Worked Analysis Of Summary-Statistic Genetic Correlation

From Two Association Results to a Genetic Correlation

Two genome-wide association studies can report evidence about many of the same variants without measuring the same trait or using the same participants. Cross-trait LD score regression (LDSC) uses the way association signals vary with linkage disequilibrium (LD) to estimate the genetic covariance between traits from their summary statistics. The genetic correlation then expresses that covariance relative to the genetic standard deviations of the two traits.

A reliable analysis is not just a regression call. Alleles must be aligned so that association statistics have comparable directions; sample sizes, trait scales, and LD reference data must be appropriate; and the intercept, slope, and uncertainty must be interpreted separately. The examples below use entirely hypothetical data. They illustrate the analysis rather than report results from a real study.

Definition: For variant $j$, let $z_{1j}$ and $z_{2j}$ be the signed association statistics for traits 1 and 2, after both statistics have been aligned to the same effect allele. Let $l_j$ be that variant's LD score, the sum of its squared correlations with variants in a specified reference set. In the standard cross-trait LDSC model, $$ E[z_{1j}z_{2j}\mid l_j] = \frac{\sqrt{N_1N_2}}{M}C_g l_j + a_{12}. $$ Here $N_1,N_2$ are the study sample sizes, $M$ is the number of variants defining the genetic-variance scale, $C_g$ is the genetic covariance on the traits' stated scales, and $a_{12}$ is the cross-trait regression intercept. The slope is $b_{12}=\sqrt{N_1N_2}C_g/M$; thus $\widehat C_g=\widehat b_{12}M/\sqrt{N_1N_2}$.

The intercept represents the expected cross-trait product not explained by the LD-score-dependent genetic component. Sample overlap and correlated confounding can contribute to it; it is not a direct estimate of sample overlap alone. The slope captures the component that increases with LD score under the model. An intercept above zero therefore does not, by itself, show that a genetic correlation exists.

The usual analysis fits a weighted regression of $z_{1j}z_{2j}$ on $l_j$, with an intercept. Its weights are intended to account for the different precision of the product statistics and LD scores. The regression assumes that the expected product follows the stated linear relationship over the analyzed variants, and that the reference LD scores represent the LD structure of the study samples. In practice, one also needs suitable association statistics, adequate variant matching, and an appropriate treatment of sample overlap and ancestry. Standard errors are commonly obtained with a block jackknife over genomic regions, which reflects dependence among nearby variants better than treating every SNP as independent.

Harmonize Before Fitting

Each summary dataset records an effect allele and an estimated effect or signed $z$-statistic. For a variant included in both studies, the two signs must refer to the same allele. If one dataset reports the other allele, reverse the sign of its effect estimate and $z$-statistic before forming cross-trait products. Allele frequency can help identify strand and allele mismatches, but it does not resolve every case. For strand-ambiguous A/T and C/G variants, frequency information may be insufficient, so excluding variants with uncertain alignment is often safer.

Worked Example: Checking Allele Direction

Suppose two hypothetical studies report the following signed statistics for three variants. The goal here is to check alignment and the resulting products, not to estimate a genome-wide regression from three variants.

Variant	Trait 1 effect allele; $z_1$	Trait 2 effect allele; $z_2$	Alignment decision
rsA	A; +3.0	A; −1.5	Already aligned
rsB	C; −2.0	T; +2.0	Reverse trait 2 to C
rsC	A; +1.2	T; −0.8	Exclude: strand-ambiguous and alignment uncertain

For rsA, both statistics refer to allele A, so $z_1z_2=(+3.0)(-1.5)=-4.5$. For rsB, the trait 2 statistic refers to T, the allele opposite to the C used for trait 1. Its aligned statistic is therefore $z_2=-2.0$, giving $z_1z_2=(-2.0)(-2.0)=4.0$. For rsC, the allele labels are strand-ambiguous and the hypothetical frequency information does not resolve the orientation. Excluding it avoids assigning a potentially incorrect sign. A sign error can change cross-trait products and distort the fitted slope, so harmonization must precede regression.

Fit, Interpret, and Convert the Regression

After harmonization, the analysis uses a common set of well-matched variants with LD scores from an appropriate reference panel. The studies' sample sizes and the definition of $M$ must be recorded consistently. For traits analyzed on standardized quantitative scales, the covariance and heritabilities below refer to those scales. For binary traits, the analyst must state whether estimates are on the observed or liability scale and use compatible estimates; these scales are not interchangeable without the relevant assumptions and conversion.

The fitted slope is not itself the genetic correlation. It first gives a genetic covariance after rescaling for sample sizes and $M$. To obtain the correlation, combine that covariance with univariate heritability estimates $h_1^2$ and $h_2^2$:

\widehat r_g = \frac{\widehat C_g}{\sqrt{\widehat h_1^2\widehat h_2^2}}.

The genetic correlation is dimensionless and describes the standardized relationship between additive genetic effects for the two traits on the scales analyzed. A positive value indicates that their additive genetic effects tend to covary in the same direction; a negative value indicates a tendency to covary in opposite directions. It is not the phenotypic correlation, and it does not establish that changing one trait would cause a change in the other.

Justification: Under the corresponding univariate LDSC models, the univariate slopes satisfy $b_1=N_1h_1^2/M$ and $b_2=N_2h_2^2/M$. Therefore, $$ \frac{b_{12}}{\sqrt{b_1b_2}} = \frac{\sqrt{N_1N_2}C_g/M} {\sqrt{(N_1h_1^2/M)(N_2h_2^2/M)}} = \frac{C_g}{\sqrt{h_1^2h_2^2}} = r_g. $$ This shows why covariance from the cross-trait slope and heritabilities on compatible scales are needed to form the genetic correlation.

Worked Example: Estimating the Slope and Genetic Covariance

Consider an intentionally small, hypothetical illustration with five equally weighted LD-score bins. Real LDSC analyses use many variants and their prescribed regression weights; this calculation shows how the slope and intercept relate to the fitted points. Suppose the bin values are:

LD score $l$	Mean cross-trait product $z_1z_2$
5	0.0320
10	0.0331
15	0.0339
20	0.0351
25	0.0359

For these points, $\bar l=15$ and $\bar y=0.0340$, where $y$ denotes the cross-trait product. The least-squares slope is the sum of centered cross-products divided by the sum of centered squared LD scores:

\widehat b_{12} = \frac{(-10)(-0.0020)+(-5)(-0.0009)+(0)(-0.0001)+(5)(0.0011)+(10)(0.0019)} {(-10)^2+(-5)^2+0^2+5^2+10^2} = \frac{0.0490}{250} = 0.0001960.

The intercept is $\widehat a_{12}=\bar y-\widehat b_{12}\bar l=0.0340-(0.0001960)(15)=0.03106$. Suppose the hypothetical studies have $N_1=10{,}000$, $N_2=9{,}000$, and use $M=1{,}000{,}000$. Then $\sqrt{N_1N_2}=\sqrt{90{,}000{,}000}=9{,}486.8$, so

\widehat C_g = \frac{(0.0001960)(1{,}000{,}000)}{9{,}486.8} = 0.02066.

This covariance is on the traits' stated scales. The estimated intercept is the model's baseline cross-trait product not captured by the LD-score slope; it should not be interpreted as genetic covariance. The five-bin calculation is only illustrative: it does not supply the variant-level information or uncertainty assessment needed for a study analysis.

Uncertainty and Sensitivity

Uncertainty in $r_g$ comes from the cross-trait covariance and both univariate heritability estimates. A standard approach is to estimate the uncertainty jointly, for example by a genomic-block jackknife that recomputes the quantities and $r_g$ after omitting each block. This retains dependence among estimates induced by using the same genome-wide data. A delta-method calculation can also clarify how uncertainty propagates, but treating the component estimates as independent when they are not can give an incorrect standard error.

Worked Example: Propagating Uncertainty and Checking Heritability Sensitivity

Continue the hypothetical analysis with $\widehat C_g=0.02066$, $\widehat h_1^2=0.20$, and $\widehat h_2^2=0.15$. Suppose the standard errors are 0.006 for $\widehat C_g$, 0.03 for $\widehat h_1^2$, and 0.02 for $\widehat h_2^2$. For this illustration only, assume their estimated covariances are zero. First,

\widehat r_g = \frac{0.02066}{\sqrt{(0.20)(0.15)}} = \frac{0.02066}{0.1732} = 0.1193.

For $r=C/\sqrt{H_1H_2}$, the derivatives are $\partial r/\partial C=1/\sqrt{H_1H_2}$, $\partial r/\partial H_1=-r/(2H_1)$, and $\partial r/\partial H_2=-r/(2H_2)$. With the stated inputs, these are approximately 5.774, −0.2983, and −0.3977. Under the illustrative zero-covariance assumption, the delta-method variance is

\widehat{\operatorname{Var}}(\widehat r_g) \approx (5.774)^2(0.006)^2 + (-0.2983)^2(0.03)^2 + (-0.3977)^2(0.02)^2 = 0.001343, \qquad SE(\widehat r_g)\approx 0.0366.

A normal-approximation 95% interval is $0.1193\pm1.96(0.0366)$, or approximately 0.047 to 0.191. Under the hypothetical model and uncertainty assumptions, this interval quantifies uncertainty in the estimated genetic correlation; it is not a statement that there is a 95% probability that the fixed parameter lies in this particular interval. In an actual LDSC analysis, a joint jackknife estimate is preferable to assuming these component estimates have zero covariance.

As a simple point-estimate sensitivity check, suppose an alternative, justified univariate analysis gave $\widehat h_2^2=0.12$, while the covariance and $h_1^2$ stayed unchanged. Then $\widehat r_g=0.02066/\sqrt{(0.20)(0.12)}=0.1334$. The change illustrates why a genetic correlation can be sensitive to uncertain or small heritability estimates: heritabilities appear in its denominator. This is not a complete sensitivity analysis, since a real alternative analysis should refit the relevant estimates and uncertainty consistently.

Sensitivity checks are useful when a result depends on particular data-processing or modeling choices. They should be motivated by plausible risks, not chosen only because they produce a preferred estimate. A practical set of checks includes:

Confirm that allele alignment, variant inclusion, sample-size inputs, and the LD reference are appropriate for both studies.
Compare the fitted intercept and slope under justified regression specifications, including whether an intercept constraint is defensible. A forced-zero intercept changes the model and should not be used merely to obtain a desired slope.
Assess whether results are materially affected by influential variants, LD-score ranges, or reasonable changes to variant filters.
Check that the univariate heritabilities are sufficiently well estimated for the ratio to be informative, and that their scales match the cross-trait covariance.
When samples overlap or confounding may be correlated across studies, interpret the intercept in light of the design and evaluate whether the analysis handles those features appropriately.

These checks do not turn the regression into a causal analysis. They test whether an estimated genome-wide genetic correlation is robust to defensible choices and whether its interpretation is supported by the data and model assumptions.

Common Pitfall: Reading the Intercept as the Correlation

A common mistake is to see a nonzero cross-trait intercept and conclude that it is evidence of genetic correlation, or to report the regression slope directly as $r_g$. The correction is to keep the quantities distinct: the intercept is the baseline component of the expected cross-trait $z$-product; the slope, after scaling by $M/\sqrt{N_1N_2}$, estimates genetic covariance; and that covariance must be divided by the square root of the product of compatible heritability estimates to obtain genetic correlation. Uncertainty in all these quantities matters, especially when heritability estimates are imprecise.

Key takeaway: A defensible summary-statistic genetic-correlation analysis aligns allele directions, fits cross-trait LDSC with appropriate LD scores and sample information, interprets slope and intercept separately, and reports $r_g$ with uncertainty and relevant sensitivity checks.

Check Your Understanding

Use the definitions and calculations in this tutorial to answer the following questions.

A variant's $z$-statistic is positive in one study and negative in the other. What must you check before interpreting their product?
In cross-trait LDSC, which part of the fitted line estimates the LD-score-dependent component, and which part is the intercept?
If $\widehat b_{12}$, $M$, $N_1$, and $N_2$ are known, write the formula for $\widehat C_g$.
Why can a change in a univariate heritability estimate change $\widehat r_g$, even when the cross-trait covariance estimate is unchanged?
Why is a joint jackknife generally preferable to assuming that the covariance and heritability estimates are independent?