Causal Graphical Models in Quantitative Genetics and Genomics

Transcription

1 Causal Graphical Models in Quantitative Genetics and Genomics Guilherme J. M. Rosa Department of Animal Sciences Department of Biostatistics & Medical Informatics

2 OUTLINE Introduction: Correlation and Causation; Randomized and Observational Studies Structural Equation Models (SEM) Estimating Phenotypic Functional Relationships Phenotype Network Reconstruction QTL Mapping; Mendelian Randomization Genomic Information Concluding Remarks

3 ASSOCIATION vs. CAUSATION z y z is associated with y x z y z y z y z causes y y causes z x causes z & y alcohol and drunkenness # police officers and criminality water consumption and dehydration

4 CAUSALITY Any factor that produces a change in the severity or frequency of the outcome (Dohoo et al. 2003)

5 RANDOMIZED EXPERIMENTS ð Testing the effect of z on y. x x z y z y Causal relationship between variables Effect of randomization applied to variable z

6 OBSERVATIONAL STUDIES ð Lack of randomization due to legal, ethical, or logistics reasons ð Potential bias and confounding effects ð Example (Parenthood and life expectancy; Agerbo et al. JECH 2013) The Economist, Dec 2012 Observation: childless die young Question: early mortality caused by absence of children or state of mind that leads some people not to have children Sample: 21,276 couples who sought IVF Results: women without children had annual rate of death four times greater; childless men had death rate twice greater Discussion: It is possible that those who cannot conceive through IVF are more at risk of death for reasons related to that inability

7 STRUCTURAL EQUATION MODELS ð Causal structure represented as a Directed Graph x 1 e 1 x 2 y 2 y 1 y 3 x 3 e 2 e 3 Example of a causal structure, in which y s represent measurements on three phenotypic traits, x s and e s represent known explanatory variables and residual factors affecting y s, respectively.

8 STRUCTURAL EQUATION MODELS ð Graph represented by a set of structural equations y y y = β = λ = λ 1 x y y + e β + λ 2 32 x 2 y 2 + e 2 + β 3 x 3 + e 3 Matrix notation: y = Λy + Xβ + e observations matrix of coefficients λ 0 Λ = λ λ λ residuals fixed effects exogenous covariates

9 STRUCTURAL EQUATION MODELS ð Reduced Model : ), N( ~, ) (, ) ( ), N( ~, *' * * * * * 1 * * * Λ DΛ 0 Λ e e e µ y Λ I Λ Λ e Xβ Λ y e Xβ y Λ I e Xβ Λy y D 0 e e Xβ Λy y = + = = + = + = + = + + =

10 STRUCTURAL EQUATION MODELS ð Reduced Model : Toy Example z y = = 2y 3 z y z y = = 6 3

11 EXAMPLE OF APPLICATION ð Completely randomized experiment to compare two treatments in terms of their effect on ribeye area in beef cattle. ð Data collected on ribeye area (RIB) and body weight (BW). ð Results indicate a significant effect of treatment on RIB. ð Results indicate also a significant effect of treatment on BW ð However, when BW is included as a covariate in the model for RIB, the treatment effect on RIB becomes non-significant. Trt Trt BW RIB BW RIB

12 STRUCTURAL EQUATION MODELS ð Competing networks may be compared using some model selection criteria (e.g., LRT, AIC, BIC, or Bayesian approaches) ð SEM intensively used in many fields, such as economics, psychometrics, social statistics, and biological sciences. ð More recently, in quantitative genetics in the context of mixed model analysis (e.g., Gianola and Sorensen 2004)

13 SEM IN QUANTITATIVE GENETICS (Gianola and Sorensen, 2004) y = Xβ + u + e r Y1,Y 2 = h 1 h 2 r U1,U 2 + e 1 e 2 r E1,E 2 u ~ N( 0, A 2 σ a )

14 SEM IN QUANTITATIVE GENETICS (Gianola and Sorensen, 2004) y = Λy + Xβ + u + e

15 EXAMPLE OF APPLICATION Maturana EL, Wu X-L, Gianola D, Weigel KA and Rosa GJM. Exploring biological relationships between calving traits in primiparous cattle with a Bayesian recursive model. Genetics 181: , 2009.

16 EXAMPLE OF APPLICATION

17 INFERRING CAUSAL PHENOTYPE NETWORKS ð d-separation ( directed separation ) concept: Verma and Pearl (1988), Pearl (1998), Geiger et al. (1990) Two variables X and Y are said to be d-separated by Q if there is no active path between any X and Y conditionally on Q. Example 1: y 2 is not a collider y 3 is not independent of y 1 Conditioned on y 2, y 3 becomes independent of y 1 Example 2: y 2 is a collider y 1 and y 3 unconditionally independent y 1 and y 3 not independent, conditioned on y 2

18 THE IC ALGORITHM (Inductive Causation; Verma and Pearl 1991) Step 1: Undirected graph (search for d-separations; connect adjacent variables) Step 2: Partially oriented graph (search for colliders) Step 3: Attempt to orient remaining undirected edges such that no new colliders or cycles are generated

19 INFERRING CAUSAL PHENOTYPE NETWORKS ð Valente et al. (2010): Correlated genetic effects act as an additional source of phenotypic covariance, which may confound the search for causal structures. (Valente et al. 2010) Example of network involving five phenotypic (observable) traits, and their corresponding additive genetic (u s) and residual (e s) effects; arcs connecting u s represents genetic correlations.

20 INFERRING CAUSAL PHENOTYPE NETWORKS ð Valente et al. (2010): To restore the connection between causal structures and joint density, the joint distribution of phenotypes conditionally on additive genetic effects is used. 1. Fit a multiple trait model where additive genetic effects could be predicted based on pedigree information. 2. Apply the IC algorithm to this matrix, returning a class of equivalent causal structures (i.e. causal structures that results in the same conditional independencies in the joint probability distribution). 3. Fit final structural equation model using the selected causal structure. Valente BD, Rosa GJM, de los Campos G, Gianola D and Silva MA. Searching for recursive causal structures in multivariate quantitative genetics mixed models. Genetics 185: , 2010.

21 ð Valente et al. (2010): Simulation study 5 phenotypic traits 300 inbred lines 6 individuals per line Model from which simulated data were drawn. Undirected acyclic graph resulting from step 1 of the IC algorithm Partially oriented graph retrieved by the IC algorithm Structure chosen given prior beliefs on the direction of the edge between y 1 and y 2 (pointing towards y 2 )

22 EXAMPLE OF APPLICATION Reproductive Traits in European Quail 892 females with observations Pedigree: 10,291 animals 5 phenotypic traits: Birth weight Weight at 35d Age at sexual maturity (1 st egg) Average egg weight Rate of lay (# eggs) Valente BD, Rosa GJM, Silva MA, Teixeira RB and Torres RA. Searching for phenotypic causal networks involving complex traits: an application to European quails. Genet. Sel. Evol. 43:37, 2011.

23 EXAMPLE OF APPLICATION BW W35 AFE NE AEW BW W35 AFE NE AEW Time Birth weight (BW) Weight at 35d (W35) Sexual maturity: Age at 1 st egg (AFE) Average egg weight (AEW) and Number of eggs (NE)

24 QTL INFORMATION AND THE RANDOMIZATION OF ALLELES ð Meiosis randomizes alleles to gametes; QTL information can be used to aid causal inference ð Schadt et al. (2005): Independent Causal Reactive Likelihood-based causality model selection (LCMS): C : p(g, t,c) = p(g)p(t g)p(c t) R : p(g, t,c) = p(g)p(c g)p(t c) I : p(g, t,c) = p(g)p(t g)p(c g, t)

25 QTL INFORMATION AND THE RANDOMIZATION OF ALLELES ð Li et al. (2006): Extension of Schadt et al. (2005) s method for multiple loci and phenotypic traits, and more general causal relationships among them. (adapted from Li et al. 2006) Causal relationships among a QTL (g) and two correlated phenotypes (y 1 and y 2 ). Arrows: causal effects; dotted lines: unresolved associations.

26 QTL INFORMATION AND THE RANDOMIZATION OF ALLELES ð Five steps (Li et al. 2006): 1. Single locus genome scans are run for each individual phenotype 2. Conditional genome scans are performed using one trait as a covariate in the analysis of another trait 3. Construction of an initial path model and its respective SEM representation 4. Path models are assessed in terms of goodness-of-fit 5. Additional step to refine the model

27 QTL INFORMATION AND THE RANDOMIZATION OF ALLELES ð Chaibub Neto et al. (2008): QTL information used to orient edges connecting phenotypes; two main steps: 1. Association network is constructed using either an undirected dependency graph (UDG; Shipley 2002) or a skeleton derived from the PC algorithm (Spirtes et al. 2000) 2. LOD score tests are used to determine causal direction for every edge that connects a pair of phenotypes, conditional on QTLs affecting the phenotypes.

28 QTL INFORMATION AND THE RANDOMIZATION OF ALLELES ð Chaibub Neto et al. (2010): Traditional single-trait QTL mapping approaches ignore the phenotype network and result in poorly estimated genetic architecture of phenotypes. Example of network with five phenotypes and three QTLs. Expected output of single-trait QTL analyses. Solid and dashed arrows represent direct and indirect effects of QTLs on phenotypes, respectively.

29 QTL INFORMATION AND THE RANDOMIZATION OF ALLELES ð Chaibub Neto et al. (2010): Methodology that simultaneously infers a causal phenotype network and its associated genetic architecture. Their approach is based on jointly modeling phenotypes and QTLs using homogeneous conditional Gaussian regression models and a graphical criterion for model equivalence. The concept of randomization of alleles during meiosis and the unidirectional relationship from genotype to phenotype are used to infer causal effects of QTLs on phenotypes. Subsequently, causal relationships among phenotypes are inferred using the QTL nodes, which might enable distinguishing among phenotype networks that would otherwise be distribution equivalent.

30 COMMENTS QTL can be used as parent nodes on putative networks, facilitating inferences on the remaining of the network QTL mapping methodology, however, low power and inaccurate location estimates Moreover, complex traits generally affected by a fairly large number of genes (of small effects), and polygenic component masks phenotypic causal relationships Ongoing research: 1. Genomic relationship matrix 2. Intermediate phenotypes (e.g., gene expression data) 3. Multi-trait GWAS using SEM 4. Multi-trait selection

31 GENOMIC INFORMATION High Density DNA Polymorphism Chips (SNP chips) Genomic Relationship Matrix (G) to Account for Polygenes Extension of Valente et al. (2010) s method Model: phenotypic traits functional relationships y = Λy + Xβ + u + e environmental effects residuals polygenes u ~ N(0,Gσ u 2 )

32 GENE EXPRESSION Genetical Genomics (eqtl Mapping) Growth and Carcass Traits in Pigs; F2 Population (Duroc x Pietran) 1,000 F2 progeny 25 growth traits 30 carcass traits 144 microsatellite loci SNPs and Microarrays Steibel JP, Bates RO, Rosa GJM, et al. Genome-wide linkage analysis of global gene expression in loin muscle tissue identifies candidate genes in pigs. PLoS One 6(2): e16766, 2011.

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34 GENOMIC INFORMATION High Density DNA Polymorphism Chips (SNP chips) GWAS for Detection of Major Genes Genomic Relationship Matrix (G) to Account for Polygenes Model: phenotypic traits functional relationships y = Λy + Xβ + Wq + Zu + e environmental effects major genes residuals polygenes u ~ N(0,Gσ u 2 )

35 EXAMPLE OF APPLICATION (Ongoing Research) Milk Fatty Acid Composition in Dairy Cattle 1,905 First Lactation Cows ( DIM) Paternal Half-sib Families, 398 Commercial Farms Fatty Acids (14 Total): C:4:0-C18:0, C10:1-C18:1, CLA 50K SNP Chip GWAS Model: DIM, Age, Season, Herd, Genetic Relationships SNPassoc in R and ASReml; FDR < 0.05

36 GWAS (Bouwman et al. BMC Genetics 12: 43, 2011)

37 GWAS (Bouwman et al. BMC Genetics 12: 43, 2011)

38 Milk Fatty Acids Network Schematic overview of major milk fatty acids synthesis pathways Bouwman et al. Structural equation models to study causal relationships between bovine milk fatty acids. 63rd EAAP, Bratislava, Slovakia, August 27-31, 2012.

39 SEM vs. MTM in ANIMAL BREEDING Litter weight at weaning (y 3 ) Prediction Litter size, 0 days (y 1 ) Litter size, 5 days (y 2 ) y1 = u1+ e1 y = λ y + u + e y = λ y + u + e Valente BD, Rosa GJM, Gianola D, Wu XL and Weigel KA. Is structural equation modeling advantageous for the genetic improvement of multiple traits? Genetics 2013 (in press)

40 SEM vs. MTM in ANIMAL BREEDING Litter weight at weaning (y 3 ) y1 = u1+ e1 y2 = c y = λ c+ u + e Litter size, 0 days (y 1 ) Litter size, 5 days (y 2 ) Cross fostering SEM MTM for predictions under stable conditions SEM superior if with modified scenarios (i.e., interventions)

41 CONCLUDING REMARKS ð SEM: flexible and powerful analysis strategy ð Assumptions: Markov condition, faithfulness, causal sufficiency ð QTL information (Mendelian Randomization) ð Genomic information (DNA polymorphism and gene expression) ð Prior knowledge ð Causal relationships knowledge to predict behavior of complex systems, e.g. biological pathways ð In agriculture, optimal decision-making strategies Rosa GJM et al. (2011) Inferring causal phenotype networks using structural equation models. Genetics Selection Evolution 43:6. Rosa GJM and BD Valente (2013) Inferring causal effects from observational data in livestock. Journal of Animal Science 91: