1. Fitting Bivariate Models October 21, 2014 Elizabeth Prom-Wormley & Hermine Maes ecpromwormle@vcu.edu 804-828-8154

2. The Problem(s) BMI may not be an appropriate measure for use in studying the genetics of obesity – Do height and weight share the same genetic/environmental influences? Smoking is a risk factor for cardiovascular disease, but how ? – Does smoking in adolescence lead to cardiovascular disease in adulthood?

3. A Solution Bivariate Genetic Analysis Two or more traits can be correlated because they share common genes or common environmental influences – e.g. Are the same genetic/environmental factors influencing the traits? With twin data on multiple traits it is possible to partition the covariation into its genetic and environmental components – Goal: to understand what factors make sets of variables correlate or co-vary

4. Bivariate Analysis A Roadmap 1- Use the data to test basic assumptions inherent to standard genetic models Saturated Bivariate Model 2- Estimate the contributions of genetic and environmental factors to the covariance between two traits Bivariate Genetic Models (Bivariate Genetic Analysis via Cholesky Decomposition)

5. Getting a Feel for the Data Phenotypic and Twin Correlations MZ DZ Height2Weight2 Height2Weight2 Height10.880.41 0.440.23 Weight10.490.84 0.170.33 Open R script for today r Weight/Height = 0.47

6. Getting a Feel for the Data Phenotypic and Twin Correlations MZ DZ Height2Weight2 Height2Weight2 Height10.880.41 0.440.23 Weight10.490.84 0.170.33 r Weight/Height = 0.47 Expectations 1- rMZ > rDZ (cross-twin/cross-trait) Genetic effects contribute to the relationship between height and weight 2- Cross-twin cross-variable correlations are not as big as the correlations between twins within variables. Variable specific genetic effects on height and/or weight

7. So…How Can We be Sure?

8. Building the Bivariate Genetic Model Sources of Information Cross-trait covariance within individuals – Within-Twin Covariance Cross-trait covariance between twins – Cross-Twin Covariance MZ:DZ ratio of cross-trait covariance between twins

9. 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 – Equal covariances between Twin 1 and Twin 2 in MZ and DZ twins 9

10. Getting a Feel for the Data Means ht1wt1ht2wt2 MZ 16.305.6716.295.65 DZ 16.415.8216.335.77

11. Getting a Feel for the Data Variances-Covariances ht1wt1ht2wt2 ht1 wt1 ht2 wt2 ht1wt1ht2wt2 ht1 wt1 ht2 wt2 MZ Twins DZ Twins Twin 1Twin 2 Twin 1 Twin 2 Twin 1Twin 2 Twin 1 Twin 2

12. Getting a Feel for the Data Variances-Covariances Variances

13. Observed Variance-Covariance Matrix P1P 2P1P 2 P 1 Variance Twin 1- P1 P2 Variance Twin 1- P2 P 1 Variance Twin2- P1 P 2 Variance Twin 2- P2 Twin 1Twin 2 Twin 1 Twin 2 Variances

14. Getting a Feel for the Data Variances-Covariances ht1wt1ht2wt2 ht1 0.43 wt1 0.73 ht2 0.44 wt2 0.78 ht1wt1ht2wt2 ht1 0.48 wt1 0.75 ht2 0.46 wt2 0.86 MZ Twins DZ Twins Twin 1Twin 2 Twin 1 Twin 2 Twin 1Twin 2 Twin 1 Twin 2

15. Getting a Feel for the Data Variances-Covariances Cross-Trait / Within-Twin Covariance

16. Observed Variance-Covariance Matrix P1P 2P1P 2 P 1 Variance Twin 1- P1 Covariance Twin 1 P1/P2 P2 Covariance Twin 1 P1/P2 Variance Twin 1- P2 P 1 Variance Twin2- P1 Covariance Twin 2 P1/P2 P 2 Covariance Twin 2 P1/P2 Variance Twin 2- P2 Twin 1Twin 2 Twin 1 Twin 2 Variances + Cross-Trait Within-Twin Covariances

17. Getting a Feel for the Data Variances-Covariances ht1wt1ht2wt2 ht1 0.430.28 wt1 0.280.73 ht2 0.440.26 wt2 0.260.78 ht1wt1ht2wt2 ht1 0.480.26 wt1 0.260.75 ht2 0.460.28 wt2 0.280.86 MZ Twins DZ Twins Twin 1Twin 2 Twin 1 Twin 2 Twin 1Twin 2 Twin 1 Twin 2

18. Getting a Feel for the Data Variances-Covariances Cross-Trait / Within-Twin Covariance

19. Observed Variance-Covariance Matrix P1P 2P1P 2 P 1 Variance Twin 1- P1 Covariance Twin 1 P1/P2 P1 Within Trait T1 /T2 Covariance P2 Covariance Twin 1 P1/P2 Variance Twin 1- P2 P2 Within Trait T1 /T2 Covariance P 1 P1 Within Trait T1 /T2 Covariance Variance Twin2- P1 Covariance Twin 2 P1/P2 P 2 P2 Within Trait T1 /T2 Covariance Twin 2 P1/P2 Variance Twin 2- P2 Twin 1Twin 2 Twin 1 Twin 2 Variances + Cross-Trait Within-Twin Covariances + Within-Trait Cross-Twin Covariances

20. Getting a Feel for the Data Variances-Covariances ht1wt1ht2wt2 ht1 0.430.280.38 wt1 0.280.730.64 ht2 0.380.440.26 wt2 0.640.260.78 ht1wt1ht2wt2 ht1 0.480.260.21 wt1 0.260.750.27 ht2 0.210.460.28 wt2 0.270.280.86 MZ Twins DZ Twins Twin 1Twin 2 Twin 1 Twin 2 Twin 1Twin 2 Twin 1 Twin 2

21. Getting a Feel for the Data Variances-Covariances Cross-Trait / Cross-Twin Covariance

22. Observed Variance-Covariance Matrix P1P 2P1P 2 P 1 Variance Twin 1- P1 Covariance Twin 1 P1/P2 P1 Within Trait T1 /T2 Covariance P1/P2 Cross-Trait T1 /T2 Covariance P2 Covariance Twin 1 P1/P2 Variance Twin 1- P2 P2/P1 Cross-Trait T1 /T2 Covariance P2 Within Trait T1 /T2 Covariance P 1 P1 Within Trait T1 /T2 Covariance P2/P1 Cross-Trait T1 /T2 Covariance Variance Twin2- P1 Covariance Twin 2 P1/P2 P 2 P1/P2 Cross-Trait T1 /T2 Covariance P2 Within Trait T1 /T2 Covariance Twin 2 P1/P2 Variance Twin 2- P2 Twin 1Twin 2 Twin 1 Twin 2 Variances + Cross-Trait Within-Twin Covariances + Within-Trait Cross-Twin Covariances

23. Observed Variance-Covariance Matrix P1P 2P1P 2 P 1 Variance Twin 1- P1 Covariance Twin 1 P1/P2 P1 Within Trait T1 /T2 Covariance P1/P2 Cross-Trait T1 /T2 Covariance P2 Covariance Twin 1 P1/P2 Variance Twin 1- P2 P2/P1 Cross-Trait T1 /T2 Covariance P2 Within Trait T1 /T2 Covariance P 1 P1 Within Trait T1 /T2 Covariance P2/P1 Cross-Trait T1 /T2 Covariance Variance Twin2- P1 Covariance Twin 2 P1/P2 P 2 P1/P2 Cross-Trait T1 /T2 Covariance P2 Within Trait T1 /T2 Covariance Twin 2 P1/P2 Variance Twin 2- P2 Twin 1Twin 2 Twin 1 Twin 2 Variances + Cross-Trait Within-Twin Covariances + Within-Trait Cross-Twin Covariances + Cross-Twin Cross-Trait Covariances

24. Observed Variance-Covariance Matrix P1P 2P1P 2 P 1 Variance P1 P2 Covariance P1-P2 Variance P2 P 1 Within- trait P1 Cross-traitVariance P1 P 2 Cross-trait Within- trait P2 Covarianc e P1-P2 Variance P2 Twin 1Twin 2 Twin 1 Twin 2 Within-twin covariance Within-twin covariance Cross-twin covariance

25. Getting a Feel for the Data Variances-Covariances ht1wt1ht2wt2 ht1 0.430.280.380.24 wt1 0.280.730.280.64 ht2 0.380.280.440.26 wt2 0.240.640.260.78 ht1wt1ht2wt2 ht1 0.480.260.210.15 wt1 0.260.750.110.27 ht2 0.210.110.460.28 wt2 0.150.270.280.86 MZ Twins DZ Twins Twin 1Twin 2 Twin 1 Twin 2 Twin 1Twin 2 Twin 1 Twin 2

26. Within-twin cross-trait covariances imply common etiological influences Cross-twin cross-trait covariances imply familial common etiological influences MZ/DZ ratio of cross-twin cross-trait covariances reflects whether common etiological influences are genetic or environmental Cross-Trait Covariances

27. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X2X2 Y2Y2 X1X1 V X1 C X1Y1 C X1X2 C X1Y2 Y1Y1 C Y1X1 V Y1 C Y1X2 C Y1Y2 X2X2 C X2X1 C X2Y1 V X2 C X2Y2 Y2Y2 C Y2X1 C Y2Y1 C Y2X2 V Y2 twin 1 twin 2 Variances of X & Y same across twins and zygosity groups Variances of X & Y same across twins and zygosity groups

28. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X2X2 Y2Y2 X1X1 V X1 C X1Y1 C X1X2 C X1Y2 Y1Y1 C Y1X1 V Y1 C Y1X2 C Y1Y2 X2X2 C X2X1 C X2Y1 V X2 C X2Y2 Y2Y2 C Y2X1 C Y2Y1 C Y2X2 V Y2 twin 1 twin 2 Covariances of X & Y same across twins and zygosity groups Covariances of X & Y same across twins and zygosity groups

29. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X2X2 Y2Y2 X1X1 V X1 C X1Y1 C X1X2 C X1Y2 Y1Y1 C Y1X1 V Y1 C Y1X2 C Y1Y2 X2X2 C X2X1 C X2Y1 V X2 C X2Y2 Y2Y2 C Y2X1 C Y2Y1 C Y2X2 V Y2 twin 1 twin 2 Cross-Twin Within- Trait Covariances differ by zygosity

30. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X2X2 Y2Y2 X1X1 V X1 C X1Y1 C X1X2 C X1Y2 Y1Y1 C Y1X1 V Y1 C Y1X2 C Y1Y2 X2X2 C X2X1 C X2Y1 V X2 C X2Y2 Y2Y2 C Y2X1 C Y2Y1 C Y2X2 V Y2 twin 1 twin 2 Cross-Twin Cross- Trait Covariances differ by zygosity

31. Saturated Model Testing

32. Bivariate Analysis A Roadmap 1- Use the data to test basic assumptions inherent to standard genetic models Saturated Bivariate Model 2- Estimate the contributions of genetic and environmental factors to the covariance between two traits Bivariate Genetic Models (Bivariate Genetic Analysis via Cholesky Decomposition)

33. Genetic Modeling with Twin Data acac MZ = 1 DZ = 0.5 1 A C Twin 1 Height Twin 2 Height ee EEC A

34. a 11 a 21 c 11 c 22 c 21 e 11 e 21 e 22 a 11 a 22 a 21 c 11 c 22 c 21 e 11 e 21 e 22 MZ = 1 DZ = 0.5 MZ = 1 DZ = 0.5 1 1 C1C2C1C2 Twin 1 Height Twin 1 Weight Twin 2 Height Twin 2 Weight E1 E2 E1 E2 a 22 Bivariate Genetic Modeling A1 A2 A1A2

35. Building the Bivariate Genetic Model Sources of Information Cross-trait covariance within individuals – Within-Twin Covariance Cross-trait covariance between twins – Cross-Twin Covariance MZ:DZ ratio of cross-trait covariance between twins

36. Alternative Representations

38. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X1X1 V X1 C X1Y1 Y1Y1 C Y1X1 V Y1 twin 1

39. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X1X1 a 11 2 C X1Y1 Y1Y1 C Y1X1 V Y1 twin 1

40. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X1X1 a 11 2 C X1Y1 Y1Y1 a 21 *a 11 V Y1 twin 1

41. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X1X1 a 11 2 a 21 *a 11 Y1Y1 V Y1 twin 1

42. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X1X1 a 11 2 a 21 *a 11 Y1Y1 a 22 2 +a 21 2 twin 1

43. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X1X1 a 11 2 + e 11 2 a 21 *a 11 + e 21 *e 11 Y1Y1 a 22 2 +a 21 2 +e 22 2 +e 21 2 twin 1

44. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X2X2 C X2X1 C X2Y1 Y2Y2 C Y2X1 C Y2Y1 twin 1 twin 2

45. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X2X2 1/0.5 * a 11 2 C X2Y1 Y2Y2 C Y2X1 C Y2Y1 twin 1 twin 2

46. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X2X2 1/0.5 * a 11 2 C X2Y1 Y2Y2 1/0.5 * a 21 *a 11 C Y2Y1 twin 2 twin 1

47. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X2X2 1/0.5 * a 11 2 1/0.5 * a 21 *a 11 Y2Y2 C Y2Y1 twin 2 twin 1

48. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X2X2 1/0.5* a 11 2 1/0.5 * a 21 *a 11 Y2Y2 1/0.5 *a 22 2 + 1/0.5 *a 21 2 twin 2 twin 1

49. Bivariate Twin Covariance Matrix X1X1 Y1Y1 X2X2 Y2Y2 X1X1 V X1 C X1Y1 C X1X2 C X1Y2 Y1Y1 C Y1X1 V Y1 C Y1X2 C Y1Y2 X2X2 C X2X1 C X2Y1 V X2 C X2Y2 Y2Y2 C Y2X1 C Y2Y1 C Y2X2 V Y2 twin 1 twin 2

50. Predicted Twin Covariance Matrix X1X1 Y1Y1 X2X2 Y2Y2 X1X1 a 11 2 + e 11 2 a 21 *a 11 + e 21 *e 11 1/0.5* a 11 2 1/0.5 * a 21 *a 11 Y1Y1 a 21 *a 11 + e 21 *e 11 a 22 2 +a 21 2 + e 22 2 +e 21 2 1/0.5 * a 21 *a 11 1/0.5*a 22 2 + 1/0.5*a 21 2 X2X2 1/0.5* a 11 2 1/0.5 * a 21 *a 11 a 11 2 + e 11 2 a 21 *a 11 + e 21 *e 11 Y2Y2 1/0.5 * a 21 *a 11 1/0.5*a 22 2 + 1/0.5*a 21 2 a 21 *a 11 + e 21 *e 11 a 22 2 +a 21 2 + e 22 2 +e 21 2 twin 1 twin 2

51. Predicted MZ Twin Covariance X1X1 Y1Y1 X2X2 Y2Y2 X1X1 a 11 2 + e 11 2 a 21 *a 11 + e 21 *e 11 a 11 2 a 21 *a 11 Y1Y1 a 21 *a 11 + e 21 *e 11 a 22 2 +a 21 2 + e 22 2 +e 21 2 a 21 *a 11 a 22 2 +a 21 2 X2X2 a 11 2 a 21 *a 11 a 11 2 + e 11 2 a 21 *a 11 + e 21 *e 11 Y2Y2 a 21 *a 11 a 22 2 +a 21 2 a 21 *a 11 + e 21 *e 11 a 22 2 +a 21 2 + e 22 2 +e 21 2 twin 1 twin 2

52. Predicted DZ Twin Covariance X1X1 Y1Y1 X2X2 Y2Y2 X1X1 a 11 2 + e 11 2 a 21 *a 11 + e 21 *e 11 0.5*a 11 2 0.5*a 21 *a 11 Y1Y1 a 21 *a 11 + e 21 *e 11 a 22 2 +a 21 2 + e 22 2 +e 21 2 0.5*a 21 *a 11 0.5*a 22 2 + 0.5* a 21 2 X2X2 0.5*a 11 2 0.5*a 21 *a 11 a 11 2 + e 11 2 a 21 *a 11 + e 21 *e 11 Y2Y2 0.5*a 21 *a 11 0.5*a 22 2 + 0.5* a 21 2 a 21 *a 11 + e 21 *e 11 a 22 2 +a 21 2 + e 22 2 +e 21 2 twin 1 twin 2

53. Predicted Covariance Matrix X1X1 Y1Y1 X2X2 Y2Y2 X1X1 V X1 C X1Y1 C X1X2 C X1Y2 Y1Y1 C Y1X1 V Y1 C Y1X2 C Y1Y2 X2X2 C X2X1 C X2Y1 V X2 C X2Y2 Y2Y2 C Y2X1 C Y2Y1 C Y2X2 V Y2 twin 1 twin 2 Variances of X & Y same across twins and zygosity groups Variances of X & Y same across twins and zygosity groups

54. Predicted Covariance Matrix X1X1 Y1Y1 X2X2 Y2Y2 X1X1 V X1 C X1Y1 C X1X2 C X1Y2 Y1Y1 C Y1X1 V Y1 C Y1X2 C Y1Y2 X2X2 C X2X1 C X2Y1 V X2 C X2Y2 Y2Y2 C Y2X1 C Y2Y1 C Y2X2 V Y2 twin 1 twin 2 Covariances of X & Y same across twins and zygosity groups Covariances of X & Y same across twins and zygosity groups

55. Predicted Covariance Matrix X1X1 Y1Y1 X2X2 Y2Y2 X1X1 V X1 C X1Y1 C X1X2 C X1Y2 Y1Y1 C Y1X1 V Y1 C Y1X2 C Y1Y2 X2X2 C X2X1 C X2Y1 V X2 C X2Y2 Y2Y2 C Y2X1 C Y2Y1 C Y2X2 V Y2 twin 1 twin 2 Cross-Twin Within- Trait Covariances differ by zygosity

56. Predicted Covariance Matrix X1X1 Y1Y1 X2X2 Y2Y2 X1X1 V X1 C X1Y1 C X1X2 C X1Y2 Y1Y1 C Y1X1 V Y1 C Y1X2 C Y1Y2 X2X2 C X2X1 C X2Y1 V X2 C X2Y2 Y2Y2 C Y2X1 C Y2Y1 C Y2X2 V Y2 twin 1 twin 2 Cross-Twin Cross- Trait Covariances differ by zygosity

57. OpenMx Specification X1X1 Y1Y1 X2X2 Y2Y2 X1X1 V X1 C X1Y1 C X1X2 C X1Y2 Y1Y1 C Y1X1 V Y1 C Y1X2 C Y1Y2 X2X2 C X2X1 C X2Y1 V X2 C X2Y2 Y2Y2 C Y2X1 C Y2Y1 C Y2X2 V Y2 OpenMx script

58. Read in and Transform Variable(s) transform variables to make variances with similar order of magnitudes # Load Data data(twinData) describe(twinData) twinData[,'ht1'] <- twinData[,'ht1']*10 twinData[,'ht2'] <- twinData[,'ht2']*10 twinData[,'wt1'] <- twinData[,'wt1']/10 twinData[,'wt2'] <- twinData[,'wt2']/10 # Select Variables for Analysis Vars <- c('ht','wt') nv <- 2 # number of variables ntv <- nv*2 # number of total variables selVars <- paste(Vars,c(rep(1,nv),rep(2,nv)),sep="") #c('ht1','wt1,'ht2','wt2') # Select Data for Analysis mzData <- subset(twinData, zyg==1, selVars) dzData <- subset(twinData, zyg==3, selVars)

59. # Set Starting Values svMe <- c(15,5) # start value for means laMe <- paste(selVars,"Mean",sep="_") svPa <-.6 # start value for parameters svPas <- diag(svPa,nv,nv) laA <- paste("a",rev(nv+1-sequence(1:nv)),rep(1:nv,nv:1),sep="") # c("a11","a21","a22") laD <- paste("d",rev(nv+1-sequence(1:nv)),rep(1:nv,nv:1),sep="") laC <- paste("c",rev(nv+1-sequence(1:nv)),rep(1:nv,nv:1),sep="") laE <- paste("e",rev(nv+1-sequence(1:nv)),rep(1:nv,nv:1),sep="") Start Values, Labels Means & Parameters create numbered labels to fill the lower triangular matrices with the first number corresponding to the variable being pointed to and the second number corresponding to the factor

60. Within-Twin Covariance [A] Path Tracing:Matrix Algebra: Lower 2x2 A 1 A 2 P1P2P1P2

61. # Matrices declared to store a, c, and e Path Coefficients pathA <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=T, values=svPas, label=laA, name="a" ) pathD <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=T, values=svPas, label=laD, name="d" ) pathC <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=F, values=0, label=laC, name="c" ) pathE <- mxMatrix( type="Lower", nrow=nv, ncol=nv, free=T, values=svPas, label=laE, name="e" ) # Matrices generated to hold A, C, and E computed Variance Components covA <- mxAlgebra( expression=a %*% t(a), name="A" ) covD <- mxAlgebra( expression=d %*% t(d), name="D" ) covC <- mxAlgebra( expression=c %*% t(c), name="C" ) covE <- mxAlgebra( expression=e %*% t(e), name="E" ) Path Coefficients Variance Components regular multiplication of lower triangular matrix and its transpose A 1 A 2 P1P2P1P2 a %*% t(a)

62. Within-Twin Covariance [A]+[E] + using matrix addition to generate total within-twin covariance

63. # Algebra to compute total variances and standard deviations (diagonal only) covP <- mxAlgebra( expression=A+D+C+E, name="V" ) matI <- mxMatrix( type="Iden", nrow=nv, ncol=nv, name="I") invSD <- mxAlgebra( expression=solve(sqrt(I*V)), name="iSD") # Algebras generated to hold Parameter Estimates and Derived Variance Components rowVars <- rep('vars',nv) colVars <- rep(c('A','D','C','E','SA','SD','SC','SE'),each=nv) estVars <- mxAlgebra( cbind(A,D,C,E,A/V,D/V,C/V,E/V), name="Vars", dimnames=list(rowVars,colVars)) Total Variances Variance Components each of covariance matrices is of size nv x nv

64. Cross-Twin Covariances [A] & 0.5[A] + using Kronecker product to multiple every element of matrix by scalar

65. # Algebra for expected Mean and Variance/Covariance Matrices in MZ & DZ twins meanG <- mxMatrix( type="Full", nrow=1, ncol=nv, free=T, values=svMe, labels=laMe, name="Mean" ) meanT <- mxAlgebra( cbind(Mean,Mean), name="expMean" ) covMZ <- mxAlgebra( rbind( cbind(V, A+D+C), cbind(A+D+C, V )), name="expCovMZ" ) covDZ <- mxAlgebra( rbind( cbind(V, 0.5%x%A+0.25%x%D+C), cbind(0.5%x%A+0.25%x%D+C, V )), name="expCovDZ" ) Expected Means & Covariances cbind creates two nv x ntv row matrices rbind turns them into to ntv x ntv matrix

66. # Data objects for Multiple Groups dataMZ <- mxData( observed=mzData, type="raw" ) dataDZ <- mxData( observed=dzData, type="raw" ) # Objective objects for Multiple Groups objMZ <- mxFIMLObjective( covariance="expCovMZ", means="expMean", dimnames=selVars ) objDZ <- mxFIMLObjective( covariance="expCovDZ", means="expMean", dimnames=selVars ) # Combine Groups pars <- list( pathA, pathD, pathC, pathE, covA, covD, covC, covE, covP, matI, invSD, estVars, meanG, meanT ) modelMZ <- mxModel( pars, covMZ, dataMZ, objMZ, name="MZ" ) modelDZ <- mxModel( pars, covDZ, dataDZ, objDZ, name="DZ" ) minus2ll <- mxAlgebra( expression=MZ.objective + DZ.objective, name="m2LL" ) obj <- mxAlgebraObjective( "m2LL" ) BivAceModel <- mxModel( "BivACE", pars, modelMZ, modelDZ, minus2ll, obj ) Data, Objectives & Model Objects expected covariances and meansobserved data

67. # Run Bivariate ACE model BivAceFit <- mxRun(BivAceModel) BivAceSum <- summary(BivAceFit) BivAceSum$pa round(BivAceFit@output$estimate,4) round(BivAceFit$Vars@result,4) BivAceSum$Mi # Generate Output with Functions source("GenEpiHelperFunctions.R") parameterSpecifications(BivAceFit) expectedMeansCovariances(BivAceFit) tableFitStatistics(BivAceFit) Parameter Estimates Variance Components two ways to get parameter estimates print pre-calculated unstandardized variance components and standardized variance components

68. Three Important Results from Bivariate Genetic Analysis 1. Variance Decomposition -> Heritability, (Shared) environmental influences 2. Covariance Decomposition -> The influences of genes and environment on the covariance between the two variables “How much of the phenotypic correlation is accounted for by genetic and environmental influences?” 3. Genetic and Environmental correlations -> the overlap in genes and environmental effects “Is there a large overlap in genetic/ environmental factors?

69. From Cholesky to Genetic Correlation standardized solution = correlated factors solution

70. Genetic Covariance to Genetic Correlation calculated by dividing genetic covariance by square root of product of genetic variances of two variables

71. # Calculate genetic and environmental correlations corA <- mxAlgebra( expression=solve(sqrt(I*A))%&%A), name ="rA" ) corD <- mxAlgebra( expression=solve(sqrt(I*D))%&%E), name ="rD" ) corC <- mxAlgebra( expression=solve(sqrt(I*C))%&%C), name ="rC" ) corE <- mxAlgebra( expression=solve(sqrt(I*E))%&%E), name ="rE" ) Genetic Correlation Algebra

72. 72 Contribution to Phenotypic Correlation if rg=1, then two sets of genes overlap completely if however, a11 and a22 are near to zero, genes do not contribute much to phenotypic correlation contribution to phenotypic correlation is function of both heritabilities and rg

73. Interpreting Results High genetic correlation = large overlap in genetic effects on the two phenotypes Does it mean that the phenotypic correlation between the traits is largely due to genetic effects? No: the substantive importance of a particular r G depends the value of the correlation and the value of the  A 2 paths i.e. importance is also determined by the heritability of each phenotype

74. Interpretation of Correlations Consider two traits with a phenotypic correlation (r P ) of 0.40 : h 2 P1 = 0.7 and h 2 P2 = 0.6 with r G =.3 Correlation due to additive genetic effects = ? Proportion of phenotypic correlation attributable to additive genetic effects = ? h 2 P1 = 0.2 and h 2 P2 = 0.3 with r G = 0.8 Correlation due to additive genetic effects = ? Proportion of phenotypic correlation attributable to additive genetic effects = ? Correlation due to A: Divide by r P to find proportion of phenotypic correlation.

75. Bivariate CholeskyMultivariate Cholesky