Using SEM for Genetic Effects Partitioning
How Structural Equation Modeling (SEM) can partition genetic effects of individual SNPs into maternal and fetal components. See real-world applications in birthweight GWAS studies and learn about disentangling mother and child effects through SEM. Discover the intricacies of conditional analysis, model building, and path tracing rules within a genetic context.
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Presentation Transcript
Using SEM to partition genetic effects of individual SNPs into maternal and fetal components David Evans University of Queensland
Post-doctoral Position in Statistical Genetics/Genomics
Objectives Illustrate the flexibility of SEM Show how SEM can be used to investigate molecular mechanisms Illustrate model building Illustrate the concept of identification Revise key concepts from the week
Birthweight GWAS (EGG and UKBB) Nicole Warrington UKBB and EGG consortium Birthweight GWAS reflects a mixture of maternal and fetal genetic effects Unrelated individuals*
Conditional Analysis of Genotyped Mother- Offspring Duos BWi = mSNPmi + cSNPci + i Conditional Regression: 1 SNP1 SNPm m 0.5 Structural Equation Model: c 1 1 SNPc BW SNP2 BUT- not many cohorts in the world with these data! Are twins suitable for these analyses?
Disentangling Mother and Child Effects on Birth Weight in UKBB UKBB contains self-reported birthweight and reported birthweight of first offspring GM m SNP = GM + SNP 0.5 1 BW c 1 BW = mGM + cSNP + 0.75 SNP SNP m BWO = cGO + mSNP + O 1 BWO O 0.5 c 0.75 GO
Tracing Rules of Path Analysis Find All Distinct Chains between Variables: Go backwards along zero or more single-headed arrows Change direction at one and only one Double-headed arrow Trace forwards along zero or more Single-headed arrows Multiply path coefficients in a chain Sum the results of step 2 For covariance of a variable with itself (Variance), chains are distinct if they have different paths or a different order
Building The Model: Path Tracing Rules SNP BW BWO GG m 0.5 c + m m + c SNP 1 BW c 1 0.75 SNP SNP c + m m2 + c2 + mc + var( ) m2 + c2 + mc + mc + m2 + c2 + mc + var( O) BW m 1 BWO O 0.5 c 0.75 GO m + c m2 + c2 + mc + mc + BWO
Building the Model: Covariance Algebra SNP = GG + SNP BW = mGG + cSNP + BWO = cGO + mSNP + O GG m 0.5 1 BW c 1 0.75 SNP SNP (1) cov(cX, Y) = c x cov(X, Y) m 1 BWO O 0.5 (2) cov(X + Y, Z) = cov(X, Z) + cov(Y, Z) c 0.75 GO (3) cov(X, X) = var(X) var(SNP)=cov (SNP, SNP) = cov( GG + SNP , GG + SNP ) = cov( GG , GG) + cov( GG , SNP ) + cov( SNP , GG) + cov( SNP , SNP ) = cov(GG , GG) + 0 + 0 + cov( SNP , SNP ) = var(GG) + var( SNP) =
Building the Model: Covariance Algebra SNP = GG + SNP BW = mGG + cSNP + BWO = cGO + mSNP + O GG m 0.5 1 BW c 1 0.75 SNP SNP (1) cov(cX, Y) = c x cov(X, Y) m 1 BWO O 0.5 (2) cov(X + Y, Z) = cov(X, Z) + cov(Y, Z) c 0.75 GO (3) cov(X, X) = var(X) cov(BW, SNP)=cov(mGG + cSNP + , GG + SNP ) = cov(mGG, GG) + cov(mGG, SNP) + cov(cSNP, GG) + cov(cSNP, SNP) + cov( , GG) + cov( , SNP) = mcov(GG,GG) + mcov(GG, SNP) + ccov(SNP,GG) + ccov(SNP, SNP) + cov( ,GG) + cov( , SNP) = mvar(GG) + 0 + c + c + 0 + 0 = m + c
Understanding SEM = ( ) Observed Sample Covariance Matrix Expected Covariance Matrix Expected covariance matrix a function of model parameters Parameters chosen to minimize the difference between observed and expected covariance matrices
Identification Means that all parameters in a model can be estimated uniquely given the data A necessary (but not sufficient condition) for identifiability is that you have the same (or more) observed statistics than parameters you want to estimate If all parameters in a model are identified, then the model as a whole is identified Even though the model as a whole may be unidentified some parameters may be identified
Identified or Not? (1) 1 + 2 = 10 (2) 1 + 2 = 10 1 - 2 = 0 (3) 1 + 2 = 10 2 1 +2 2 = 20
Identification in Twin Models OBSERVED EXPECTED VARMZ-T1 VA + VC + VE MZ = ( )MZ = COVMZVARMZ-T2 VA + VC + VE VA + VC VARDZ-T1 VA + VC + VE DZ = ( )DZ = VARDZ-T2 COVDZ VA + VC + VE VA + VC How many observed statistics? How many parameters? Why can t we model VA, VC, VD, VE
Identified or Not? SNP BW BWO GG m c + m m + c SNP 0.5 1 BW c c + m m2 + c2 + mc + var( ) m2 + c2 + mc + mc + m2 + c2 + mc + var( O) 1 BW 0.75 SNP SNP m 1 BWO O 0.5 c m + c m2 + c2 + mc + mc + BWO 0.75 GO How many observed statistics? How many parameters?
Identified or Not? SNP BW BWO GG m c + m m + c SNP 0.5 1 BW c c + m m2 + c2 + mc + var( ) m2 + c2 + mc + mc + m2 + c2 + mc + var( O) 1 BW 0.75 SNP SNP m 1 BWO O 0.5 c m + c m2 + c2 + mc + mc + BWO 0.75 GO = var(SNP)
Identified or Not? - c and m SNP BW BWO GG m c + m m + c SNP 0.5 1 BW c c + m m2 + c2 + mc + var( ) m2 + c2 + mc + mc + m2 + c2 + mc + var( O) 1 BW 0.75 SNP SNP m 1 BWO O 0.5 c m + c m2 + c2 + mc + mc + BWO 0.75 GO c + m = cov(BW, SNP) m + c = cov(BWO,SNP)
Identified or Not? var() SNP BW BWO GG m c + m m + c SNP 0.5 1 BW c c + m m2 + c2 + mc + var( ) m2 + c2 + mc + mc + m2 + c2 + mc + var( O) 1 BW 0.75 SNP SNP m 1 BWO O 0.5 c m + c m2 + c2 + mc + mc + BWO 0.75 GO m2 + c2 + mc + var( ) = var(BW)
Identified or Not? var(O) SNP BW BWO GG m c + m m + c SNP 0.5 1 BW c c + m m2 + c2 + mc + var( ) m2 + c2 + mc + mc + m2 + c2 + mc + var( O) 1 BW 0.75 SNP SNP m 1 BWO O 0.5 c m + c m2 + c2 + mc + mc + BWO 0.75 GO m2 + c2 + mc + var( O) = var(BWO)
Identified or Not? SNP BW BWO GG m c + m m + c SNP 0.5 1 BW c c + m m2 + c2 + mc + var( ) m2 + c2 + mc + mc + m2 + c2 + mc + var( O) 1 BW 0.75 SNP SNP m 1 BWO O 0.5 c m + c m2 + c2 + mc + mc + BWO 0.75 GO m2 + c2 + mc + mc + = cov(BW, BWO)
Disentangling Mother and Child Effects on Birth Weight SNP SBP SBPm GG SNP m 0.5 SBPm m c SBP 0.75 Gm m SBP 0.5 SBPm c 0.75 SNP
Disentangling Mother and Child Effects on Birth Weight SNP SBP SBPm GG c + m c + m SNP m 0.5 SBPm m c c + m m2 + c2 + mc + var( ) c2 + m2 + mc + mc + SBP 0.75 Gm m SBP 0.5 c + m m2 + c2 + mc + var( m) c2 + m2 + mc + mc + SBPm c 0.75 SNP
Identified or Not? SNP SBP SBPm GG c + m c + m SNP m 0.5 SBPm m c c + m m2 + c2 + mc + var( ) c2 + m2 + mc + mc + SBP 0.75 Gm m SBP 0.5 c + m m2 + c2 + mc + var( m) c2 + m2 + mc + mc + SBPm c 0.75 SNP How many observed statistics? How many parameters?
Identified or Not? SNP SBP SBPm GG c + m c + m SNP m 0.5 SBPm m c c + m m2 + c2 + mc + var( ) c2 + m2 + mc + mc + SBP 0.75 Gm m SBP 0.5 c + m m2 + c2 + mc + var( m) c2 + m2 + mc + mc + SBPm c 0.75 SNP
Identified or Not? c,m? SNP SBP SBPm GG c + m c + m SNP m 0.5 SBPm m c c + m m2 + c2 + mc + var( ) c2 + m2 + mc + mc + SBP 0.75 Gm m SBP 0.5 c + m m2 + c2 + mc + var( m) c2 + m2 + mc + mc + SBPm c 0.75 SNP
Intuition To estimate maternal effects, we need individuals with observed genotypes who have reported their offspring s phenotype In the first situation where we examine birthweight we have this In the second situation where we examine blood pressure we do not
