Presentation is loading. Please wait.

Presentation is loading. Please wait.

Introduction to Multivariate Genetic Analysis Kate Morley and Frühling Rijsdijk 21st Twin and Family Methodology Workshop, March 2008.

Published by Modified over 8 years ago

Similar presentations

Presentation on theme: "Introduction to Multivariate Genetic Analysis Kate Morley and Frühling Rijsdijk 21st Twin and Family Methodology Workshop, March 2008."— Presentation transcript:

1 Introduction to Multivariate Genetic Analysis Kate Morley and Frühling Rijsdijk 21st Twin and Family Methodology Workshop, March 2008

2 Aim and Rationale Aim: to examine the source of factors that make traits correlate or co-vary Rationale:  Traits may be correlated due to shared genetic factors (A) or shared environmental factors (C or E)  Can use information on multiple traits from twin pairs to partition covariation into genetic and environmental components

3 Example 1 Why do traits correlate/covary? How can we explain the association?  Additive genetic factors (r G )  Shared environment (r C )  Non-shared environment (r E ) Kuntsi et al. (2004) Am J Med Genet B, 124:41 ADHD E1E1 E2E2 A1A1 A2A2  A11 2 IQ rGrG C1C1 C2C2 rCrC 1111 1 1 rErE  C11 2  A22 2  C22 2  E11 2  E22 2

4 Example 2 Associations between phenotypes over time  Does anxiety in childhood lead to depression in adolescence? How can we explain the association?  Additive genetic factors (a 21 )  Shared environment (c 21 )  Non-shared environment (e 21 )  How much is not explained by prior anxiety? Rice et al. (2004) BMC Psychiatry 4:43 Childhood anxiety A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Adolescent depression C1C1 C2C2 c 11 c 21 c 22 1 1

5 Sources of Information As an example: two traits measured in twin pairs Interested in:  Cross-trait covariance within individuals  Cross-trait covariance between twins  MZ:DZ ratio of cross-trait covariance between twins

6 Observed Covariance Matrix Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2

7 Observed Covariance Matrix Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance

8 Observed Covariance Matrix Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance

9 SEM: Cholesky Decomposition Twin 1 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 22 e 11 e 22 1 1 1 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 22 1 1

10 SEM: Cholesky Decomposition Twin 1 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1

11 SEM: Cholesky Decomposition Twin 1 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 Twin 2 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 2 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 1/0.5 1 1

12 Why Fit This Model? Covariance matrices must be positive definite If a matrix is positive definite, it can be decomposed into the product of a triangular matrix and its transpose:  A = X*X’ Many other multivariate models possible  Depends on data and hypotheses of interest

13 Cholesky Decomposition Path Tracing

14 Within-Twin Covariances (A) A1A1 A2A2 a 11 a 21 a 22 1 1 Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 Twin 1 Phenotype 2

15 Within-Twin Covariances (A) A1A1 A2A2 a 11 a 21 a 22 1 1 Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 Twin 1 Phenotype 2

16 Within-Twin Covariances (A) A1A1 A2A2 a 11 a 21 a 22 1 1 Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 Twin 1 Phenotype 2

17 Within-Twin Covariances (A) A1A1 A2A2 a 11 a 21 a 22 1 1 Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 Twin 1 Phenotype 2

18 Within-Twin Covariances (C) C1C1 C2C2 c 11 c 21 c 22 1 1 Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 Twin 1 Phenotype 2

19 Within-Twin Covariances (E) Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 +e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Phenotype 1 E1E1 E2E2 e 11 e 21 e 22 1 1 Twin 1 Phenotype 2

20 Cross-Twin Covariances (A) Twin 1 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 1 Phenotype 2 Twin 2 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 2 Phenotype 2 1/0.5 Phenotype 1Phenotype 2 Phenotype 1 Phenotype 2 +c 11 c 21 +c 22 2 +c 21 2 Twin 1 Twin 2

21 Cross-Twin Covariances (A) Twin 1 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 1 Phenotype 2 Twin 2 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 2 Phenotype 2 1/0.5 Phenotype 1Phenotype 2 Phenotype 1 1/0.5a 11 2 + Phenotype 2 +c 11 c 21 +c 22 2 +c 21 2 Twin 1 Twin 2

22 Cross-Twin Covariances (A) Twin 1 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 1 Phenotype 2 Twin 2 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 2 Phenotype 2 1/0.5 Phenotype 1Phenotype 2 Phenotype 1 1/0.5a 11 2 +c 11 2 Phenotype 2 1/0.5a 11 a 21 +c 11 c 21 +c 22 2 +c 21 2 Twin 1 Twin 2

23 Cross-Twin Covariances (A) Twin 1 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 1 Phenotype 2 Twin 2 Phenotype 1 A1A1 A2A2 a 11 a 21 a 22 1 1 Twin 2 Phenotype 2 1/0.5 Phenotype 1Phenotype 2 Phenotype 1 1/0.5a 11 2 +c 11 2 Phenotype 2 1/0.5a 11 a 21 +c 11 c 21 1/0.5a 22 2 +1/0.5a 21 2 +c 22 2 +c 21 2 Twin 1 Twin 2

24 Cross-Twin Covariances (C) Twin 1 Phenotype 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 Twin 2 Phenotype 1 Twin 2 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 1 1 Phenotype 1Phenotype 2 Phenotype 1 1/0.5a 11 2 +c 11 2 Phenotype 2 1/0.5a 11 a 21 +c 11 c 21 1/0.5a 22 2 +1/0.5a 21 2 +c 22 2 +c 21 2 Twin 1 Twin 2

25 Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 a 11 2 +c 11 2 +e 11 2 Phenotype 2 a 11 a 21 +c 11 c 21 + e 11 e 21 a 22 2 +a 21 2 +c 22 2 + c 21 2 +e 22 2 +e 21 2 Phenotype 1 1/.5a 11 2 +c 11 2 a 11 2 +c 11 2 +e 11 2 Phenotype 2 1/.5a 11 a 21 + c 11 c 21 1/.5a 22 2 +1/.5 a 21 2 +c 22 2 +c 21 2 a 11 a 21 +c 11 c 21 + e 11 e 21 a 22 2 +a 21 2 +c 22 2 +c 21 2 +e 22 2 +e 21 2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance

26 Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance

27 Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance Variance of P1 and P2 the same across twins and zygosity groups

28 Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance Covariance of P1 and P2 the same across twins and zygosity groups

29 Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance within each trait differs by zygosity

30 Predicted Model Phenotype 1Phenotype 2Phenotype 1Phenotype 2 Phenotype 1 Variance P1 Phenotype 2 Covariance P1-P2 Variance P2 Phenotype 1 Within-trait P1 Cross-traitVariance P1 Phenotype 2 Cross-traitWithin-trait P2 Covariance P1-P2 Variance P2 Twin 1 Twin 2 Within-twin covariance Cross-twin covariance Cross-twin cross-trait covariance differs by zygosity

31 Example Covariance Matrix P1P2P1P2 P1 1 P2.301 P1 0.790.491 P2 0.500.590.291 P1P2P1P2 P1 1 P2 0.301 P1 0.390.251 P2 0.240.430.311 Twin 1 Twin 2 Twin 1 Twin 2 MZ DZ Within-twin covariance Cross-twin covariance

32 Example Covariance Matrix P1P2P1P2 P1 1 P2.301 P1 0.790.491 P2 0.500.590.291 P1P2P1P2 P1 1 P2 0.301 P1 0.390.251 P2 0.240.430.311 Twin 1 Twin 2 Twin 1 Twin 2 MZ DZ Within-twin covariance Cross-twin covariance

33 Example Covariance Matrix P1P2P1P2 P1 1 P2.301 P1 0.790.491 P2 0.500.590.291 P1P2P1P2 P1 1 P2 0.301 P1 0.390.251 P2 0.240.430.311 Twin 1 Twin 2 Twin 1 Twin 2 MZ DZ Within-twin covariance Cross-twin covariance

34 Example Covariance Matrix P1P2P1P2 P1 1 P2.301 P1 0.790.251 P2 0.240.590.291 P1P2P1P2 P1 1 P2 0.301 P1 0.390.251 P2 0.240.430.311 Twin 1 Twin 2 Twin 1 Twin 2 MZ DZ Within-twin covariance Cross-twin covariance

35 Example Covariance Matrix P1P2P1P2 P1 1 P2.301 P1 0.790.011 P2 0.010.590.291 P1P2P1P2 P1 1 P2 0.301 P1 0.390.011 P2 0.010.430.311 Twin 1 Twin 2 Twin 1 Twin 2 MZ DZ Within-twin covariance Cross-twin covariance

36 Summary Within-individual cross-trait covariance implies common aetiological influences Cross-twin cross-trait covariance implies common aetiological influences are familial Whether familial influences genetic or environmental shown by MZ:DZ ratio of cross-twin cross-trait covariances

37 Cholesky Decomposition Specification in Mx

38 Mx Parameter Matrices #define nvar 2 Begin Matrices; X lower nvar nvar free! Genetic coefficients Y lower nvar nvar free! C coefficients Z lower nvar nvar free! E coefficients G Full 1 nvar free! Means H Full 1 1 fix! 0.5 for DZ A covar End Matrices; Begin Algebra; A=X*X’;! A var/cov C=Y*Y’;! C var/cov E=Z*Z’; ! E var/cov P=A+C+E End Algebra;

39 Within-Twin Covariance P1 P2 A1A1 A2A2 a 11 a 21 a 22 1 1 Path Tracing:

40 Within-Twin Covariance P1 P2 A1A1 A2A2 a 11 a 21 a 22 1 1 Path Tracing: X Lower 2 x 2: P1 P2 A1A2

41 Within-Twin Covariance P1 P2 A1A1 A2A2 a 11 a 21 a 22 1 1 Path Tracing: X Lower 2 x 2: P1 P2 A1A2 a 11 a 21 a 22

42 Total Within-Twin Covar. Using matrix addition, the total within-twin covariance for the phenotypes is defined as:

43 Cross-Twin Covariance (DZ) P11 P21 A1A1 A2A2 a 11 a 21 a 22 1 1 P12 P22 A1A1 A2A2 a 11 a 21 a 22 1 1 0.5 Twin 1Twin 2

44 Cross-Twin Covariance (DZ) P11 P21 A1A1 A2A2 a 11 a 21 a 22 1 1 P12 P22 A1A1 A2A2 a 11 a 21 a 22 1 1 0.5 Twin 1Twin 2 Within-traits P11-P12 = 0.5a 11 2 P21-P22 = 0.5a 22 2 +0.5a 21 2

45 Cross-Twin Covariance (DZ) P11 P21 A1A1 A2A2 a 11 a 21 a 22 1 1 P12 P22 A1A1 A2A2 a 11 a 21 a 22 1 1 0.5 Twin 1Twin 2 Path Tracing: Within-traits P11-P12 = 0.5a 11 2 P21-P22 = 0.5a 22 2 +0.5a 21 2 Cross-traits P11-P22 = 0.5a 11 a 21 P21-P12 = 0.5a 21 a 11

46 Additive Genetic Cross-Twin Covariance (DZ) P11 P21 A1A1 A2A2 a 11 a 21 a 22 1 1 P12 P22 A1A1 A2A2 a 11 a 21 a 22 1 1 0.5 Twin 1Twin 2 Path Tracing: Within-traits P11-P12 = 0.5a 11 2 P21-P22 = 0.5a 22 2 +0.5a 21 2 Cross-traits P11-P22 = 0.5a 11 a 21 P21-P12 = 0.5a 21 a 11 N.B. in Mx  = @

47 Additive Genetic Cross-Twin Covariance (MZ) P11 P21 A1A1 A2A2 a 11 a 21 a 22 1 1 P12 P22 A1A1 A2A2 a 11 a 21 a 22 1 1 1 1 Twin 1Twin 2

48 Common Environment Cross- Twin Covariance P11 P21 C1C1 C2C2 c 11 c 21 c 22 1 1 P12 P22 C1C1 C2C2 c 11 c 21 c 22 1 1 1 1 Twin 1Twin 2

49 Covariance Model for Twin Pairs MZ:  CovarianceA+C+E | A+C_ A+C | A+C+E / DZ:  CovarianceA+C+E | H@A+C_ H@A+C | A+C+E / N.B. H Full 1 1 Fixed = 0.5

50 Obtaining Standardised Estimates

51 Correlated Factors Solution Phenotype 1 E1E1 E2E2 A1A1 A2A2  A11 2 Phenotype 2 rGrG C1C1 C2C2 rCrC 1111 1 1 rErE  C11 2  A22 2  C22 2  E11 2  E22 2 Each variable decomposed into genetic/environmental components Correlations across variables estimated Results from Cholesky can be converted to this model

52 Using matrix algebra notation: Covariance to Correlation

53 Genetic Correlations

54 Specification in Mx 1.Matrix function in Mx: R = \stnd(A); 2.R = \sqrt(I.A)˜ * A * \sqrt(I.A)˜; Where I is an identity matrix: and I.A =

55 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

56 Example ADHD A1A1 A2A2  h P1 2 IQ rGrG 11  h P2 2 N.B.  h P1 2 =  A11 2 Proportion of r P due to additive genetic factors:

57 Standardised Results Begin algebra; K = A%P|C%P|E%P; End algebra; % is the Mx operator for element division Proportion of the phenotypic correlation due to genetic effects

58 Example Mx Output Matrix K:  Additive Genetic Component of ADHD = 63%, for IQ = 33%  % of covariance between ADHD and IQ due to A = 84% Matrix K Estimates [A%P|C%P|E%P] 1 2 3 4 5 6 1 0.6286 0.8360 0.0000 0.0000 0.3714 0.1640 2 0.8360 0.3338 0.0000 0.3666 0.1640 0.2996 [S=\STND(P)] [R=\STND(A)] 1 21 2 1 1.0000 -0.2865 1 1.0000 -0.2865 2 -0.2865 1.0000 2 -0.5248 1.0000 Phenotypic correlation Genetic correlation

59 Interpretation of Correlations Consider two traits with a phenotypic correlation 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.

60 Interpretation of Correlations Consider two traits with a phenotypic correlation of 0.40 : h 2 P1 = 0.7 and h 2 P2 = 0.6 with r G =.3 Correlation due to additive genetic effects = 0.19 Proportion of phenotypic correlation attributable to additive genetic effects = 0.49 h 2 P1 = 0.2 and h 2 P2 = 0.3 with r G = 0.8 Correlation due to additive genetic effects = 0.20 Proportion of phenotypic correlation attributable to additive genetic effects = 0.49 Weakly heritable traits can still have a large portion of their correlation attributable to genetic effects.

61 More Variables… Twin 1 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 Twin 1 Phenotype 3 E3E3 e 33 e 31 e 32 C3C3 c 33 1 A3A3 a 33 c 32 a 31 c 31 a 32 1 1

62 More Variables… Twin 1 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 1 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 1/0.5 1 1 Twin 1 Phenotype 3 E3E3 e 33 e 31 e 32 C3C3 c 33 1 A3A3 a 33 c 32 a 31 c 31 a 32 1 1 Twin 2 Phenotype 1 A1A1 A2A2 E1E1 E2E2 a 11 a 21 a 22 e 11 e 21 e 22 1 1 1 1 Twin 2 Phenotype 2 C1C1 C2C2 c 11 c 21 c 22 1 1 Twin 2 Phenotype 3 E3E3 e 33 e 31 e 32 C3C3 c 33 1 A3A3 a 33 c 32 a 31 c 31 a 32 1 1 1/0.5 1

63 Mx Parameter Matrices #define nvar 3 Begin Matrices; X lower nvar nvar free! Genetic coefficients Y lower nvar nvar free! C coefficients Z lower nvar nvar free! E coefficients G Full 1 nvar free! Means H Full 1 1 fix! 0.5 for DZ A covar End Matrices; Begin Algebra; A=X*X’;! Gen var/cov C=Y*Y’;! C var/cov E=Z*Z’; ! E var/cov P=A+C+E End Algebra;

64 Expanded Matrices X Lower 3 x 3 Y Lower 3 x 3 Z Lower 3 x 3

Download ppt "Introduction to Multivariate Genetic Analysis Kate Morley and Frühling Rijsdijk 21st Twin and Family Methodology Workshop, March 2008."

Similar presentations

Multivariate Twin Analysis

Multivariate Twin Analysis
SEM PURPOSE Model phenomena from observed or theoretical stances

SEM PURPOSE Model phenomena from observed or theoretical stances
Bivariate analysis HGEN619 class 2007.

Bivariate analysis HGEN619 class 2007.
Fitting Bivariate Models October 21, 2014 Elizabeth Prom-Wormley & Hermine Maes 804-828-8154.

Fitting Bivariate Models October 21, 2014 Elizabeth Prom-Wormley & Hermine Maes
Multivariate Mx Exercise D Posthuma Files: \\danielle\Multivariate.

Multivariate Mx Exercise D Posthuma Files: \\danielle\Multivariate.
Estimating “Heritability” using Genetic Data David Evans University of Queensland.

Estimating “Heritability” using Genetic Data David Evans University of Queensland.
Factor analysis Caroline van Baal March 3 rd 2004, Boulder.

Factor analysis Caroline van Baal March 3 rd 2004, Boulder.
Path Analysis Danielle Dick Boulder 2006. Path Analysis Allows us to represent linear models for the relationships between variables in diagrammatic form.

Path Analysis Danielle Dick Boulder Path Analysis Allows us to represent linear models for the relationships between variables in diagrammatic form.
Multivariate Analysis Nick Martin, Hermine Maes TC21 March 2008 HGEN619 10/20/03.

Multivariate Analysis Nick Martin, Hermine Maes TC21 March 2008 HGEN619 10/20/03.
Introduction to Multivariate Analysis Frühling Rijsdijk & Shaun Purcell Twin Workshop 2004.

Introduction to Multivariate Analysis Frühling Rijsdijk & Shaun Purcell Twin Workshop 2004.
Multivariate Genetic Analysis: Introduction(II) Frühling Rijsdijk & Shaun Purcell Wednesday March 6, 2002.

Multivariate Genetic Analysis: Introduction(II) Frühling Rijsdijk & Shaun Purcell Wednesday March 6, 2002.
Developmental models. Multivariate analysis choleski models factor models –y =  f + u genetic factor models –P j = h G j + c C j + e E j –common pathway.

Developmental models. Multivariate analysis choleski models factor models –y =  f + u genetic factor models –P j = h G j + c C j + e E j –common pathway.
Univariate Analysis in Mx Boulder, 2004. Group Structure Title Type: Data/ Calculation/ Constraint Reading Data Matrices Declaration Assigning Specifications/

Univariate Analysis in Mx Boulder, Group Structure Title Type: Data/ Calculation/ Constraint Reading Data Matrices Declaration Assigning Specifications/
Multivariate Analysis Hermine Maes TC19 March 2006 HGEN619 10/20/03.

Multivariate Analysis Hermine Maes TC19 March 2006 HGEN619 10/20/03.
Biometrical Genetics Pak Sham & Shaun Purcell Twin Workshop, March 2002.

Biometrical Genetics Pak Sham & Shaun Purcell Twin Workshop, March 2002.
Univariate Analysis Hermine Maes TC19 March 2006.

Univariate Analysis Hermine Maes TC19 March 2006.
Genetic Growth Curve Models Practica Boulder, March 2008 Irene Rebollo & Gitta Lubke & Mike Neale VU Amsterdam NL & Notre Dame, US & VCU, US.

Genetic Growth Curve Models Practica Boulder, March 2008 Irene Rebollo & Gitta Lubke & Mike Neale VU Amsterdam NL & Notre Dame, US & VCU, US.
Gene x Environment Interactions Brad Verhulst (With lots of help from slides written by Hermine and Liz) September 30, 2014.

Gene x Environment Interactions Brad Verhulst (With lots of help from slides written by Hermine and Liz) September 30, 2014.
Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK.

Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK.
Mx Practical TC18, 2005 Dorret Boomsma, Nick Martin, Hermine H. Maes.

Mx Practical TC18, 2005 Dorret Boomsma, Nick Martin, Hermine H. Maes.

Similar presentations

Ads by Google