Summary-Statistic Conditional Analysis

Why Reconstruct Conditional Results From Summary Statistics?

A marginal association estimate describes a variant’s relationship with an outcome when that variant is tested on its own, subject to the covariates in the model. A conditional estimate instead describes the variant’s association while the other included variants are held fixed. When individual-level genotypes are unavailable, marginal estimates and the correlations among variants can sometimes recover the joint estimates. The calculation is useful only when those summaries refer to compatible data and models.

This tutorial develops the calculation for a quantitative outcome analyzed with linear regression. The key distinction is that the conditional point estimates follow from a linear-algebra identity, while conditional standard errors and tests also require a suitable residual-variance estimate and compatible sampling assumptions.

Model, Assumptions, and Genotype Scaling

Begin with the linear model

Here \(\beta\) is the vector of conditional variant effects, measured per genotype-allele count if \(G\) contains allele counts. For the usual standard-error and test calculations, assume the model is correctly specified and the residuals are independent with common variance; normal residuals give exact finite-sample \(t\)-tests, while large samples often justify normal approximations. The covariates and all variants must be observed in the same analytic sample, and the marginal results must use the same outcome, covariate adjustment, and allele coding.

Let \(M_C=I-C(C'C)^{-1}C'\) be the residual-maker matrix, assuming \(C\) has full column rank. Require \(C\) to include an intercept. Define the covariate-residualized outcome and genotypes by \(y^*=M_Cy\) and \(W=M_CG\). The intercept means that every column of \(W\) is centered. Let \(D\) be the diagonal matrix whose \(j\)th diagonal entry is \(d_j=\sqrt{W_j'W_j}\), the Euclidean norm of residualized genotype \(j\). Each \(d_j\) must be positive.

With these definitions, \(R\) is the Pearson correlation matrix of the covariate-residualized genotype columns: its diagonal entries are one. The entries of \(D\) are genotype norms, not standard deviations. For example, the sample standard deviation of residualized genotype \(j\), using divisor \(N-1\), is \(d_j/\sqrt{N-1}\). This distinction depends on the intercept and centering; without an intercept, the normalized Gram matrix need not be a Pearson correlation matrix.

Deriving the Conditional Estimates

The marginal regression coefficient for variant \(j\), after adjusting for \(C\), is

Collect the marginal coefficients in \(\hat b\), and define \(X=WD^{-1}\). Since the \(j\)th column of \(X\) has unit length, the vector of marginal cross-products is \(X'y^*=D\hat b\). In the joint regression, the coefficient vector for \(X\) is \(\hat\theta=(X'X)^{-1}X'y^*=R^{-1}D\hat b\). Since \(G\beta\), after removing the covariates, is \(W\beta=XD\beta\), the joint coefficient for \(X\) equals \(D\hat\beta_{\mathrm{cond}}\). Therefore:

This identity requires \(R\) to be invertible. If variants are in exact linear dependence after covariate adjustment, the joint coefficients are not uniquely identified by ordinary least squares. The formula is an exact reconstruction of the joint point estimates when the marginal coefficients come from the same sample and adjustment set. In practice, the genotype matrix is often not available, so the formula is applied to reported summaries and an externally supplied or otherwise reconstructed \(R\); compatibility then becomes a substantive assumption.

Standard Errors and Tests From Summary Statistics

Under the linear model, the covariance matrix of the joint coefficient estimates is

Consequently, if \(\hat\sigma\) is an appropriate estimate of the residual standard deviation from the joint model, the conditional standard error for variant \(j\) is \(\hat\sigma\sqrt{(R^{-1})_{jj}}/d_j\). A marginal standard error is approximately \(\hat\sigma/d_j\) when the marginal and joint residual-variance estimates are close. This is often a useful large-sample approximation, but it is not an exact identity: marginal and joint regressions generally have different residual sums of squares and degrees of freedom.

If \(z_j\) is the marginal association statistic, the commonly used summary-statistic approximation is

To see the scaling, under a common residual-variance approximation the marginal statistics satisfy \(z\approx D\hat b/\hat\sigma\). The conditional estimate is \(D^{-1}R^{-1}D\hat b\), and its standard error is \(\hat\sigma\sqrt{(R^{-1})_{jj}}/d_j\). Dividing estimate by standard error gives the displayed expression. The diagonal denominator matters: \(R^{-1}z\) alone is not a standardized conditional test statistic.

For two variants with correlation \(r\), the formula simplifies to

The norm ratios are needed when genotype scales differ. Omitting them silently assumes \(d_1=d_2\), which need not hold even if both variants are encoded as allele counts.

Near-Collinearity and Interpretation

When two variants are highly correlated, \(R\) is close to singular. The factor \(1/(1-r^2)\) then grows, so small differences between marginal signals can produce large changes in the separate conditional estimates. At the same time, the diagonal elements of \(R^{-1}\) grow, increasing conditional standard errors. This is loss of precision in separating the variants’ associations, not evidence that either estimate is necessarily wrong.

Practical Conditions and a Common Mistake

Before applying the formulas, verify that the marginal statistics and \(R\) refer to the same allele orientation, variant set, covariate adjustment, and compatible participants. The matrix must describe correlations of the appropriately covariate-adjusted genotypes for the analytic sample. If it does not, the algebra may still produce numbers, but those numbers need not reproduce the joint regression. Likewise, the standard-error approximation can fail when sample sizes differ across variants, residual variances differ materially, or the outcome model is not the stated linear model.

A common mistake is to calculate \(R^{-1}z\) and report its entries directly as conditional \(z\)-statistics. This omits the uncertainty scaling. For example, in the equal-scale example, the first entry of \(R^{-1}z\) is \(3.4375\), whereas the conditional statistic is \(3.4375/\sqrt{1.5625}=2.75\). The correction is to divide each transformed entry by \(\sqrt{(R^{-1})_{jj}}\), and to interpret the resulting test in light of the common-variance and sample-compatibility assumptions.

The central lesson is that marginal summary statistics do not, by themselves, determine conditional results. They do so only together with a compatible variant-correlation matrix and the scaling information needed for effect estimates. Conditional association remains a statement about statistical adjustment within the specified model; it does not identify a causal variant.