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Univariate Twin Analysis- Saturated Models for Continuous and Categorical Data September 2, 2014 Elizabeth Prom-Wormley & Hermine Maes

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Overall Questions to be Answered Does the data satisfy the assumptions of the classical twin study? Does a trait of interest cluster among related individuals? 2

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Family & Twin Study Designs Family Studies Classical Twin Studies Adoption Studies Extended Twin Studies 3

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The Data Please open twinSatConECPW Fall2014.R Australian Twin Register years old, males and females Work from this session will focus on Body Mass Index (weight/height2) in females only Sample size – MZF = 534 complete pairs (zyg = 1) – DZF = 328 complete pairs (zyg = 3) total MZ pairs 351 total DZ pairs 569 total MZ pairs 351 total DZ pairs

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A Quick Look at the Data 5

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Classical Twin Studies Basic Background The Classical Twin Study (CTS) uses MZ and DZ twins reared together – MZ twins share 100% of their genes – DZ twins share on average 50% of their genes Expectation- Genetic factors are assumed to contribute to a phenotype when MZ twins are more similar than DZ twins 6

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Classical Twin Study Assumptions MZ twins are genetically identical Equal Environments of MZ and DZ pairs 7

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Basic Data Assumptions MZ and DZ twins are sampled from the same population, therefore we expect :- – Equal means/variances in Twin 1 and Twin 2 – Equal means/variances in MZ and DZ twins Further assumptions would need to be tested if we introduce male twins and opposite sex twin pairs 8

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“ Old Fashioned ” Data Checking 9 MZDZ T1T2T1T2 mean variance covariance (T1- T2) Nice, but how can we actually be sure that these means and variances are truly the same?

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Univariate Analysis A Roadmap Use the data to test basic assumptions (equal means & variances for twin 1/twin 2 and MZ/DZ pairs) Saturated Model 2- Estimate contributions of genetic and environmental effects on the total variance of a phenotype ACE or ADE Models 3- Test ACE (ADE) submodels to identify and report significant genetic and environmental contributions AE or CE or E Only Models 10

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Saturated Twin Model 11

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Saturated Code Deconstructed 12 mMZ1 mMZ2 mDZ1 mDZ2 mean MZ = 1 x 2 matrixmean DZ = 1 x 2 matrix meanMZ <- mxMatrix( type="Full", nrow=1, ncol=ntv, free=TRUE, values=meVals, labels=c("mMZ1","mMZ2"), name=”meanMZ" ) meanDZ <- mxMatrix( type="Full", nrow=1, ncol=ntv, free=TRUE, values=meVals, labels=c("mDZ1","mDZ2"), name=”meanDZ" )

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Saturated Code Deconstructed 13 vMZ1cMZ21 vMZ2 T1 T2 T1 T2 vDZ1cDZ21 vDZ2 T1T2 T1 T2 covMZ = 2 x 2 matrix covDZ = 2 x 2 matrix covMZ <- mxMatrix( type="Symm", nrow=ntv, ncol=ntv, free=TRUE, values=cvVals, lbound=lbVals, labels=c("vMZ1","cMZ21","vMZ2"), name=”covMZ" ) covDZ <- mxMatrix( type="Symm", nrow=ntv, ncol=ntv, free=TRUE, values=cvVals, lbound=lbVals, labels=c("vDZ1","cDZ21","vDZ2"), name=”covDZ" )

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Time to Play... Continue with the File twinSatConECPW Fall2014.R 14

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Estimated Values T1T2T1T2 Saturated Model meanMZDZ covT1 T2 10 Total Parameters Estimated Standardize covariance matrices for twin pair correlations (covMZ & covDZ) mMZ1, mMZ2, vMZ1,vMZ2,cMZ21 mDZ1, mDZ2, vDZ1,vDZ2,cDZ21

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Estimated Values Total Parameters Estimated Standardize covariance matrices for twin pair correlations (covMZ & covDZ) mMZ1, mMZ2, vMZ1,vMZ2,cMZ21 mDZ1, mDZ2, vDZ1,vDZ2,cDZ21 T1T2T1T2 Saturated Model meanMZ DZ covT10.73T10.77 T T

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Fitting Nested Models Saturated Model – likelihood of data without any constraints – fitting as many means and (co)variances as possible Equality of means & variances by twin order – test if mean of twin 1 = mean of twin 2 – test if variance of twin 1 = variance of twin 2 Equality of means & variances by zygosity – test if mean of MZ = mean of DZ – test if variance of MZ = variance of DZ 17

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Estimated Values T1T2T1T2 Equate Means & Variances across Twin Order meanMZDZ covT1 T2 Equate Means Variances across Twin Order & Zygosity meanMZDZ covT1 T2

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Estimates 19 T1T2T1T2 Equate Means & Variances across Twin Order meanMZ21.35 DZ21.45 covT10.76T10.79 T T Equate Means Variances across Twin Order & Zygosity meanMZ21.39 DZ21.39 covT10.78T10.78 T T

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Stats 20 Modelep-2lldfAIC diff - 2ll diff df p Saturated mT1=mT mT1=mT2 & varT1=VarT Zyg MZ=DZ No significant differences between saturated model and models where means/variances/covariances are equal by zygosity and between twins

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Working with Binary and Ordinal Data Elizabeth Prom-Wormley and Hermine Maes Special Thanks to Sarah Medland

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Transitioning from Continuous Logic to Categorical Logic Ordinal data has 1 less degree of freedom compared to continuous data MZcov, DZcov, Prevalence No information on the variance Thinking about our ACE/ADE model 4 parameters being estimated A/ C/ E/ mean ACE/ADE model is unidentified without adding a constraint

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Two Approaches to the Liability Threshold Model Traditional – Maps data to a standard normal distribution – Total variance constrained to be 1 Alternate – Fixes an alternate parameter (usually E) – Estimates the remaining parameters

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Time to Look at the Data! Please open BinaryWarmUp.R

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Observed Binary BMI is Imperfect Measure of Underlying Continuous Distribution Mean (bmiB2) = 0.39 SD (bmiB2) = 0.49 Prevalence “ low ” BMI = 60.6% We are interested in the liability of risk for being in the “ high ” BMI category

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It ’ s Helpful to Rescale Raw Data (Unstandardized) mean=0.49, SD=0.39 -Data not mapped to a standard normal -No easy conversion to % -Difficult to compare between groups Since the scaling is now arbitrary Standard Normal (Standardized) mean=0, SD=1 Area under the curve between two z-values is interpreted as a probability or percentage

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Binary Review Threshold calculated using the cumulative normal distribution (CND) -We used frequencies and inverse CND to do our own estimation of the threshold qnorm(0.816) = Threshold is the Z Value that corresponds with the proportion of the population having “ low BMI ”

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Moving to Ordinal Data!

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Getting a Feel for the Data Open twinSatOrd.R Calculate the frequencies of the 5 BMI categories for the second twins of the MZ pairs CrossTable(mzDataOrdF$bmi2)

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Estimating MZ Twin 2 Thresholds by Hand T1 = qnorm(0.124) T1 = T2 = qnorm( ) T2 = T3 = qnorm( ) T3 =0.388 T4 = qnorm( ) T4 = Estimate Twin Pair Correlations for the Liabilities Too!

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Translating Back to the SEM Approach in OpenMx

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Handling Ordinal Data in OpenMx 1- Determine the 1 st threshold 2- Determine displacements between 1 st threshold and subsequent thresholds 3- Add the 1 st threshold and the displacement to obtain the subsequent thresholds

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Ordinal Saturated Code Deconstructed Defining Threshold Matrices threM <- mxMatrix( type="Full", nrow=nth, ncol=ntv, free=TRUE, values=thVal, lbound=thLB, labels=thLabMZ, name="ThreMZ" ) threD <- mxMatrix( type="Full", nrow=nth, ncol=ntv, free=TRUE, values=thVal, lbound=thLB, labels=thLabDZ, name="ThreDZ" ) t1MZ1t1MZ2 t2MZ1t2MZ2 t3MZ1t3MZ2 t4MZ1t4MZ2 t1DZ1t1DZ2 t2DZ1t2DZ2 t3DZ1t3DZ2 t4DZ1t4DZ2 1 L T1 1 cov MZ L T2 Variance Constraint Threshold Model 1 L T1 1 cov DZ L T2 1 μ MZT1 0 μ MZT2 0 1 μ DZT1 0 μ DZT2 0

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Ordinal Saturated Code Deconstructed Defining Threshold Matrices- ThreMZ 1- Determine the 1st threshold Tw1Tw Determine displacements between1 st thresholds and subsequent thresholds

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Double Check- Moving from Frequencies to Displacements Frequency BMI T2 Cumulative Frequency Z ValueDisplacement

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Ordinal Saturated Code Deconstructed Estimating Expected Threshold Matrices threMZ <- mxAlgebra( expression= Inc %*% ThreMZ, name="expThreMZ" ) Inc <- mxMatrix( type="Lower", nrow=nth, ncol=nth, free=FALSE, values=1, name="Inc" ) % * % = Add the 1st threshold and the displacement to obtain the subsequent thresholds

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Ordinal Saturated Code Deconstructed Estimating Correlations & Fixing Variance corMZ <- mxMatrix( type="Stand", nrow=ntv, ncol=ntv, free=TRUE, values=corVals, lbound=lbrVal, ubound=ubrVal, labels="rMZ", name="expCorMZ" ) corDZ <- mxMatrix( type="Stand", nrow=ntv, ncol=ntv, free=TRUE, values=corVals, lbound=lbrVal, ubound=ubrVal, labels="rDZ", name="expCorDZ" )

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How Many Parameters in this Ordinal Model? MZ correlation- rMZ DZ correlation- rDZ Thresholds – t1MZ1,t2MZ1,t3MZ1,t4MZ1 – t1MZ2,t2MZ2,t3MZ2,t4MZ2 – t1DZ1,t2DZ1,t3DZ1,t4DZ1 – t1DZ2,t2DZ2,t3DZ2,t4DZ2

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Problem Set 1 Open twinSatOrdA.R and twinSatOrd.R – What do these scripts do? – Looking at the scripts only: How are they similar? How are they different? – What do these differences in the scripts reflect regarding conceptual differences in the two models? Run either script and double check against your previously hand-calculated values of thresholds. Report your results. If you can’t get it to match up, don’t panic…do . Run twinSatOrd.R – Is testing an ACE model with the usual model assumptions justified? Why or why not?

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Univariate Analysis with Ordinal Data A Roadmap 1- Use the data to test basic assumptions inherent to standard ACE (ADE) models Saturated Model 2- Estimate contributions of genetic and environmental effects on the liability of a trait ADE or ACE Models 3- Test ADE (ACE) submodels to identify and report significant genetic and environmental contributions AE or E Only Models

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Questions? 41

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