Centering, Scaling, and LD Pruning Before PCA

Why preprocessing changes PCA

Principal-component analysis (PCA) finds directions of variation in a matrix, but it does not decide which genetic variation should count most. Before PCA, the way genotype columns are centered, scaled, and filtered determines how strongly each variant can contribute. A run using centered dosages alone can therefore give different leading components from a run using variance-scaled dosages or a set pruned for linkage disequilibrium (LD).

This is not just a computational detail. If many nearby variants carry nearly redundant information, they can collectively give one genomic region more influence than a region represented by fewer variants. If rare genotypes are scaled aggressively, a small number of carriers can also have substantial influence. Preprocessing choices should reflect the goal of the analysis and be reported alongside the PCA.

Genotype dosage, centering, and scaling

Suppose there are \(n\) individuals and \(m\) biallelic variants. Let \(g_{ij}\) be the genotype dosage for individual \(i\) at variant \(j\), coded as the number of copies of a consistently chosen allele: \(0\), \(1\), or \(2\). Let \(\hat{p}_j\) be the sample frequency of that allele among called chromosomes at variant \(j\). If all \(n\) individuals have called genotypes, then \(\hat{p}_j=\sum_i g_{ij}/(2n)\).

Centering makes the average of each complete genotype column zero. It removes the column mean, not the variation between individuals. Scaling changes the relative weight of columns. With centered but unscaled dosages, a variant with greater genotype variance can contribute more total squared variation. Under the HWE scaling above, variants have equal expected genotype variance under the model used to set the denominator.

The scaling formula assumes a diploid, biallelic dosage coded from zero to two and an allele frequency strictly between zero and one. Its variance interpretation follows under HWE: each individual’s two allele copies are modeled as independent Bernoulli variables with success probability \(p_j\). The formula is a convention for weighting genotypes, not a claim that every variant or population actually satisfies HWE. Population structure, relatedness, and other departures can make observed genotype variation differ from the HWE expectation.

For a centered genotype matrix \(X\), the matrix \(XX^\mathsf{T}\) is the sum of the individual-level contributions from each variant. Multiplying a column by a constant changes that variant’s contribution to this matrix. PCA then acts on the resulting set of contributions; it cannot undo a preprocessing choice.

What the HWE scaling does—and does not do

Under the stated HWE model, a genotype dosage \(G\) is the sum of two independent allele-copy indicators. Each indicator has variance \(p(1-p)\), so the variance of their sum is \[ \operatorname{Var}(G)=p(1-p)+p(1-p)=2p(1-p). \] Subtracting the mean \(2p\) does not change the variance. Dividing by \(\sqrt{2p(1-p)}\) therefore gives a variable with variance one under this model:

This result explains the scaling denominator; it does not guarantee equal observed variances across variants in a finite or structured sample. In particular, using the sample allele frequency in the denominator does not make each column’s empirical variance exactly one. Another possible convention is to divide each centered column by its observed sample standard deviation. That produces unit empirical variance for nonconstant columns, but it is a different choice and can also amplify rare or unusual patterns. Analyses should specify which convention was used.

For unscaled centered dosages, HWE predicts variance \(2p(1-p)\), which is largest at \(p=0.5\) and decreases toward zero as the allele becomes rare. Scaling removes this frequency dependence under the model. That can be useful when the aim is to let variants contribute on a more comparable expected-variance scale. But for a rare variant, the denominator is small, so a dosage difference can translate into a large scaled value. Equal expected variance is not the same as equal precision or equal biological importance.

LD pruning limits repeated information

LD is non-random association between alleles at different variants. When many nearby variants are strongly correlated, their genotype columns can encode much of the same pattern across individuals. If all are included, that pattern may contribute repeatedly to PCA. LD pruning reduces this redundancy by retaining a subset of variants that are not too strongly correlated according to a chosen rule.

A typical pruning procedure considers variants within genomic windows and removes one or more variants when pairwise \(r^2\) exceeds a specified threshold. The precise window size, step, threshold, and variant retained can vary by software and analysis. Pruning is a pragmatic way to reduce repeated signal; it does not make the retained variants independent, nor does it remove all LD. It also does not identify causal variants.

The weighting effect can be seen directly. If two centered columns are identical, \(x_1=x_2=x\), their joint contribution to the individual-level matrix is \((x_1x_1^\mathsf{T}+x_2x_2^\mathsf{T})=2xx^\mathsf{T}\). One copy contributes \(xx^\mathsf{T}\). Thus, exact duplicates double the contribution of that pattern before considering any other variants. Highly correlated, but not identical, columns can have a similar repeated-weighting effect.

Allele-frequency filters and missing genotypes

A minor-allele-frequency (MAF) filter removes variants whose less frequent allele is below a chosen frequency threshold. In a biallelic sample, MAF is the smaller of the two allele frequencies. Filtering can exclude nearly invariant variants and reduce the influence of very low-count alleles, which are especially sensitive to genotyping error and can receive large values under variance scaling. It also changes which genetic variation is represented: a stringent threshold removes genuine rare variation as well as potentially unreliable observations.

There is no universally correct MAF threshold for PCA. The choice depends on sample size, genotype quality, population composition, and the purpose of the analysis. A frequency filter does not guarantee that all retained variants are well measured or free of population-specific frequency patterns. Report the threshold and whether frequencies were calculated across the full sample or within a defined group; pooled frequencies can differ from within-group frequencies when the sample is structured.

Missing genotypes also require a stated policy. One common simple choice is to impute a missing dosage with the variant’s mean dosage, \(2\hat{p}_j\), so its centered value is zero. This avoids treating missingness as an additional allele state, but it also treats the missing observation as contributing no deviation from the column mean. More elaborate imputation may be appropriate in other settings. Whatever method is used, unexplained or ancestry-related missingness can create patterns that preprocessing alone does not resolve.

A practical sequence and common pitfall

A common mistake is to assume that variance scaling is automatically more fair because it gives each variant unit expected variance. That reasoning overlooks the small denominator for rare alleles and the influence of a few carriers. Another is to assume that a large number of variants guarantees balanced genomic representation; a dense LD block can supply many redundant columns and be counted repeatedly. The correction is to treat scaling, filtering, and pruning as explicit modeling choices, justify them for the aim of the PCA, and report them so the weighting can be understood.