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Causal graphical models in systems genetics: A unified framework for joint inference of causal network and genetic architecture for correlated phenotypes

Elias Chaibub Neto, Mark P. Keller, Alan D. Attie, Brian S. Yandell

arXiv:1010.1402v1stat.AP

TL;DR

Existing approaches may estimate QTLs without accounting for phenotype-network structure, even though genetic architectures can be complex. QTLnet jointly infers the network and genetic architecture using QTLs and homogeneous conditional Gaussian regression. The authors report that this approach avoids false detection of indirect-effect QTLs and improves phenotype-network inference.

  • Problem

    Existing methods assume QTLs are known or infer them without regard to phenotype-network structure, although genetic architectures may be complex and poorly estimated architectures may compromise causal inference.

    Source: §1 · p. 2

  • Method

    QTLnet jointly infers phenotype-network structure and genetic architecture with homogeneous conditional Gaussian regression, using an iterative MCMC approach.

    Source: §1 · p. 2

  • Results

    Tailoring QTL mapping to inferred network structure avoids false detection of QTLs with indirect effects and improves phenotype-network structure inference.

    Source: §1 · p. 2

  • Takeaways & Limitations

    For co-mapping traits, QTLnet is expected to be most fruitful when strong phenotype correlation reflects a strong causal signal.

    Source: §8 · p. 18

    The approach is restricted to directed acyclic graphs and can infer phenotype causality only under assumptions of no latent variables and no measurement error.

    Source: §8 · p. 19

Abstract

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Causal inference approaches in systems genetics exploit quantitative trait loci (QTL) genotypes to infer causal relationships among phenotypes. The genetic architecture of each phenotype may be complex, and poorly estimated genetic architectures may compromise the inference of causal relationships among phenotypes. Existing methods assume QTLs are known or inferred without regard to the phenotype network structure. In this paper we develop a QTL-driven phenotype network method (QTLnet) to jointly infer a causal phenotype network and associated genetic architecture for sets of correlated phenotypes. Randomization of alleles during meiosis and the unidirectional influence of genotype on phenotype allow the inference of QTLs causal to phenotypes. Causal relationships among phenotypes can be inferred using these QTL nodes, enabling us to distinguish among phenotype networks that would otherwise be distribution equivalent. We jointly model phenotypes and QTLs using homogeneous conditional Gaussian regression models, and we derive a graphical criterion for distribution equivalence. We validate the QTLnet approach in a simulation study. Finally, we illustrate with simulated data and a real example how QTLnet can be used to infer both direct and indirect effects of QTLs and phenotypes that co-map to a genomic region.

1. Introduction.

Existing QTL-mapping methods can misestimate phenotype genetic architectures when they ignore the phenotype network, compromising causal inference. QTLnet jointly infers the network and genetic architecture, using QTL nodes to help distinguish otherwise distribution-equivalent phenotype networks.

1. Introduction.

  • Ignoring phenotype networks during QTL mapping can incorrectly infer indirect-effect QTLs, biasing genetic architectures and potentially compromising causal-relationship inference.The paper aims to sort out direct and indirect effects when phenotypes co-map to genomic regions.Source: §1 · p. 2
  • QTLnet jointly infers phenotype-network structure and genetic architecture, iteratively updating both with MCMC and summarizing network structures by Bayesian model averaging.Genetic architecture is inferred conditional on the phenotype network.Source: §1 · p. 2
  • Allele randomization during meiosis and genotype-to-phenotype direction support causal QTL inference, while QTL nodes help distinguish otherwise distribution-equivalent phenotype networks.The paper combines randomized-experiment reasoning for QTL–phenotype links with conditional-probability reasoning for phenotype relationships.Source: §1 · p. 2
  • For co-mapping phenotypes, QTLnet aims to distinguish direct and indirect QTL and phenotype effects in situations where a hot spot may reflect an upstream phenotype.The paper describes this as an objective of QTLnet, not as a general claim about all hot spots.Source: §1 · p. 2

2. HCGR genetic model.

The genetic model jointly represents QTL genotypes and continuous phenotypes with homogeneous conditional Gaussian regression. It models phenotype dependence through the phenotype network while allowing trait-specific genetic architectures.

2. HCGR genetic model.

  • The model jointly represents phenotypes and QTL genotypes with homogeneous conditional Gaussian regression, with phenotype-network structure explicitly determining dependence among traits.Conditional on QTL genotypes and covariates, phenotypes follow a multivariate normal distribution.Source: §2 · p. 3
  • Given QTL genotypes, marker genotypes provide no additional information about phenotypes; the recombination model p(q | m) is not estimated in this paper.The model partitions the problem into phenotype-to-QTL genetic modeling and QTL-to-marker recombination modeling.Source: §2 · p. 3
  • Each trait’s mean includes genetic-effect predictors and covariates, plus effects from its parent phenotype nodes in the network.Genetic effects may follow Cockerham’s model but are not restricted to it.Source: §2 · p. 4
  • The covariance structure depends on phenotype relationships rather than genetic architectures, and the model permits cyclic as well as acyclic networks, though the paper focuses on acyclic networks.Different phenotypes may have different genetic architectures.Source: §2 · p. 4

3. A causal framework for systems genetics.

QTL nodes can break equivalence among phenotype-network models, supporting causal orientation beyond what phenotype distributions alone distinguish. For the HCGR family, distribution equivalence has a graphical criterion based on skeletons and v-structures.

3. A causal framework for systems genetics.

  • Adding causal QTL nodes can break phenotype-network Markov equivalence by creating new conditional-independence relationships among nodes.For HCGR models, Markov equivalence and distribution equivalence coincide.Source: §3 · pp. 5, 8
  • Allele randomization and genotype precedence support causal QTL-to-phenotype claims, whereas phenotype-to-phenotype causal claims additionally rely on conditional independence.Phenotype precedence cannot generally be determined a priori.Source: §3 · p. 6
  • When each connected phenotype pair has a QTL affecting one phenotype but not the other, adding QTL nodes makes the extended graphs non-Markov-equivalent.This condition can reduce an equivalence class, possibly to a single network.Source: §3 · pp. 7–8
  • In the illustrative three-phenotype case, a QTL affecting only the middle phenotype yields extended models with different v-structures despite retaining the same skeleton.The original phenotype models share the conditional independence Y1 ⊥⊥ Y3 | Y2.Source: §3 · p. 7
  • For the HCGR family, two DAGs are distribution equivalent if and only if they have the same skeletons and v-structures.This criterion permits checking equivalence from graph structure rather than algebraic manipulation of joint distributions.Source: §3 · p. 8

4. QTL mapping and phenotype network structure.

Conditional LOD scores provide a formal test of conditional independence between phenotypes and QTLs, including with interactive covariates. Mapping conditioned on the phenotype network can separate direct from indirect QTL effects, while conditioning on downstream phenotypes can produce misleading causal directions.

4. QTL mapping and phenotype network structure.

  • Conditional LOD scores formally measure conditional independence between phenotypes and QTLs and can handle interactive covariates.Unlike partial correlations, they do not require multinormality to test independence.Source: §4 · pp. 8–9
  • Single-trait QTL mapping can detect both direct effects and indirect effects from QTLs acting through upstream phenotypes.The example figure distinguishes direct from indirect QTL–phenotype relationships with dashed and pointed arrows.Source: §4 · p. 9, Figure 2 · p. 9
  • When the phenotype structure is correct, mapping each phenotype conditional on its parents avoids detecting indirect QTLs as direct effects.For Y5 conditional on Y2, Y3, and Y4, only Q5 is detected.Source: §4 · p. 9
  • Conditioning on downstream phenotypes can make downstream QTLs appear causal for upstream phenotypes, but model selection generally favors the correct causal order.The example identifies Q5 incorrectly for Y4 when conditioning on Y5; the reversed model generally has lower marginal likelihood.Source: §4 · p. 10

5. QTLnet algorithm.

QTLnet jointly infers phenotype-network structure and genetic architecture, using iterative network and QTL updates. Its algorithm searches network structures with Metropolis–Hastings sampling and averages causal relationships across models.

5. QTLnet algorithm.

  • QTLnet jointly infers phenotype-network structure and genetic architecture rather than treating QTL inference and network reconstruction separately.The method uses genetic architecture conditional on the phenotype network and iterates between updating both.Source: §5 · p. 10
  • The QTL mapping strategy can incorporate QTLs with indirect effects into phenotype genetic architectures.Source: §5 · p. 11
  • Because phenotype-network graph space grows super-exponentially, exhaustive search is impractical even for small networks.The algorithm uses Metropolis–Hastings proposals that add or drop an edge or change causal direction, remapping affected phenotypes.Source: §5 · p. 11
  • For a directed acyclic graph, the likelihood factors into regression models for each phenotype given its QTLs and parent phenotypes.QTL genotypes enter through the model mean.Source: §5 · p. 11
  • Bayesian model averaging produces an averaged network because the highest-posterior structure may still have low probability.Causal relationships are included when they have maximum posterior probability or exceed a predetermined threshold.Source: §5 · pp. 11–12

6. Simulations.

QTLnet recovered important features of the simulated causal networks, with stronger recovery under strong signals than weak ones. Conditioning QTL mapping on phenotype parents improved genetic-architecture detection and helped explain apparent hot spots as indirect effects.

6. Simulations.

  • The simulations used 1,000 five-phenotype data sets with four causal QTLs, contrasting strong and weak genetic and phenotype-link signals.Each data set represented an F2 population of 500 individuals; Y3 had no QTL.Source: §6 · p. 12
  • Conditioning QTL mapping on phenotype parents improved detection of the correct genetic architecture in both strong- and weak-signal settings.In strong-signal simulations, unconditional mapping often detected indirect QTLs; conditioning on parent nodes increased detection of true architectures.Source: §6 · pp. 12–13, Table 1 · p. 13
  • In strong-signal simulations, the true model had the highest posterior probability in most runs, while weak-signal model rankings were widespread.Table 2 summarizes the frequency of each rank across 1,000 simulations.Source: §6 · p. 13, Table 2 · p. 14
  • Strong-signal simulations recovered the correct causal relationships with high probability; weak-signal results were weaker but pointed in the correct direction.These relationships were assessed by the most probable direction or absence of a direct relation for each phenotype pair.Source: §6 · pp. 13–14, Table 3 · p. 14
  • In strong-signal simulations, Y5 mapped to upstream Q1 more often than to Q5 when phenotype links and the two QTL effects and variances were similar.Under these conditions, cor(y5,q1) can exceed cor(y5,q5), making an indirect QTL appear associated with Y5.Source: §6 · pp. 14–15
  • Unconditional mapping produced an apparent Q1 hot spot for all phenotypes in the strong-signal setting, whereas weak-signal simulations mostly mapped phenotypes to their respective QTLs.No hot spot appeared in additional simulations with strong QTL–phenotype relationships but weak phenotype–phenotype relations.Source: §6 · p. 15

7. Network inference for a liver hot spot.

In the liver-expression example, QTLnet inferred a model-averaged network in which Il16 is upstream of the other transcripts and is the only transcript directly mapping to the chromosome 2 locus. The result offers a hypothesis that another genetic factor at that locus may drive the pathway, rather than the expected cis transcript Pscdbp.

7. Network inference for a liver hot spot.

  • The example included 14 transcripts mapping to the same chromosome 2 region, with strong correlations that remained after adjusting for the peak marker.The authors note that causal relationships could explain the correlations, but environmental or unobserved latent factors are other possibilities.Source: §7 · pp. 15–16
  • QTLnet used 129 mice without missing transcript data and Bayesian model averaging over posterior samples from a 1,000,000-iteration Markov chain.The authors report that diagnostic plots and measures supported chain convergence.Source: §7 · p. 16
  • Il16 was inferred to be upstream of all other transcripts and was the only one directly mapping to the chromosome 2 locus.The model-averaged network summarizes the most probable causal relation for each phenotype pair; arrow darkness represents posterior probability.Source: §7 · pp. 16–17, Figure 4 · p. 16
  • Although Pscdbp was expected to be the upstream phenotype, the inferred network made it causal to only two other transcripts.The authors suggest another genetic factor on chromosome 2 may drive the pathway; the network provides hypotheses for future mouse experiments.Source: §7 · p. 17

8. Discussion.

QTLnet jointly infers genetic architecture and causal phenotype structure, using causal QTLs to distinguish networks that would otherwise be likelihood equivalent. The discussion highlights co-mapping settings and limits in population scope, graph assumptions, and computational scalability.

8. Discussion.

  • QTLnet jointly infers genetic architecture and causal phenotype structure; incorporating causal QTLs can break likelihood equivalence, while ignoring network structure may introduce indirect QTL effects and reduce power to detect causal relationships.The framework grounds QTL-to-phenotype causality in allele randomization during meiosis and unidirectional genotype influence.Source: §8 · pp. 17, 19
  • Earlier systems-genetics approaches often inferred genetic architecture and phenotype structure separately; Liu et al. (2008) partially incorporated network structure, but local models could include indirect eQTLs.Their local structural models fit at most two candidate regulators per target transcript; the discussion also describes Zhu et al. (2008)'s network reconstruction from Bayesian networks and single-trait QTLs.Source: §8 · pp. 17–18
  • Against Winrow et al. (2009), QTLnet performed better at high signal-to-noise, while weak-signal results depended on whether a direct causal link existed and were comparable on average.QTLnet more often detected direct links in the weak-signal setting, whereas Winrow's approach did better when no direct link existed; Winrow's method could miss direct QTLs under strong phenotype effects.Source: §8 · p. 18
  • QTLnet may be most fruitful when co-mapping traits have strong causal correlation, a setting in which single-trait analysis can produce a well-defined hot-spot pattern.The authors connect this setting to a path-analysis argument and their simulations and real-data example.Source: §8 · p. 18
  • QTLnet alternates phenotype-network proposals with genetic-architecture fitting conditional on each proposed network, because each inferred component can affect the other's marginal-likelihood assessment.A poorly inferred genetic architecture may compromise the marginal likelihood of a network whose phenotype structure is close to the truth.Source: §8 · p. 18
  • Single-edge Metropolis–Hastings proposals mix slowly, requiring long chains and wide thinning windows and creating a scalability bottleneck.The authors propose investigating more extensive edge-reversal moves and Markov-blanket decomposition.Source: §8 · pp. 18–19
  • The real-data analysis took over 20 hours, the current implementation handles up to 20 phenotype nodes, and development of network-structure priors is ongoing.The authors note that DAG counts grow super-exponentially with phenotype-node count and that code optimization is under development.Source: §8 · p. 19
  • The framework focuses on experimental crosses with inbred founders and directed acyclic graphs, and its causal inference assumes no latent variables or measurement error.Outbred-population extensions would require random genotypic effects and a variance-components formulation; latent variables and measurement error can affect reconstruction.Source: §8 · p. 19
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