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Univariate Model Fitting Sarah Medland QIMR – openMx workshop Brisbane 16/08/10

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Univariate Twin Models = using the twin design to assess the aetiology of one trait (univariate) 1. Path Diagrams 2. Basic ACE R Script

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1. Path Diagrams for Univariate Models

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Basic Twin Model - MZ Twin 1 Trait ACE c e a Twin 2 Trait ECA c a e

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Basic Twin Model - DZ Twin 1 Trait ACE c e a Twin 2 Trait ECA c a e

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Basic Twin Model Twin 1 Trait ACE c e a Twin 2 Trait ECA c a e rMZ = 1.0; rDZ = 1.0 rMZ = 1.0; rDZ =

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The Variance Since the variance of a variable is the covariance of the variable with itself, the expected variance will be the sum of all paths from the variable to itself, which follow Wright’s rules

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Variance for Twin 1 - A Twin 1 Trait ACE c e a 11 1 a*1*a = a 2

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Variance for Twin 1 - C a*1*a = a 2 c*1*c = c 2 Twin 1 Trait ACE c e a 11 1

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Variance for Twin 1 - E a*1*a = a 2 c*1*c = c 2 e*1*e = e 2 Twin 1 Trait ACE c e a 11 1

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Total Variance for Twin 1 a*1*a = a 2 c*1*c = c 2 e*1*e = e 2 Twin 1 Trait ACE c e a 11 1 Total Variance = a 2 + c 2 + e 2

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Covariance - MZ Twin 1 Trait ACE c e a Twin 2 Trait ECA c a e a2 + c2a2 + c2

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Covariance - DZ Twin 1 Trait ACE c e a Twin 2 Trait ECA c a e a 2 + c 2

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Variance-Covariance Matrices MZTwin 1Twin 2 Twin 1a 2 + c 2 + e 2 a 2 + c 2 Twin 2a 2 + c 2 a 2 + c 2 + e 2

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Variance-Covariance Matrices DZTwin 1Twin 2 Twin 1a 2 + c 2 + e 2 0.5a 2 + c 2 Twin 20.5a 2 + c 2 a 2 + c 2 + e 2

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Variance-Covariance Matrices MZ Twin 1Twin 2 Twin 1a 2 + c 2 + e 2 a 2 + c 2 Twin 2a 2 + c 2 a 2 + c 2 + e 2 DZ Twin 1Twin 2 Twin 1a 2 + c 2 + e 2 0.5a 2 + c 2 Twin 20.5a 2 + c 2 a 2 + c 2 + e 2

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Why isn’t e 2 included in the covariance? Because, e 2 refers to environmental influences UNIQUE to each twin. Therefore, this cannot explain why there is similarity between twins. Why is a 2 only.5 for DZs but not MZs? Because DZ twins share on average half of their genes, whereas MZs share all of their genes.

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2. Basic openMx ACE Script

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Overview OpenMx script Running the script Describing the output

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ACE model # Fit ACE Model with RawData and Matrices Input # univACEModel <- mxModel("univACE", mxModel("ACE", # Matrices a, c, and e to store a, c, and e path coefficients mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=10, label="a11", name="a" ), mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=10, label="c11", name="c" ), mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=10, label="e11", name="e" ), # Matrices A, C, and E compute variance components mxAlgebra( expression=a %*% t(a), name="A" ), mxAlgebra( expression=c %*% t(c), name="C" ), mxAlgebra( expression=e %*% t(e), name="E" ), # Algebra to compute total variances and standard deviations (diagonal only) mxAlgebra( expression=A+C+E, name="V" ), Specify the matrices you need To build the model Twin 1 Trait ACE c e a 11 1 Do some algebra to get the variances

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Start values? a 2 = additive genetic variance (A) c 2 = Shared E variance (C) e 2 = Non-shared E variance (E) Sum is modelled to be expected Total Variance Start Values for a, c, e: (Total Variance / 3) Twin 1 Trait ACE c e a 11 1

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Standardize parameter estimates The regression coefficient a is standardized by: (a * SD(A)) / SD(Trait) where SD(Trait) is the standard deviation of the dependent variable, and SD(A) is the standard deviation of the predictor, the latent factor ‘A’ (=1) Twin 1 Trait A a 1 V1 * = V VV SD inv = 1/SD a * = a/SD

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Standardize parameter estimates # Algebra to compute total variances and standard deviations (diagonal only) mxAlgebra( expression=A+C+E, name="V" ), mxMatrix( type="Iden", nrow=nv, ncol=nv, name="I"), mxAlgebra( expression=solve(sqrt(I*V)), name="iSD"), # Algebra to compute standardized path estimares and variance components mxAlgebra( expression=a%*%iSD, name="sta"), mxAlgebra( expression=c%*%iSD, name="stc"), mxAlgebra( expression=e%*%iSD, name="ste"), mxAlgebra( expression=A/V, name="h2"), mxAlgebra( expression=C/V, name="c2"), mxAlgebra( expression=E/V, name="e2"), The heritability ‘h 2 ’ is the proportion of the total variance due to A (additive genetic effects; = A / V. Note: this will be “sta” squared. The standardized variance components for C and E are: C / V; E / V N Twin 1 Trait A sta 1 VA / = “h2 ” “sta” 2

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Standardising data V = A + C + EA/V=73/233=.31 V = [73] + [90] + [70]C/V=90/233=.39 V = [233]E/V=70/233=.30 a = 8.55 c = 9.49 e = 8.35 SD = sqrt(V)=15.26 sta = 8.55 / = 0.56squared =.31 stc = 9.49 / = 0.62 squared =.39 ste = 8.35 / = 0.55 squared =.30

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# Algebra for expected variance/covariance matrix in MZ mxAlgebra( expression= rbind ( cbind(A+C+E, A+C), cbind(A+C, A+C+E)), name="expCovMZ" ), # Algebra for expected variance/covariance matrix in DZ mxAlgebra( expression= rbind ( cbind(A+C+E, 0.5%x%A+C), cbind(0.5%x%A+C, A+C+E)), name="expCovDZ" ) MZTwin 1Twin 2 Twin 1a 2 + c 2 + e 2 a 2 + c 2 Twin 2a 2 + c 2 a 2 + c 2 + e 2 DZTwin 1Twin 2 Twin 1a 2 + c 2 + e 2 0.5a 2 + c 2 Twin 20.5a 2 + c 2 a 2 + c 2 + e 2 Twin 1 Trait ACE c e a Twin 2 Trait ECA c a e 1/

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mxModel("MZ", mxData( observed=mzData, type="raw" ), mxFIMLObjective( covariance="ACE.expCovMZ", means="ACE.expMean", dimnames=selVars ) ), mxModel("DZ", mxData( observed=dzData, type="raw" ), mxFIMLObjective( covariance="ACE.expCovDZ", means="ACE.expMean", dimnames=selVars ) ), mxAlgebra( expression=MZ.objective + DZ.objective, name="m2ACEsumll" ), mxAlgebraObjective("m2ACEsumll"), mxCI(c('ACE.A', 'ACE.C', 'ACE.E')) ) univACEFit <- mxRun(univACEModel, intervals=T) univACESumm <- summary(univACEFit) univACESumm

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Sub-models # Fit AE model # univAEModel <- mxModel(univACEFit, name="univAE", mxModel(univACEFit$ACE, mxMatrix( type="Lower", nrow=1, ncol=1, free=FALSE, values=0, label="c11", name="c" ) ) ) univAEFit <- mxRun(univAEModel) univAESumm <- summary(univAEFit) univAESumm You can fit sub-models to test the significance of your parameters -you simply drop the parameter and see if the model fit is significantly worse than full model Twin 1 Trait AE Twin 2 Trait EA

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Sub-models # Fit CE model # univCEModel <- mxModel(univACEFit, name="univCE", mxModel(univACEFit$ACE, mxMatrix( type="Lower", nrow=1, ncol=1, free=FALSE, values=0, label="a11", name="a" ) ) ) univCEFit <- mxRun(univCEModel) univCESumm <- summary(univCEFit) univCESumm Twin 1 Trait CE Twin 2 Trait EC # Fit E model # univEModel <- mxModel(univAEFit, name="univE", mxModel(univAEFit$ACE, mxMatrix( type="Lower", nrow=1, ncol=1, free=FALSE, values=0, label="a11", name="a" ) ) ) univEFit <- mxRun(univEModel) univESumm <- summary(univEFit) univESumm Twin 1 Trait E Twin 2 Trait E The E parameter can never not be dropped because it includes measurement error

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OpenMx Output

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univACESumm free parameters: name matrix row col Estimate 1 a11 ACE.a c11 ACE.c e11 ACE.e mean ACE.Mean observed statistics: 2198 estimated parameters: 4 degrees of freedom: log likelihood: saturated -2 log likelihood: NA number of observations: 1110 chi-square: NA p: NA AIC (Mx): BIC (Mx): adjusted BIC: RMSEA: NA

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tableFitStatistics models compared to saturated model Name ep -2LL df AIC diffLL diffdf p M1 : univTwinSat M2 : univACE M3 : univAE M4 : univCE M5 : univE Smaller -2LL means better fit. -2LL of sub-model is always higher (worse fit). The question is: is it significantly worse. Chi-sq test: dif in -2LL is chi-square distributed. Evaluate sig of chi-sq test. A non-sig p-value means that the model is consistent with the data.

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Nested.fit models compared to ACE model Name ep -2LL df diffLL diffdf p univACE NA NA NA univAE univCE univE Smaller -2LL means better fit. -2LL of sub-model is always higher (worse fit). The question is: is it significantly worse. Chi-sq test: dif in -2LL is chi-square distributed. Evaluate sig of chi-sq test. Critical Chi-sq value for 1 DF = 3.84 A non-sig p-value means that the dropped parameter(s) are non-significant.

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Estimates ACE model > univACEFit$ACE.h2 [,1] [1,] > univACEFit$ACE.c2 [,1] [1,] > univACEFit$ACE.e2 [,1] [1,]

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