Deriving the Joint Summary-Statistic Likelihood

Why a Joint Likelihood Needs Cohort-Specific Structure

Association summaries from multiple ancestry groups can be analyzed together, but they are not interchangeable rows of one larger summary-statistic table. Each cohort has its own pattern of linkage disequilibrium (LD), its own sample size, and often its own standard errors. A likelihood that combines the cohorts should retain these differences rather than replacing them with one pooled LD matrix or one average standard error.

The starting point is a likelihood for the vector of marginal effect estimates in each cohort. We will use a Gaussian working model: the expected marginal estimates depend on the cohort’s LD matrix, and their covariance depends on that cohort’s standard errors and LD. Multiplying cohort likelihoods is justified when the cohort data—and therefore their sampling errors—are independent, conditional on the model parameters.

Definition: For cohort \(a\), let \(\hat{\boldsymbol b}_a\) be the vector of estimated marginal genetic effects for \(m\) harmonized variants, \(R_a\) the \(m\times m\) genotype correlation (LD) matrix, and \(S_a=\operatorname{diag}(SE_{a1},\ldots,SE_{am})\) the diagonal matrix of their standard errors. In the working model used here, \[ \hat{\boldsymbol b}_a\mid\boldsymbol\beta \sim N\!\left(R_a\boldsymbol\beta,\;S_aR_aS_a\right). \] The vector \(\boldsymbol\beta\) contains effects on a common, standardized genotype and trait scale. This specification assumes a common effect vector across cohorts; it is an assumption of this likelihood, not a consequence of combining the data.

From Marginal Estimates to a Cohort Likelihood

The mean \(R_a\boldsymbol\beta\) arises because a marginal association at one variant can reflect the effects of correlated variants. For example, if the effect vector has a nonzero entry only at one variant, the expected marginal estimate at another variant is proportional to its LD with that variant. The proportionality and the pattern of correlations are specific to cohort \(a\).

The covariance \(S_aR_aS_a\) is a working covariance model for the marginal estimates. Its diagonal entries are \(SE_{aj}^2\), and its off-diagonal entries are \(SE_{aj}SE_{ak}R_{a,jk}\). Thus, correlated genotypes generally produce correlated estimation errors. Standard errors incorporate precision, which is affected by sample size and may also vary across variants because of missingness or other cohort-specific features.

For this Gaussian density, assume positive standard errors and a positive-definite \(R_a\). Let \(\boldsymbol d_a=\hat{\boldsymbol b}_a-R_a\boldsymbol\beta\) denote the residual from the model mean. The cohort likelihood is

L_a(\boldsymbol\beta) =(2\pi)^{-m/2}|S_aR_aS_a|^{-1/2} \exp\left\{-\frac{1}{2} \boldsymbol d_a^{\mathsf T}(S_aR_aS_a)^{-1}\boldsymbol d_a\right\}.

The determinant term accounts for the volume and scale of the sampling uncertainty; the quadratic term measures how far the observed summaries are from their model mean, accounting for their covariance. Because \(S_a\) is diagonal,

|S_aR_aS_a|=|S_a|^2|R_a|, \qquad (S_aR_aS_a)^{-1}=S_a^{-1}R_a^{-1}S_a^{-1}.

Equivalently, standardize the residuals as \(\boldsymbol u_a=S_a^{-1}\boldsymbol d_a\). The quadratic term becomes \(\boldsymbol u_a^{\mathsf T}R_a^{-1}\boldsymbol u_a\). This form makes clear that standardized residuals must be evaluated using the LD matrix for the same cohort.

Worked Example: Evaluating One Cohort’s Likelihood

Suppose a hypothetical cohort reports two marginal estimates, in standardized effect units, \(\hat{\boldsymbol b}=(0.30,0.10)^{\mathsf T}\), with standard errors \(0.20\) and \(0.25\). Its LD matrix has correlation \(0.50\), and consider the candidate effect vector \(\boldsymbol\beta=(0.40,0)^{\mathsf T}\). Assume the stated standard errors and LD matrix satisfy the working model.

The model mean is \(R\boldsymbol\beta=(0.40,0.20)^{\mathsf T}\). Thus the residual is \(\boldsymbol d=(-0.10,-0.10)^{\mathsf T}\), and the standardized residual is \(\boldsymbol u=(-0.50,-0.40)^{\mathsf T}\). For two variants with correlation \(r=0.50\), the inverse LD matrix gives

\boldsymbol u^{\mathsf T}R^{-1}\boldsymbol u =\frac{u_1^2+u_2^2-2ru_1u_2}{1-r^2} =\frac{0.25+0.16-2(0.50)(0.20)}{0.75} =0.280.

The covariance determinant is \((0.20^2)(0.25^2)(1-0.50^2)=0.001875\). Substituting into the log-density,

\ell(\boldsymbol\beta) =-\log(2\pi)-\frac{1}{2}\log(0.001875)-\frac{1}{2}(0.280) \approx 1.162.

This is a log density for the observed vector of estimates at the specified candidate effects, not a probability that the candidate is true. The example uses the cohort’s correlation to assess the joint residual; treating the two residuals as independent would give a different quadratic term.

Combining Independent Cohorts

Suppose there are \(A\) cohorts. If their underlying samples do not overlap and their sampling errors are independent conditional on the parameters, their joint density is the product of their cohort densities. Under the common-effect assumption in the definition, the same \(\boldsymbol\beta\) appears in every factor, while \(R_a\) and \(S_a\) can differ by cohort.

L_{\mathrm{joint}}(\boldsymbol\beta) =\prod_{a=1}^{A}L_a(\boldsymbol\beta), \qquad \ell_{\mathrm{joint}}(\boldsymbol\beta) =\sum_{a=1}^{A}\ell_a(\boldsymbol\beta).

The addition of log likelihoods follows directly from taking the logarithm of the product. Independence is doing essential work here: if cohorts overlap, their sampling errors can be correlated, so the product is generally not the correct joint likelihood. In that case, the cross-cohort covariance must be modeled rather than ignored.

Worked Example: Summing Independent Cohort Contributions

For a simple hypothetical illustration, suppose two independent cohorts each report one harmonized variant. At a fixed candidate effect, cohort A has estimate \(0.18\) and standard error \(0.20\); cohort B has estimate \(0.008\) and standard error \(0.010\). Let the candidate predict a marginal estimate of zero in both cohorts. Each cohort has \(R_a=(1)\), and assume the cohorts are independent.

For one estimate \(y\) with predicted mean \(\mu\) and standard error \(s\), the log likelihood is

\ell=-\frac{1}{2}\log(2\pi s^2)-\frac{1}{2}\left(\frac{y-\mu}{s}\right)^2.

For cohort A, the standardized residual is \(0.18/0.20=0.90\), so

\ell_A=-\frac{1}{2}\log(2\pi(0.20)^2)-\frac{1}{2}(0.90)^2 \approx 0.2855.

For cohort B, the standardized residual is \(0.008/0.010=0.80\), so

\ell_B=-\frac{1}{2}\log(2\pi(0.010)^2)-\frac{1}{2}(0.80)^2 \approx 3.366.

Because the cohorts are independent, the joint log likelihood is \(0.2855+3.3662\approx 3.6517\), or \(3.652\) to three decimal places. A log density can be positive, particularly when the measurement scale is small; it is not a probability and need not lie between zero and one.

Rounding should be done after adding the unrounded contributions. For example, if two evaluated contributions are \(0.291768\) and \(3.340665\), their sum is \(3.632433\), which rounds to \(3.632\), not \(3.633\), to three decimal places. Adding separately rounded values can change the final reported digit.

Keeping LD and Precision Specific to Each Cohort

The product form does not require the cohorts to have the same LD matrix or standard errors. It requires each cohort’s own covariance model to be used in its own likelihood factor. The next example shows how different LD patterns change both the mean marginal summaries and the likelihood normalization.

Worked Example: Two Variants With Different Cohort LD

Consider two hypothetical independent cohorts and two harmonized variants. For both, use the common candidate effect vector \(\boldsymbol\beta=(0.20,0)^{\mathsf T}\). Cohort A has \(R_A\) with correlation \(0.20\) and standard errors \(0.20\) for each variant. Cohort B has correlation \(0.80\) and standard errors \(0.10\) for each. Suppose the observed estimates equal the model means in each cohort:

\hat{\boldsymbol b}_A=(0.20,0.04)^{\mathsf T}, \qquad \hat{\boldsymbol b}_B=(0.20,0.16)^{\mathsf T}.

These are the predicted means because \(R_A\boldsymbol\beta=(0.20,0.20(0.20))^{\mathsf T}=(0.20,0.04)^{\mathsf T}\), while \(R_B\boldsymbol\beta=(0.20,0.80(0.20))^{\mathsf T}=(0.20,0.16)^{\mathsf T}\). The residuals are zero, so both quadratic terms are zero. For two variants, the covariance determinant is \(SE_1^2SE_2^2(1-r^2)\).

For cohort A, this determinant is \((0.20)^4(1-0.20^2)=0.001536\). Its contribution is

\ell_A=-\log(2\pi)-\frac{1}{2}\log(0.001536)\approx 1.4014.

For cohort B, the determinant is \((0.10)^4(1-0.80^2)=0.000036\), giving

\ell_B=-\log(2\pi)-\frac{1}{2}\log(0.000036)\approx 3.2781.

The joint log likelihood is therefore approximately \(1.4014+3.2781=4.6795\). The second marginal estimate differs across cohorts because the LD differs, even though the candidate effect vector is the same. The distinct standard errors also change the density scale. Neither feature should be erased by substituting a pooled LD matrix or common standard error.

Derivation and Conditions for the Joint Model

A short derivation explains why LD enters the mean. Let \(X_a\) be the cohort’s genotype matrix with columns standardized to mean zero and variance one, and let \(y_a\) be a standardized trait vector. Under the linear model \(y_a=X_a\boldsymbol\beta+\boldsymbol\varepsilon_a\), the vector of marginal regression estimates is approximately \(\hat{\boldsymbol b}_a=X_a^{\mathsf T}y_a/n_a\). Conditional on the genotype matrix, its expectation is

E(\hat{\boldsymbol b}_a\mid X_a) =\frac{X_a^{\mathsf T}X_a}{n_a}\boldsymbol\beta \approx R_a\boldsymbol\beta.

The approximation uses the empirical genotype correlation matrix \(X_a^{\mathsf T}X_a/n_a\) as an estimate of \(R_a\). If the residuals are independent with common variance \(\sigma^2\), the conditional covariance is

\operatorname{Cov}(\hat{\boldsymbol b}_a\mid X_a) =\frac{\sigma^2}{n_a}\frac{X_a^{\mathsf T}X_a}{n_a} \approx \frac{\sigma^2}{n_a}R_a.

This special case has a common standard error across standardized variants. In practice, the working covariance \(S_aR_aS_a\) allows standard errors to differ, for example because of variant-specific sample sizes. It should be treated as an assumption to assess, not as an identity guaranteed by having standard errors and an LD reference. The Gaussian approximation also presumes the estimates are sufficiently well approximated by a multivariate normal distribution.

For the joint product to represent the data, allele directions and effect units must be harmonized, the LD reference must correspond adequately to the cohort, and the same variants must be aligned in the same order. The samples should be independent, or any cross-cohort covariance must be represented. The common \(\boldsymbol\beta\) further assumes effects are on a comparable scale and shared across cohorts. If that assumption is not appropriate, a different parameterization is required; the product across independent cohorts alone does not establish effect equality.

Specify each cohort’s summaries.
Align variants and effect alleles, and record \(\hat{\boldsymbol b}_a\), \(SE_{aj}\), and a matched \(R_a\).

State the mean and covariance model.
For the working model here, use \(R_a\boldsymbol\beta\) and \(S_aR_aS_a\), and state the common-effect and Gaussian assumptions.

Evaluate each cohort likelihood.
Use that cohort’s residual, standard errors, and LD matrix in its determinant and quadratic terms.

Combine only under the appropriate sampling model.
For conditionally independent cohorts, add the unrounded log-likelihood contributions; otherwise model the dependence.

Common Mistakes and Practical Checks

Using one LD matrix for every cohort. This discards ancestry-specific correlation patterns. Use \(R_a\) in the likelihood for cohort \(a\).
Assuming independent SNP estimates within a cohort. Marginal estimates are generally correlated when genotypes are in LD. The covariance term \(S_aR_aS_a\) represents that dependence.
Multiplying likelihoods for overlapping samples without adjustment. Overlap can induce correlated sampling errors. A product of cohort likelihoods is justified only when the required conditional independence holds.
Confusing independent cohorts with a shared effect assumption. Independence concerns the sampling errors; a common \(\boldsymbol\beta\) is an additional model assumption.
Rounding before summing. Add the cohort log likelihoods at full available precision, then round the total. Otherwise the displayed total may not match the correctly rounded sum.
Using an ordinary inverse for singular LD. Perfectly correlated variants can make \(R_a\) singular, so the displayed density formula is not directly defined. The model or variant representation must be adjusted, for example by removing redundant variants or using a justified lower-dimensional formulation.

Key takeaway: A joint summary-statistic likelihood is a product of cohort-specific Gaussian likelihoods when cohort sampling errors are independent. Each factor retains its own LD matrix and standard errors; the validity of the product and of any shared-effect parameterization rests on explicit assumptions.

Check Your Understanding

Use the definitions and assumptions above to answer the following.

In the stated working model, what determines the expected marginal estimate vector in cohort \(a\)?
Why does the covariance matrix contain both \(S_a\) and \(R_a\), rather than only the squared standard errors on its diagonal?
Two cohorts have non-overlapping samples but different LD matrices. Which quantities differ in their likelihood factors, and why?
What assumption justifies adding the cohort log likelihoods? What changes if the samples overlap?
Why is a common effect vector an assumption distinct from independence of the cohorts?