Joint Outlier Detection With Cell-Level QC Metrics

Why Combine Cell-Level QC Metrics?

A barcode with unusually few counts may also have unusually few detected genes. Those two observations are related: both partly reflect the amount of captured RNA. Counting them as two independent pieces of evidence can therefore exaggerate how unusual the barcode is. Mitochondrial fraction adds another perspective, but it too may vary with cell state, cell type, and library size. Joint outlier detection aims to summarize these metrics together while accounting for their dependence.

The goal is a screening flag, not an automatic verdict that a barcode is invalid. A flag identifies a cell whose QC profile is unusual relative to an appropriate reference. It does not establish why the profile is unusual, nor does it by itself justify removing the cell. The earlier material on per-cell counts, detected genes, and mitochondrial fraction defines these metrics; the introductory material on adaptive mitochondrial thresholds explains why a single mitochondrial cutoff can be inappropriate across different cell groups.

Robust Standardized Deviations

Counts and detected-gene numbers are often right-skewed, so a small number of high values can pull the mean and standard deviation away from the bulk of the cells. A median and median absolute deviation (MAD) are less affected by such extremes. Before calculating deviations, it is often useful to transform the count metrics. For example, define \(u_i=\log_{10}(1+\mathrm{UMI}_i)\) and \(v_i=\log_{10}(1+\mathrm{genes}_i)\) for barcode \(i\). The logarithm compresses large values while preserving their ordering.

For mitochondrial fraction, use a consistent definition and scale. If fractions are strictly between zero and one and the range warrants it, a logit transformation can make the scale less bounded: \(w_i=\log\{f_i/(1-f_i)\}\), where \(f_i\) is the fraction. Exact zeros or ones require a prespecified adjustment, or a different transformation. One may also robustly standardize the fraction on its original scale when that scale is suitable. The choice of transformation should be made before inspecting which barcodes will be flagged and applied consistently to the reference and candidate cells.

The reference set matters. If biologically distinct cell groups have different typical QC profiles, a pooled median and MAD can make ordinary cells in one group appear unusual. Reference cells should therefore be chosen to be comparable in sample, assay, and relevant biological composition. A reference dominated by poor-quality barcodes can also shift its center and scale, masking the very outliers of interest. If a MAD is zero, the formula is undefined; do not divide by zero or silently replace it with an arbitrary number. Investigate whether the metric is constant or highly discrete, and use a suitable alternative scale or analysis.

Combining Metrics Without Double-Counting

A simple rule might flag a barcode if any one of several robust deviations exceeds a cutoff. Such a rule is easy to interpret, but it does not describe how unusual the joint profile is, and the chance of crossing at least one cutoff increases as more metrics are checked. A multivariate distance provides a different summary: it measures distance from the typical joint profile while accounting for correlations.

Let \(z_i\) be the vector of robust standardized deviations for the selected metrics, such as transformed UMI count, transformed detected-gene count, and mitochondrial fraction. Let \(R\) be a robust estimate of their correlation matrix in the reference cells. Then define the squared robust Mahalanobis distance as follows.

One robust approach estimates center and covariance from a subset of the reference cells whose covariance determinant is small, as in the minimum covariance determinant (MCD) method, with suitable correction for scale. The selected subset must be large enough to estimate a nonsingular covariance matrix. Regardless of estimator, the reference must represent the cells to which it will be applied, and the resulting correlation matrix must be invertible. If two metrics are nearly redundant, the matrix may be unstable or singular; reduce redundancy, use a carefully regularized method, or analyze the metrics separately rather than trusting an unstable inverse.

Interpreting a Joint Flag

A distance gives an overall measure of unusualness but does not say which metric drove it or whether the cell is unusually high or low on that metric. Always examine the component deviations alongside the distance. For QC, direction matters: low counts and genes may indicate low capture, whereas unusually high mitochondrial fraction may indicate a different concern. A joint distance can also be large because of a combination of moderate deviations, even when no single component crosses a univariate cutoff.

Assumptions, Diagnostics, and Alternatives

Robust standardization and robust distance are useful only when their reference summaries are meaningful. The main conditions to check are:

• Comparable reference cells: The reference should reflect the assay and biological population being evaluated. Differences in cell composition or sample handling can create apparent outliers.
• Usable metric scales: Transformations should address severe skew or boundary effects without erasing meaningful differences. The same transformation must be used for reference and candidate cells.
• Reliable robust summaries: The reference should contain enough cells and variation to estimate medians, scales, and correlations. Zero MADs, unstable correlations, and nearly singular matrices are warning signs.
• Understandable direction: Look at the individual deviations as well as the joint distance. A distance alone hides whether the unusual profile reflects low counts, high mitochondrial fraction, or another combination.

A chi-square calibration for \(D^2\) is available under a multivariate normal model with known center and covariance: if a \(p\)-dimensional vector follows that model, its squared Mahalanobis distance from the true center has a chi-square distribution with \(p\) degrees of freedom. In practice, QC distances use estimated, often robust, summaries and transformed metrics that need not be multivariate normal. Therefore, do not automatically interpret a chi-square tail area as a calibrated p-value. If a formal cutoff is needed, its calibration should reflect the estimation procedure and reference data; exploratory QC often uses distance plots and sensitivity checks instead.

A fixed univariate threshold is transparent and may be appropriate when a metric has a clear, justified limit, but it can miss joint patterns and can mishandle correlated metrics. An “any metric exceeds its cutoff” rule is straightforward but its overall flag rate depends on the number and dependence of metrics. A robust multivariate distance accounts for correlation, but is less directly interpretable and can be unstable with too many redundant metrics or an unsuitable reference. None of these methods replaces examining QC distributions and the scientific consequences of flagging cells.

Common Pitfall: Treating Correlated Metrics as Independent Evidence

A tempting argument is: “The barcode is low in counts and low in detected genes, so two separate metrics confirm that it is poor quality.” This overstates the evidence when the metrics are strongly correlated; low library size naturally tends to produce fewer detected genes. The correction is to examine their joint distribution, use a correlation-aware summary when appropriate, and report the direction of each component. Even a correlation-adjusted flag remains a screening result, not proof that the barcode is invalid.