. These are not interchangeable tests: the choice depends on the scientific target, design, and assumptions. Propeller’s component-wise transformed-scale model should not be described as a joint model of all cell-type proportions.

The estimated difference is conditional on the analyzed cell population, annotation, and sample-processing pipeline. It describes relative representation among the cells included in the denominator, not absolute numbers of cells in tissue. In addition, a transformed-scale test does not establish why a proportion changed. Interpretation should distinguish evidence for a difference in the modeled quantity from a biological mechanism or causal claim.