How LD and Marker Coverage Determine Tagged Heritability

Why the Marker Set Matters

A measured SNP can carry information about an additive genetic effect even when that SNP is not itself causal. The reason is linkage disequilibrium (LD): alleles at different variants can be correlated in a population. If a measured marker is correlated with a variant that affects the trait, the marker may tag some of that variant's effect. A marker panel with different variants, or measured in a population with different LD, can therefore capture a different fraction of the same underlying additive genetic signal.

This makes it important to distinguish the additive variance due to a specified set of causal or modeled variants from the variance that a particular measured marker set can tag. Heritability is always tied to a defined variant set and population. When the measured set is used as a predictor set, its tagged heritability describes the share of phenotypic variance represented by the genetic signal linearly predictable from those markers. It need not equal the total additive heritability.

Definition: In a specified population, tagged genetic variance for a measured marker set is the variance of the population linear projection of the additive genetic signal onto those markers. Tagged heritability is this variance divided by phenotypic variance. It depends on the marker set, its LD with effect-bearing variants, and the distribution of genetic effects.

A Projection Model for Tagged Heritability

Let \(G\) denote an additive genetic signal formed by \(m\) effect-bearing variants, and let \(Z\) be the vector of \(k\) measured markers. Assume the variables have been centered and each genotype column has been standardized to variance one in the target population. Write \(C\) for the vector of standardized effect-bearing genotypes and \(\boldsymbol{\beta}\) for their additive effects, so that \(G=C^\mathsf{T}\boldsymbol{\beta}\). Let \(Y=G+E\), where \(E\) is uncorrelated with the causal genotypes and the measured markers. Thus \(\operatorname{Var}(Y)=\operatorname{Var}(G)+\operatorname{Var}(E)\).

Define the marker correlation matrix \(R_{ZZ}=\operatorname{Cov}(Z,Z)\), and define \(r_{ZG}=\operatorname{Cov}(Z,G)\). The best linear predictor of \(G\) from \(Z\) has the form \(Z^\mathsf{T}\boldsymbol{\gamma}\), with coefficients chosen to minimize mean squared prediction error. If \(R_{ZZ}\) is invertible, those coefficients are \(\boldsymbol{\gamma}=R_{ZZ}^{-1}r_{ZG}\). The variance of this predictor is the tagged genetic variance.

V_{\mathrm{tag}}(Z) = r_{ZG}^{\mathsf{T}}R_{ZZ}^{-1}r_{ZG}, \qquad h^2_{\mathrm{tag}}(Z) = \frac{V_{\mathrm{tag}}(Z)}{\operatorname{Var}(Y)}.

The definition is a population quantity, not a guarantee that a finite-sample estimate will equal it. If markers are perfectly collinear, \(R_{ZZ}\) is singular and the predictor is still defined as the projection onto the space spanned by the markers, though its coefficient vector may not be unique. The explained variance remains well-defined.

The same expression can be written in terms of LD with the effect-bearing variants. Let \(R_{CC}=\operatorname{Cov}(C,C)\) and \(R_{ZC}=\operatorname{Cov}(Z,C)\). Since \(r_{ZG}=R_{ZC}\boldsymbol{\beta}\), the tagged variance is

V_{\mathrm{tag}}(Z) = \boldsymbol{\beta}^{\mathsf{T}} R_{CZ}R_{ZZ}^{-1}R_{ZC} \boldsymbol{\beta}.

This formula shows why marker count alone is not a measure of coverage. The relevant quantities are the correlations between markers and the effect-bearing variants, the correlations among markers themselves, and the pattern of effects. Correlated markers can provide overlapping information; a large marker panel does not necessarily tag every effect well.

Worked Example: One Causal Variant and One Imperfect Tag

Suppose a standardized causal genotype \(C\) has additive effect \(\beta=0.4472\), and the phenotype is \(Y=0.4472C+E\), where \(E\) is uncorrelated with both \(C\) and the measured marker \(Z\). Let \(\operatorname{Var}(E)=0.8\), and suppose the standardized marker has correlation \(r=0.6\) with \(C\). All quantities are hypothetical.

The causal genetic variance is \(\beta^2\operatorname{Var}(C)=0.4472^2(1)\), approximately \(0.2000\). Therefore \(\operatorname{Var}(Y)\) is approximately \(0.2000+0.8=1.000\). The marker's covariance with the genetic signal is \(\operatorname{Cov}(Z,G)=0.4472(0.6)=0.2683\). Since \(Z\) has variance one, projecting onto \(Z\) gives a tagged variance of

V_{\mathrm{tag}}(Z) = \frac{\operatorname{Cov}(Z,G)^2}{\operatorname{Var}(Z)} = \frac{0.2683^2}{1} \approx 0.0720.

Thus tagged heritability is approximately \(0.0720/1.000=0.0720\), or 7.20%. The additive variance due to the causal variant is about 20.0% of phenotypic variance, but this marker captures only \(r^2=0.36\) of that signal. The difference reflects imperfect tagging, not a change in the causal effect.

How LD Translates into Captured Variance

For a single standardized causal genotype \(C\) with effect \(\beta\), and a single standardized marker \(Z\) with correlation \(r\), the genetic signal is \(G=\beta C\). The marker's covariance with that signal is \(r\beta\). Because the marker variance is one, its projection variance is \((r\beta)^2\), while the causal genetic variance is \(\beta^2\). When \(\beta\ne 0\), the fraction of this genetic variance tagged is therefore \(r^2\). This result explains why the square of the correlation—not the correlation itself—gives the fraction captured in this simple case.

With multiple causal variants, effects can reinforce or offset each other in the covariance with a marker. With multiple markers, their joint tagging depends on their LD with the genetic signal and on redundancy among the markers. In particular, treating each marker's marginal correlation with the signal as if it contributed independently can overstate the variance captured. The joint projection accounts for overlap through \(R_{ZZ}^{-1}\).

Worked Example: One Marker Tags Two Effect-Bearing Variants

Suppose two standardized effect-bearing genotypes \(C_1\) and \(C_2\) have correlation \(0.3\), with effects \(\beta_1=0.2\) and \(\beta_2=0.1\). The additive signal is \(G=0.2C_1+0.1C_2\). A single standardized measured marker \(Z\) has correlations \(0.8\) with \(C_1\) and \(0.1\) with \(C_2\). Let \(E\) be uncorrelated with these genotypes and have variance \(0.938\).

First calculate the genetic variance, including the covariance between the two effect-bearing genotypes:

\operatorname{Var}(G) = 0.2^2+0.1^2+2(0.2)(0.1)(0.3) = 0.040+0.010+0.012 = 0.062.

Hence \(\operatorname{Var}(Y)=0.062+0.938=1.000\). The marker's covariance with \(G\) is

\operatorname{Cov}(Z,G) = (0.8)(0.2)+(0.1)(0.1) = 0.16+0.01 = 0.17.

Because \(\operatorname{Var}(Z)=1\), the marker projection has variance \(0.17^2/1=0.0289\), and tagged heritability is \(0.0289/1.000=0.0289\), or 2.89%. The marker is strongly correlated with the first genotype, but it does not capture all of the genetic signal: the second variant also contributes, and its effect and LD pattern enter the covariance calculation.

Why Adding Markers Cannot Reduce Population Tagged Variance

For a fixed population and a fixed linear-projection definition, expanding the marker set cannot lower the population tagged variance. This statement concerns the best projection using the markers, not the result of a particular estimation algorithm in a finite sample. The reason is that every predictor available from the smaller set remains available from the larger set by assigning coefficient zero to each added marker.

Start with the smaller set.
Let the best linear predictor using markers \(Z_1\) have mean squared error \(Q_1\).

Expand the candidate predictors.
Let \(Z_2\) contain all markers in \(Z_1\) plus additional markers. The old predictor is still available using \(Z_2\), with zero coefficients on the additions.

Use optimality of the larger-set projection.
Since the best predictor from \(Z_2\) minimizes error over a set that includes the old predictor, \(Q_2\le Q_1\).

Translate prediction error into captured variance.
Projection residuals are uncorrelated with the predictor. Thus the projection variance is \(\operatorname{Var}(G)-Q\), so \(Q_2\le Q_1\) implies \(V_{\mathrm{tag}}(Z_2)\ge V_{\mathrm{tag}}(Z_1)\).

V_{\mathrm{tag}}(Z_2)\ge V_{\mathrm{tag}}(Z_1), \qquad h^2_{\mathrm{tag}}(Z_2)\ge h^2_{\mathrm{tag}}(Z_1),

provided the population, phenotype denominator, and projection definition are unchanged. Equality occurs when the added markers provide no new linear information about the genetic signal beyond what the original markers already provide.

Worked Example: Two Correlated Markers Improve Coverage Jointly

Suppose one standardized causal genotype \(C\) has effect \(\beta=\sqrt{0.2}\approx0.4472\), so its genetic variance is \(0.2\). Let \(\operatorname{Var}(E)=0.8\), with \(E\) uncorrelated with the genotypes, giving \(\operatorname{Var}(Y)=1.000\). Consider two standardized markers \(Z_1,Z_2\) with correlations \(0.5\) and \(0.6\) with \(C\), and correlation \(0.2\) with each other. These hypothetical correlations are compatible with a positive-definite joint correlation matrix.

With \(Z_1\) alone, the tagged fraction of the causal variance is \(0.5^2=0.25\), so tagged variance is \(0.25(0.2)=0.0500\), and tagged heritability is 5.00%. With \(Z_2\) alone, the corresponding values are \(0.6^2(0.2)=0.0720\) and 7.20%.

For the joint marker set,

R_{ZZ} = \begin{pmatrix} 1 & 0.2\\ 0.2 & 1 \end{pmatrix}, \qquad r_{ZC} = \begin{pmatrix} 0.5\\ 0.6 \end{pmatrix}.

Because \(G=\sqrt{0.2}C\), the joint tagged variance is \(0.2\,r_{ZC}^{\mathsf{T}}R_{ZZ}^{-1}r_{ZC}\). Inverting the marker correlation matrix gives

R_{ZZ}^{-1} = \frac{1}{0.96} \begin{pmatrix} 1 & -0.2\\ -0.2 & 1 \end{pmatrix}, \qquad r_{ZC}^{\mathsf{T}}R_{ZZ}^{-1}r_{ZC} = \frac{0.5(0.38)+0.6(0.50)}{0.96} = \frac{0.49}{0.96} \approx 0.5104.

Therefore the joint tagged variance is \(0.2(0.5104)=0.1021\), and tagged heritability is approximately 10.21%. The two-marker set captures more than either marker alone, but less than the full causal variance of \(0.2\). Their positive correlation means that some of their information overlaps.

Interpreting Differences Across Marker Sets

A difference in tagged heritability between marker sets does not by itself imply a difference in the underlying biological contribution of genetics. It may instead reflect how well each set tags the effect-bearing variants. More comprehensive coverage can raise the population projection variance, but adding many redundant markers may add little. Conversely, a smaller set can capture substantial variance if its markers are well correlated with the relevant effects.

LD is population-specific. Even if allele effects and causal variants were unchanged, differences in LD can alter \(R_{ZC}\) and \(R_{ZZ}\), and therefore change tagged variance. This is why ancestry matching of LD information matters when interpreting estimates based on reference data. A marker panel's coverage is not an intrinsic property of its size alone; it is coverage relative to a population and an effect pattern.

The projection formulas also clarify what a simple LD calculation can and cannot say. In the single-causal-variant, single-marker case, \(r^2\) directly gives the fraction tagged. With many variants and markers, an average \(r^2\) generally is not enough: effects may differ in size and direction, and markers may be correlated with one another. The full covariance structure and effect pattern determine the captured variance.

Common Mistakes

Equating marker-set heritability with all additive heritability. A marker set can tag only part of the genetic signal. The correction is to state the set and population, and distinguish tagged variance from total additive variance.
Assuming each added SNP must increase an estimate in a sample. The population projection variance is nondecreasing for nested marker sets, but sampling variation, model fitting, and estimation constraints can produce finite-sample estimates that do not follow that order.
Using the number of markers as a stand-in for coverage. Marker count does not reveal LD with effect-bearing variants or redundancy among markers. Coverage depends on those correlations and the effect pattern.
Interpreting a change in tagged heritability as a biological change. A change can arise because the measured set or its LD with causal variants differs. The underlying additive genetic architecture need not have changed.

Key takeaway: Tagged heritability is the phenotypic variance fraction captured by the linear projection of genetic signal onto a specified marker set. LD determines how well markers tag effects, and marker coverage changes the projection even when the underlying genetic effects remain the same.

Check Your Understanding

Use the definitions and projection results above to answer these questions.

For one standardized causal genotype and one standardized marker with correlation \(r=0.4\), what fraction of the causal genetic variance is tagged by that marker?
Why does a marker's correlation with a genetic signal need to be adjusted for correlation among markers when computing joint tagged variance?
Under what conditions is tagged variance equal to the full additive genetic variance in the model?
Explain why adding markers cannot reduce population tagged variance for nested marker sets, and give one reason a finite-sample estimate might nevertheless decrease.
Two populations have the same causal effects but different LD between measured markers and effect-bearing variants. Why can their tagged heritabilities differ?