Why Study Design Determines What Can Be Learned
A bivariate genomic model can assign cross-trait covariance to shared additive genetic effects and to residual dependence between the two measurements on the same person. These sources have different covariance patterns: genetic covariance follows genomic relatedness, whereas same-person residual covariance follows whether the observations come from one individual. A study can distinguish them only if its observed pairs of trait measurements provide enough variation in those patterns.
This is a question of identifiability, not simply sample size. A large study may still fail to identify two covariance components if the observed data make their model-implied patterns indistinguishable. Conversely, a well-chosen relatedness and overlap structure can provide the distinct information needed to separate them.
Covariance Patterns in a Bivariate Model
Suppose \(y_{1i}\) and \(y_{2i}\) denote adjusted measurements of two traits for individual \(i\). For the additive genetic effects, let \(\sigma_{g,12}\) be their cross-trait covariance, and let \(K_{ij}\) be the genomic relatedness between individuals \(i\) and \(j\). For residual effects, let \(\sigma_{e,12}\) be the cross-trait residual covariance for the same individual. The model assumes that additive genetic effects have covariance proportional to \(K\), while residual effects are independent across different individuals.
The model is a covariance description, not a claim that every observed pair has covariance equal to an estimated component without error. It also relies on the design and model being appropriate: relevant covariates have been handled, the relatedness matrix represents the intended genetic relationships, and there is no unmodelled cross-person residual dependence that is being mistaken for genetic covariance. In particular, shared environments among relatives can create cross-person resemblance not explained by additive genetic relatedness.
The model can also be written using covariance matrices. If the same \(n\) individuals have both traits, then the cross-trait block of the joint covariance matrix is $$ \operatorname{Cov}(\mathbf y_1,\mathbf y_2)=\sigma_{g,12}K+\sigma_{e,12}I_n, $$ where \(\mathbf y_1\) and \(\mathbf y_2\) are the two trait vectors and \(I_n\) is the \(n\)-dimensional identity matrix. With incomplete trait availability, only the entries corresponding to observed measurements contribute. With distinct cohorts, the genetic term instead uses the relatedness matrix between the trait-1 and trait-2 participants, and there is no same-person residual term if no person has both traits.
A Practical Identifiability Criterion
In a covariance model, a parameter is identifiable when different values of that parameter cannot produce exactly the same model-implied distribution for the observed data, after accounting for the other parameters. For the two cross-trait components here, the relevant design information is the pattern of \(K_{ij}\) values and whether each pair is the same person. Restrict these patterns to the cross-trait pairs actually observed.
The criterion concerns exact distinction in the model, not whether a study has enough information for a precise estimate. Even when the patterns are linearly independent, they may be so similar, or the informative pairs so few, that estimates are unstable. Identifiability is therefore necessary for learning a component separately, but does not guarantee good precision.
Worked Example: Both Traits Measured in the Same Participants
Consider a hypothetical study in which both traits are measured on the same four individuals. Suppose the genomic relatedness matrix has diagonal entries \(1\), a value of \(0.25\) for one relative pair, and \(0\) for an unrelated pair. For illustration, suppose the model components are \(\sigma_{g,12}=0.12\) and \(\sigma_{e,12}=0.08\). These are hypothetical component values, not estimates from observed data.
For the same individual, \(K_{ii}=1\) and \(I(i=i)=1\), so the model-implied cross-trait covariance is $$ 0.12(1)+0.08(1)=0.20. $$
For the two distinct relatives, \(K_{ij}=0.25\) and \(I(i=j)=0\), giving $$ 0.12(0.25)+0.08(0)=0.030. $$
For a distinct unrelated pair, \(K_{ij}=0\) and \(I(i=j)=0\), giving $$ 0.12(0)+0.08(0)=0. $$
These are different design patterns: same-person pairs contain both components, while distinct related pairs carry a genetic contribution without the same-person residual contribution. In an actual study the component values are unknown, but the presence of these distinct patterns can allow them to be distinguished by the model.
Why Relatedness Among Different People Matters
Same-person pairs alone do not generally distinguish genetic from residual cross-trait covariance. For each person with both traits, the cross-trait covariance is \(\sigma_{g,12}K_{ii}+\sigma_{e,12}\). If the diagonal of \(K\) is \(1\), that is the sum \(\sigma_{g,12}+\sigma_{e,12}\). The same-person measurements reveal that sum, but a separate pattern is needed to attribute some of it to genetic covariance.
Pairs of distinct people can provide that pattern because their residual cross-covariance is zero under the model, while their genetic cross-covariance is \(\sigma_{g,12}K_{ij}\). Related pairs with nonzero \(K_{ij}\) therefore supply information about genetic covariance that cannot be supplied by same-person pairs alone. If all available distinct pairs have \(K_{ij}=0\), they do not provide that genetic signal.
Worked Example: Overlap Without Informative Relatedness
Suppose a hypothetical data set contains both traits for several people, but all observed distinct-person pairs have \(K_{ij}=0\). Assume \(K_{ii}=1\). The only informative cross-trait covariance pattern is then the same-person pattern: $$ \operatorname{Cov}(y_{1i},y_{2i})=\sigma_{g,12}+\sigma_{e,12}. $$
For example, the hypothetical values \(\sigma_{g,12}=0.12\) and \(\sigma_{e,12}=0.08\) imply a same-person covariance of $$ 0.12+0.08=0.20. $$ But the different values \(\sigma_{g,12}=0.05\) and \(\sigma_{e,12}=0.15\) imply exactly the same covariance: $$ 0.05+0.15=0.20. $$
Because these two parameter pairs give the same observed cross-trait covariance pattern, the two components cannot be separated using this design information alone. Having more same-person observations of the same pattern may improve information about their sum, but it does not by itself resolve the exact confounding.
Separate Cohorts and Cross-Cohort Relatedness
A study need not measure both traits on the same people to contain information about genetic covariance. If trait 1 is measured in cohort A and trait 2 in cohort B, then cross-trait pairs are between different individuals. Under the model, their residual cross-covariance is zero, but their genetic cross-covariance is \(\sigma_{g,12}K_{ij}\). The relevant design information is therefore the set of relatedness values between participants in the two cohorts.
This requires that the cohorts be linked by usable genetic relatedness information and that the model’s assumptions about the traits and genetic effects apply across them. If all cross-cohort relatedness values are zero, the cross-trait covariance block predicted by this individual-level model is zero regardless of \(\sigma_{g,12}\); that block then provides no information to identify the genetic cross-trait covariance. Merely studying both traits in genetically unrelated samples does not create cross-person covariance information.
Worked Example: Separate but Genetically Connected Cohorts
Suppose, hypothetically, cohort A provides trait 1 and cohort B provides trait 2, with no participants in common. Assume the cross-cohort genomic relatedness matrix contains entries \(K_{12}=0.20\) for one pair and \(K_{34}=0.05\) for another. If \(\sigma_{g,12}=0.12\), the model-implied cross-trait covariances for those pairs are $$ 0.12(0.20)=0.024 \qquad\text{and}\qquad 0.12(0.05)=0.006. $$
There is no \(\sigma_{e,12}I(i=j)\) contribution because the cohorts contain different individuals. If the cross-cohort relatedness pattern is known and informative, its contribution can distinguish genetic covariance values: for example, a value of \(0.10\) would instead imply \(0.10(0.20)=0.020\) for the first pair. This is a design-based source of information, not a guarantee that an estimate will be precise.
If, by contrast, every cross-cohort \(K_{ij}\) is zero, then for every observed pair $$ \sigma_{g,12}K_{ij}=\sigma_{g,12}(0)=0. $$ The cross-trait covariance block does not change with \(\sigma_{g,12}\), so this design cannot identify that parameter through these cross-cohort pairs.
Partial Overlap and Missing Trait Measurements
Many studies lie between complete overlap and entirely separate cohorts. Some people have both traits, some have only one, and related individuals may contribute different traits. The model uses only pairs for which the relevant trait measurements are available. Thus, the key question is not simply “How many people overlap?” but “Which same-person and between-person relatedness patterns are present among the observed cross-trait pairs?”
A small overlap can be informative about residual cross-trait covariance if there are also related but distinct people whose trait measurements form cross-trait pairs. Conversely, a large overlap may leave genetic and residual covariance difficult to distinguish if the only available cross-trait pattern is the same-person diagonal. The full model can include other covariance components; for identifiability, all the relevant observed component patterns must be distinguishable jointly, not just one pair at a time.
Record which individuals have trait 1, trait 2, or both, so the observed cross-trait pairs are explicit.
For each observed pair, note \(K_{ij}\) and whether \(i=j\). Do not infer informativeness from cohort labels alone.
Check whether relatedness and same-person indicators, along with any other model components, vary independently on the observed entries.
If the patterns are distinct but nearly indistinguishable or supported by few pairs, the components may be identifiable yet estimated imprecisely.
Deriving the Pattern Criterion
The criterion follows directly from the cross-trait covariance equation. Index the observed cross-trait pairs by \(m\), and write \(k_m\) for their relatedness and \(d_m\) for their same-person indicator. Their model-implied covariances have the form $$ c_m=\sigma_{g,12}k_m+\sigma_{e,12}d_m. $$ Collect the pair-specific quantities into vectors \(\mathbf c\), \(\mathbf k\), and \(\mathbf d\). Then $$ \mathbf c=\sigma_{g,12}\mathbf k+\sigma_{e,12}\mathbf d. $$
If \(\mathbf k\) and \(\mathbf d\) are linearly independent, the coefficients in this representation are uniquely determined by \(\mathbf c\). To see why, suppose two parameter pairs produce the same \(\mathbf c\). Subtracting their equations gives $$ (\sigma_{g,12}-\sigma'_{g,12})\mathbf k+ (\sigma_{e,12}-\sigma'_{e,12})\mathbf d=\mathbf 0. $$ Linear independence forces both coefficient differences to be zero, so the parameter pairs are identical. If instead \(\mathbf k=a\mathbf d\) for some constant \(a\), then $$ \mathbf c=(a\sigma_{g,12}+\sigma_{e,12})\mathbf d, $$ which depends only on a combination of the two parameters. They cannot be uniquely separated from this pattern. With additional covariance components, the same argument applies to the full set of observed covariance-pattern vectors: no component can be isolated if its pattern is an exact linear combination of the others.
Common Mistake and Design Implications
Wrong reasoning: “Both traits were measured in the same large sample, so genetic covariance must be identifiable.” Overlap provides same-person cross-trait pairs, but those pairs can contain both genetic and residual covariance. Without other informative relatedness patterns, the two contributions may remain confounded.
Correction: Assess the covariance patterns supplied by the actual observed pairs. Same-person pairs and related-but-distinct pairs can have different roles; separate cohorts can also contribute if cross-cohort relatedness is informative. A large sample increases the number of observations, but it cannot fix an exact lack of distinction between model components.
Identifiability assessment helps researchers plan which participants to recruit and which traits to measure together. Measuring both traits in some individuals can reveal same-person covariance, while including related individuals with different trait availability can provide genetic-relatedness contrasts. The design should preserve the information needed for the intended covariance model, and the model should not attribute cross-person residual dependence to genetics without justification.
Check Your Understanding
Use the covariance model and observed-pair patterns to answer the following.
- For two different individuals with \(K_{ij}=0.10\), what is their model-implied cross-trait covariance under the stated residual assumption?
- Why do same-person pairs alone generally fail to separate genetic and residual cross-trait covariance when \(K_{ii}=1\)?
- Can separate cohorts inform genetic covariance if their cross-cohort relatedness values are all zero? Explain using the covariance formula.
- What feature of the observed covariance patterns is required to identify genetic and same-person residual cross-trait covariance separately?
- Explain why a covariance component can be identifiable but still estimated imprecisely.