Explained Variance and Scree-Plot Decisions

Why Summarize PCA Variance?

PCA replaces a collection of measured features with ordered directions that capture variation in the data. The eigenvalues quantify how much variance is associated with those directions. Explained-variance ratios and scree plots turn that ordered list into summaries that can help describe how concentrated or dispersed the variation is.

These summaries are useful, but their scope is limited. They report variance in the input matrix; they do not identify what caused that variance or whether it corresponds to a biological process of interest. In single-cell RNA sequencing, a large source of variation might reflect biology, technical differences, or both. A variance threshold is therefore a descriptive rule, not proof that retained axes are biologically meaningful.

Definitions and Conditions

Let \(X\) be an \(n\times p\) matrix of centered values, with cells in rows and features in columns, where \(n>1\). Centering means each feature column has sample mean zero. Define the sample covariance matrix as \(S=X^\mathsf{T}X/(n-1)\), and let its eigenvalues be ordered from largest to smallest: \(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_p\geq 0\). Eigenvalues are nonnegative because \(S\) is positive semidefinite. The common factor \(1/(n-1)\) gives the eigenvalues variance units.

The total \(T\) is the sum of the sample variances of the centered input features. Ratios are dimensionless: \(r_j\) describes the fraction of total variance associated with component \(j\), and \(R_k\) describes the fraction associated with the first \(k\) components. When \(T>0\), every ratio is between zero and one, the ratios sum to one, and the cumulative values cannot decrease.

The values depend on the matrix submitted to PCA. The features included, their units, and any preprocessing affect the variance being summarized. Thus, “explains 80%” means 80% of the variance in that particular input matrix, not 80% of all biological variation in the cells.

Why the Ratios Sum to One

A short linear-algebra argument establishes the interpretation. Let \(v_j\) be a unit eigenvector of \(S\) for \(\lambda_j\). The projected cell values in component \(j\) are \(Xv_j\). Their sample variance is

because \(v_j^\mathsf{T}v_j=1\). The total variance of the original features is \(\operatorname{tr}(S)\), the sum of the diagonal entries of \(S\). The trace of a matrix equals the sum of its eigenvalues, so \(\operatorname{tr}(S)=\sum_{\ell=1}^{p}\lambda_\ell=T\). Consequently, the fraction for component \(j\) is its variance divided by the total, \(\lambda_j/T\). Summing over \(j\) gives \(\sum_{j=1}^{p}r_j=T/T=1\), provided \(T>0\).

Centering and \(n>1\) make the stated sample covariance well-defined. If the centered matrix is all zeros, then \(S\) is also all zeros and \(T=0\). In that case no variance is present to apportion, so it is incorrect to report ratios of zero or to claim that they sum to one. More generally, centered data can have zero eigenvalues when its rank is less than \(p\); these components have zero ratio as long as the total variance is positive.

Reading a Scree Plot

A scree plot displays component number on the horizontal axis and either the eigenvalue or its explained-variance ratio on the vertical axis. Since every ratio is the corresponding eigenvalue multiplied by the same positive constant \(1/T\), these two versions have the same ordering and overall shape. The ratio version makes the vertical scale a fraction of total variance; the eigenvalue version retains variance units.

A common visual heuristic is to look for an “elbow”: a point after which successive eigenvalues decline more gradually. The components before a pronounced elbow account for relatively large amounts of variance compared with those after it. This is a descriptive cue, not a mathematically unique cutoff. A smooth decline may have no clear elbow, and different viewers may choose different locations.

A cumulative-variance rule works differently. It selects the smallest \(k\) for which \(R_k\) reaches a prespecified threshold, such as 0.90. The threshold gives a reproducible arithmetic criterion once the cutoff is specified, but the chosen percentage is a convention rather than a universal statistical law. It can require many components when variance is diffuse, or very few when a few components dominate.

Variance Is Not Biological Meaning

PCA ranks directions by variance, not by biological importance. A strong technical gradient can dominate the leading eigenvalue even if the scientific goal is to distinguish a subtler cell state. Conversely, a rare but important population may contribute little to total variance because it involves relatively few cells. A cutoff such as 90% can therefore retain many weak or technical directions while still failing to ensure that a scientifically relevant pattern is represented.

For example, suppose a hypothetical cell matrix has eigenvalues \(5, 2, 1, 1\), for a total of \(9\). The first two ratios are \(5/9\approx0.556\) and \(2/9\approx0.222\), so \(R_2=7/9\approx0.778\). If independent experimental records indicate that the strongest variation is associated with sequencing depth, the fact that the first component explains about 55.6% does not make it a biologically relevant axis. The eigenvalue alone cannot reveal the source of variation. Biological interpretation requires evidence beyond the variance summary.

The reverse caution matters too: a component with a modest ratio is not automatically noise or irrelevant. It may capture a structured pattern that matters for a specific question. Variance summaries are global: they aggregate squared deviations across cells and features, so they need not prioritize rare or localized structure.

Diagnostics, Alternatives, and Limitations

Before interpreting the plot, check that the eigenvalues are nonnegative up to small numerical rounding error, are listed in descending order, and correspond to the same PCA input used for interpretation. Recompute the total \(T\) and confirm that the ratios sum to approximately one and the final cumulative value is approximately one when \(T>0\). Small discrepancies can arise from rounding; substantial discrepancies suggest an indexing, normalization, or transcription error.

A zero or negative total is not a situation in which to proceed with ratios: for a valid covariance matrix the eigenvalues are nonnegative, and zero total means no variance in the analyzed matrix. Also distinguish a zero eigenvalue within a positive-total spectrum, which simply contributes zero, from a zero total, which makes every ratio undefined.

The main choices answer different questions:

These approaches are complementary, not interchangeable. A threshold or elbow is a compact description of the spectrum. Stability checks and independent biological evidence address different concerns, although they too have assumptions and limitations. No single variance summary can validate a biological interpretation. In particular, when eigenvalues are tied or nearly tied, the corresponding individual directions may not be uniquely determined; the variance associated with their shared subspace can be more stable than an interpretation attached to one direction alone.

Common Mistakes

• Dividing by a zero total. If all eigenvalues are zero, \(0/0\) is undefined; do not report zero ratios or claim they sum to one.
• Calling a threshold a guarantee. Reaching 90% means 90% of input-matrix variance, not 90% of biological information.
• Reading an elbow as an exact boundary. An elbow is a visual heuristic, and some spectra have no convincing elbow.
• Ignoring the analyzed features. Ratios refer to the matrix actually supplied to PCA; changes in features or preprocessing can change the spectrum.
• Assuming small variance means no value. A low-variance component may still contain a pattern relevant to a particular scientific question.