Copy files Go to FacultymarleenBoulder 2012Multivariate Copy all
Copy files • Go to FacultymarleenBoulder 2012Multivariate • Copy all files to your own directory • Go to FacultykeesBoulder 2012Multivariate • Copy all files to your own directory March 7, 2012 M. de Moor, Twin Workshop Boulder 1
Introduction to Multivariate Genetic Analysis (1) Marleen de Moor, Kees-Jan Kan & Nick Martin March 7, 2012 M. de Moor, Twin Workshop Boulder 2
Outline • 11. 00 -12. 30 – Lecture Bivariate Cholesky Decomposition – Practical Bivariate analysis of IQ and attention problems • 12. 30 -13. 30 LUNCH • 13. 30 -15. 00 – Lecture Multivariate Cholesky Decomposition – PCA versus Cholesky – Practical Tri- and Four-variate analysis of IQ, educational attainment and attention problems March 7, 2012 M. de Moor, Twin Workshop Boulder 3
Outline • 11. 00 -12. 30 – Lecture Bivariate Cholesky Decomposition – Practical Bivariate analysis of IQ and attention problems • 12. 30 -13. 30 LUNCH • 13. 30 -15. 00 – Lecture Multivariate Cholesky Decomposition – PCA versus Cholesky – Practical Tri- and Four-variate analysis of IQ, educational attainment and attention problems March 7, 2012 M. de Moor, Twin Workshop Boulder 4
Aim / Rationale multivariate models 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 D) or shared environmental factors (C or E) • Can use information on multiple traits from twin pairs to partition covariation into genetic and environmental components March 7, 2012 M. de Moor, Twin Workshop Boulder 5
Example • Interested in relationship between ADHD and IQ r. G 1 1 A 1 • How can we explain the association r. C 1 A 2 C 1 a 11 1 c 11 a 22 ADHD – Additive genetic factors (r. G) – Common environment (r. C) – Unique environment (r. E) C 2 c 22 IQ e 11 e 22 1 1 E 2 r. E March 7, 2012 M. de Moor, Twin Workshop Boulder 6
March 7, 2012 M. de Moor, Twin Workshop Boulder 7
Bivariate Cholesky Multivariate Cholesky March 7, 2012 M. de Moor, Twin Workshop Boulder 8
Sources of information • Two traits measured in twin pairs • Interested in: – Cross-trait covariance within individuals = phenotypic covariance – Cross-trait covariance between twins = cross-trait crosstwin covariance – MZ: DZ ratio of cross-trait covariance between twins March 7, 2012 M. de Moor, Twin Workshop Boulder 9
Observed Covariance Matrix: 4 x 4 Twin 1 Twin 2 Twin 1 Phenotype 1 Twin 2 Phenotype 1 Variance P 1 Phenotype 2 Covariance P 1 -P 2 Variance P 2 Phenotype 1 Within-trait P 1 Cross-trait Variance P 1 Cross-trait Within-trait P 2 Covariance P 1 -P 2 Phenotype 2 Variance P 2
Observed Covariance Matrix: 4 x 4 Twin 1 Phenotype 1 Twin 2 Phenotype 1 Phenotype 2 Twin 1 Within-twin covariance Phenotype 1 Variance P 1 Phenotype 2 Covariance P 1 -P 2 Variance P 2 Twin 2 Within-twin covariance Phenotype 1 Phenotype 2 Within-trait P 1 Cross-trait Variance P 1 Cross-trait Within-trait P 2 Covariance P 1 -P 2 Variance P 2
Observed Covariance Matrix: 4 x 4 Twin 1 Phenotype 1 Twin 2 Phenotype 1 Phenotype 2 Twin 1 Within-twin covariance Phenotype 1 Variance P 1 Phenotype 2 Covariance P 1 -P 2 Variance P 2 Twin 2 Cross-twin covariance Phenotype 1 Phenotype 2 Within-twin covariance Within-trait P 1 Cross-trait Variance P 1 Cross-trait Within-trait P 2 Covariance P 1 -P 2 Variance P 2
Cholesky decomposition 1 1 1/. 5 C 1 A 1 C 2 A 2 1 1 C 1 A 1 c 21 c 11 a 21 a 11 e 11 1 c 22 Twin 1 Phenotype 2 e 21 March 7, 2012 c 11 a 21 a 11 e 11 1 E 2 c 22 a 22 Twin 2 Phenotype 1 e 22 1 E 1 C 2 A 2 c 21 a 22 Twin 1 Phenotype 1 1 1 e 22 1 E 1 M. de Moor, Twin Workshop Boulder E 2 13
Now let’s do the path tracing! 1 1 1/. 5 C 1 A 1 C 2 A 2 1 1 C 1 A 1 c 21 c 11 a 21 a 11 e 11 1 c 22 Twin 1 Phenotype 2 e 21 March 7, 2012 c 11 a 21 a 11 e 11 1 E 2 c 22 a 22 Twin 2 Phenotype 1 e 22 1 E 1 C 2 A 2 c 21 a 22 Twin 1 Phenotype 1 1 1 e 22 1 E 1 M. de Moor, Twin Workshop Boulder E 2 14
Within-Twin Covariances (A) 1 1 A 1 a 11 A 2 a 21 Twin 1 Phenotype 1 a 22 Twin 1 Phenotype 1 a 112+c 112+e 112 Phenotype 2 a 11 a 21+c 11 c 21+e 11 e 21 Phenotype 2 a 222+a 212+c 222+c 212 +e 222+e 212
Within-Twin Covariances (A) 1 1 A 1 a 11 A 2 a 21 Twin 1 Phenotype 1 a 22 Twin 1 Phenotype 1 a 112+c 112+e 112 Phenotype 2 a 11 a 21+c 11 c 21+e 11 e 21 Phenotype 2 a 222+a 212+c 222+c 212 +e 222+e 212
Within-Twin Covariances (A) 1 1 A 1 a 11 A 2 a 21 Twin 1 Phenotype 1 a 22 Twin 1 Phenotype 1 a 112+c 112+e 112 Phenotype 2 a 11 a 21+c 11 c 21+e 11 e 21 Phenotype 2 a 222+a 212+c 222+c 212 +e 222+e 212
Within-Twin Covariances (A) 1 1 A 1 a 11 A 2 a 21 Twin 1 Phenotype 1 a 22 Twin 1 Phenotype 1 a 112+c 112+e 112 Phenotype 2 a 11 a 21+c 11 c 21+e 11 e 21 Phenotype 2 a 222+a 212+c 222+c 212 +e 222+e 212
Within-Twin Covariances (C) 1 1 C 1 c 11 C 2 c 21 Twin 1 Phenotype 1 c 22 Twin 1 Phenotype 1 a 112+c 112+e 112 Phenotype 2 a 11 a 21+c 11 c 21+e 11 e 21 Phenotype 2 a 222+a 212+c 222+c 212 +e 222+e 212
Within-Twin Covariances (E) Twin 1 Phenotype 2 Twin 1 Phenotype 1 e 11 1 e 22 1 E 2 Twin 1 Phenotype 1 a 112+c 112+e 112 Phenotype 2 a 11 a 21+c 11 c 21+e 11 e 21 Phenotype 2 a 222+a 212+c 222+c 212 +e 222+e 212
Cross-Twin Covariances (A) 1/0. 5 1 1 1 A 1 a 11 Twin 1 Phenotype 1 A 2 a 21 1/0. 5 1 A 1 a 11 a 22 Twin 1 Phenotype 2 Twin 2 Phenotype 1 A 2 a 21 a 22 Twin 2 Phenotype 2 Twin 1 Phenotype 2 +c 11 c 21 +c 222+c 212 Phenotype 1 Phenotype 2
Cross-Twin Covariances (A) 1/0. 5 1 1 1 A 1 a 11 Twin 1 Phenotype 1 A 2 a 21 1/0. 5 a 22 Twin 1 Phenotype 2 1 A 1 a 11 Twin 2 Phenotype 1 A 2 a 21 a 22 Twin 2 Phenotype 2 Twin 1 Phenotype 2 Phenotype 1 Phenotype 2 1/0. 5 a 112+ +c 11 c 21 +c 222+c 212
Cross-Twin Covariances (A) 1/0. 5 1 1 1 A 1 a 11 Twin 1 Phenotype 1 A 2 a 21 1/0. 5 a 22 Twin 1 Phenotype 2 1 A 1 a 11 Twin 2 Phenotype 1 A 2 a 21 a 22 Twin 2 Phenotype 2 Twin 1 Phenotype 2 Phenotype 1 Phenotype 2 1/0. 5 a 112+c 112 1/0. 5 a 11 a 21+c 11 c 21 +c 222+c 212
Cross-Twin Covariances (A) 1/0. 5 1 1 1 A 1 a 11 Twin 1 Phenotype 1 A 2 a 21 1/0. 5 a 22 Twin 1 Phenotype 2 1 A 1 a 11 Twin 2 Phenotype 1 A 2 a 21 a 22 Twin 2 Phenotype 2 Twin 1 Twin 2 Phenotype 1 Phenotype 2 1/0. 5 a 112+c 112 1/0. 5 a 11 a 21+c 11 c 21 1/0. 5 a 222+1/0. 5 a 212+c 222+c 212
Cross-Twin Covariances (C) 1 C 1 1 1 C 1 C 2 c 21 c 11 Twin 1 Phenotype 1 c 11 c 22 Twin 1 Phenotype 2 Twin 2 Phenotype 1 c 22 Twin 2 Phenotype 2 Twin 1 Twin 2 Phenotype 1 Phenotype 2 1/0. 5 a 112+c 112 1/0. 5 a 11 a 21+c 11 c 21 1/0. 5 a 222+1/0. 5 a 212+c 222+c 212
Predicted Model Twin 1 Phenotype 2 Twin 2 Phenotype 1 Phenotype 2 Twin 1 Within-twin covariance Phenotype 1 a 112+c 112+e 112 Phenotype 2 a 11 a 21+c 11 c 21+ e 11 e 21 a 222+a 212+c 222+ c 212+e 222+e 212 Twin 2 Cross-twin covariance Phenotype 1 1/. 5 a 112+c 112 Phenotype 2 1/. 5 a 11 a 21+ c 11 c 21 Within-twin covariance a 112+c 112+e 112 1/. 5 a 222+1/. 5 a 212+c 222+c 212 a 11 a 21+c 11 c 21+ e 11 e 21 a 222+a 212+c 222 +c 212+e 222+e 21
Predicted Model Twin 1 Phenotype 1 Twin 2 Phenotype 1 Phenotype 2 Twin 1 Within-twin covariance Phenotype 1 Variance P 1 Phenotype 2 Covariance P 1 -P 2 Variance P 2 Twin 2 Cross-twin covariance Phenotype 1 Phenotype 2 Within-twin covariance Within-trait P 1 Cross-trait Variance P 1 Cross-trait Within-trait P 2 Covariance P 1 -P 2 Variance P 2
Predicted Model Twin 1 Phenotype 1 Twin 2 Phenotype 1 Phenotype 2 Twin 1 Within-twin covariance Phenotype 1 Variance P 1 Phenotype 2 Covariance P 1 -P 2 Variance P 2 Twin 2 Cross-twin covariance Phenotype 1 Phenotype 2 Variance of P 1 and P 2 the same across twins and zygosity groups Within-twin covariance Within-trait P 1 Cross-trait Variance P 1 Cross-trait Within-trait P 2 Covariance P 1 -P 2 Variance P 2
Predicted Model Twin 1 Phenotype 1 Twin 2 Phenotype 1 Phenotype 2 Twin 1 Within-twin covariance Phenotype 1 Variance P 1 Phenotype 2 Covariance P 1 -P 2 Variance P 2 Twin 2 Cross-twin covariance Phenotype 1 Phenotype 2 Covariance of P 1 and P 2 the same across twins and zygosity groups Within-twin covariance Within-trait P 1 Cross-trait Variance P 1 Cross-trait Within-trait P 2 Covariance P 1 -P 2 Variance P 2
Predicted Model Twin 1 Phenotype 1 Twin 2 Phenotype 1 Phenotype 2 Twin 1 Within-twin covariance Phenotype 1 Variance P 1 Phenotype 2 Covariance P 1 -P 2 Variance P 2 Twin 2 Cross-twin covariance Phenotype 1 Phenotype 2 Cross-twin covariance within each trait differs by zygosity Within-twin covariance Within-trait P 1 Cross-trait Variance P 1 Cross-trait Within-trait P 2 Covariance P 1 -P 2 Variance P 2
Predicted Model Twin 1 Phenotype 1 Twin 2 Phenotype 1 Phenotype 2 Twin 1 Within-twin covariance Phenotype 1 Variance P 1 Phenotype 2 Covariance P 1 -P 2 Variance P 2 Twin 2 Cross-twin covariance Phenotype 1 Phenotype 2 Cross-twin cross-trait covariance differs by zygosity Within-twin covariance Within-trait P 1 Cross-trait Variance P 1 Cross-trait Within-trait P 2 Covariance P 1 -P 2 Variance P 2
Example covariance matrix MZ Twin 1 ADHD Twin 2 IQ ADHD IQ Within-twin covariance Twin 1 ADHD IQ 1 -0. 26 1 Twin 2 Cross-twin covariance Within-twin covariance ADHD 0. 64 -0. 21 1 IQ -0. 25 0. 70 -0. 31 1
Example covariance matrix DZ Twin 1 ADHD Twin 2 IQ ADHD IQ Within-twin covariance Twin 1 ADHD IQ 1 -0. 31 1 Twin 2 Cross-twin covariance Within-twin covariance ADHD 0. 20 -0. 12 1 IQ -0. 12 0. 53 -0. 27 1
Kuntsi et al. study March 7, 2012 M. de Moor, Twin Workshop Boulder 34
Summary • Within-twin cross-trait covariance (phenotypic covariance) implies common aetiological influences • Cross-twin cross-trait covariances >0 implies common aetiological influences are familial • Whether familial influences are genetic or common environmental is shown by MZ: DZ ratio of cross-twin cross-trait covariances
Specification in Open. Mx? Open. Mx script…. March 7, 2012 M. de Moor, Twin Workshop Boulder 36
Within-Twin Covariance (A) 1 Path Tracing: 1 A 1 a 11 A 2 a 21 a 22 Lower 2 x 2 matrix: P 1 P 2 a 1 a 2
Within-Twin Covariance (A) Open. Mx Vars <- c("FSIQ", "Att. Prob") nv <- length(Vars) a. Labs <- c("a 11“, "a 21", "a 22") path. A <- mx. Matrix(name = "a", type = "Lower", nrow = nv, ncol = nv, labels = a. Labs) cov. A <- mx. Algebra(name = "A", expression = a %*% t(a))
Within-Twin Covariance (A+C+E) Using matrix addition, the total within-twin covariance for the phenotypes is defined as:
Open. Mx Matrices & Algebra Open. Mx Vars <- c("FSIQ", "Att. Prob") nv <- length(Vars) a. Labs <- c("a 11“, "a 21", "a 22") c. Labs <- c("c 11", "c 22") e. Labs <- c("e 11", "e 22") # Matrices a, c, and e to store a, c, and e Path Coefficients path. A <- mx. Matrix(name = "a", type = "Lower", nrow = nv, ncol = nv, labels = a. Labs) path. C <- mx. Matrix(name = "c", type = "Lower", nrow = nv, ncol = nv, labels = c. Labs) path. E <- mx. Matrix(name = "e", type = "Lower", nrow = nv, ncol = nv, labels = e. Labs) # Matrices generated to hold A, C, and E computed Variance Components cov. A <- mx. Algebra(name = "A", expression = a %*% t(a)) cov. C <- mx. Algebra(name = "C", expression = c %*% t(c)) cov. E <- mx. Algebra(name = "E", expression = e %*% t(e)) # Algebra to compute total variances and standard deviations (diagonal only) cov. Ph <- mx. Algebra(name = "V", expression = A+C+E) mat. I <- mx. Matrix(name= "I", type="Iden", nrow = nv, ncol = nv) inv. SD <- mx. Algebra(name ="i. SD", expression = solve(sqrt(I*V)))
MZ Cross-Twin Covariance (A) Twin 1 Twin 2 1 1 1 A 1 a 11 A 2 a 21 P 1 Cross-twin within-trait: P 1 -P 1 = 1*a 112 P 2 -P 2 = 1*a 222+1*a 212 Cross-twin cross-trait: P 1 -P 2 = 1*a 11 a 21 P 2 -P 1 = 1*a 21 a 11 a 22 P 2 1 A 1 a 11 P 1 A 2 a 21 a 22 P 2
DZ Cross-Twin Covariance (A) Twin 1 Twin 2 0. 5 1 1 A 1 a 11 A 2 a 21 P 1 Cross-twin within-trait: P 1 -P 1 = 0. 5 a 112 P 2 -P 2 = 0. 5 a 222+0. 5 a 212 Cross-twin cross-trait: P 1 -P 2 = 0. 5 a 11 a 21 P 2 -P 1 = 0. 5 a 21 a 11 a 22 P 2 1 A 1 a 11 P 1 A 2 a 21 a 22 P 2
MZ/DZ Cross-Twin Covariance (C) Twin 1 Twin 2 1 1 1 C 1 c 11 C 2 c 21 P 1 Cross-twin within-trait: P 1 -P 1 = 1*c 112 P 2 -P 2 = 1*c 222+1*c 212 Cross-twin cross-trait: P 1 -P 2 = 1*c 11 c 21 P 2 -P 1 = 1*c 21 c 11 c 22 P 2 1 C 1 c 11 P 1 C 2 c 21 c 22 P 2
Covariance Model for Twin Pairs Open. Mx # Algebra for expected variance/covariance matrix in MZ exp. Cov. MZ <- mx. Algebra(name = "exp. Cov. MZ", expression = rbind (cbind(A+C+E, A+C), cbind(A+C, A+C+E) ) ) # Algebra for expected variance/covariance matrix in DZ exp. Cov. DZ <- mx. Algebra(name = "exp. Cov. DZ", expression = rbind (cbind(A+C+E, 0. 5%x%A+C), cbind(0. 5%x%A+C, A+C+E) ) )
Unstandardized vs standardized solution
Genetic correlation • It is calculated by dividing the genetic covariance by the square root of the product of the genetic variances of the two variables
Genetic correlation A 1 a 11 a 21 P 11 1/. 5 A 2 A 1 a 22 P 21 Twin 1 a 11 A 2 a 21 P 12 a 22 P 22 Twin 2 Genetic covariance Sqrt of the product of the genetic variances
Standardized Solution = Correlated Factors Solution A 1 a 11 1/. 5 A 2 A 1 a 22 P 11 P 21 Twin 1 A 2 a 11 a 22 P 12 P 22 Twin 2
Genetic correlation – matrix algebra Open. Mx cor. A <- mx. Algebra(name ="r. A", expression = solve(sqrt(I*A))%*%A%*%solve(sqrt(I*A)))
Contribution to phenotypic correlation If the rg = 1, the two sets of genes overlap completely rg A 2 A 1 a 11 a 22 P 11 P 21 If however a 11 and a 22 are near to zero, genes do not contribute much to the phenotypic correlation Twin 1 Ø The contribution to the phenotypic correlation is a function of both heritabilities and the rg
Contribution to phenotypic correlation rg 1 Proportion of r. P 1, P 2 due to additive genetic factors: 1 A 2 a. P 12 a. P 22 P 1 rg 1 1 A 1 a 11 P 1 A 2 a 21 a 22 P 2
Contribution to phenotypic correlation Open. Mx ACEcov. Matrices <- c("A", "C", "E", "V", "A/V", "C/V", "E/V") ACEcov. Labels <("cov. Comp_A", "cov. Comp_C", "cov. Comp_E", "Var", "st. Cov. Comp_A", "st. Cov. Comp_C", "st. Cov. Comp_E") format. Output. Matrices(Chol. ACEFit, ACEcov. Matrices, ACEcov. Labels, Vars, 4) Proportion of the phenotypic correlation due to genetic effects Proportion of the phenotypic correlation due to shared environmental effects Proportion of the phenotypic correlation due to unshared environmental effects
Summary / Interpretation • Genetic correlation (rg) = the correlation between two latent genetic factors – High genetic correlation = large overlap in genetic effects on the two phenotypes • Contribution of genes to phenotypic correlation = The proportion of the phenotypic correlation explained by the overlapping genetic factors – This is a function of the rg and the heritabilities of the two traits
Outline • 11. 00 -12. 30 – Lecture Bivariate Cholesky Decomposition – Practical Bivariate analysis of IQ and attention problems • 12. 30 -13. 30 LUNCH • 13. 30 -15. 00 – Lecture Multivariate Cholesky Decomposition – Practical Tri- and Four-variate analysis of IQ, educational attainment and attention problems March 7, 2012 M. de Moor, Twin Workshop Boulder 54
Practical • Replicate findings from Kuntsi et al. • 126 MZ and 126 DZ twin pairs from Netherlands Twin Register • Age 12 • FSIQ • Attention Problems (AP) [mother-report] March 7, 2012 M. de Moor, Twin Workshop Boulder 55
Practical – exercise 1 • Script Cholesky. Bivariate. R • Dataset Cholesky. dat • Run script up to saturated model March 7, 2012 M. de Moor, Twin Workshop Boulder 56
Practical – exercise 1 • Fill in the table with correlations: MZ FSIQ 1 1 AP 1 FSIQ 2 1 AP 2 1 DZ FSIQ 1 1 AP 1 FSIQ 2 AP 2 1 1 AP 2 March 7, 2012 AP 2 1 M. de Moor, Twin Workshop Boulder 57
Practical – exercise 1 - Questions • Are correlations similar to those reported by Kuntsi et al. ? • What is the phenotypic correlation between FSIQ and AP? • What are the MZ and DZ cross-twin cross-trait correlations? • What are your expectations for the common aetiological influences? – Are they familial? – If yes, are they genetic or shared environmental? March 7, 2012 M. de Moor, Twin Workshop Boulder 58
Practical – exercise 2 • Run Bivariate ACE model in the script • Look whether you understand the output. If not, ask us! • Adapt the first submodel such that you drop all C • Compare fit of AE model with ACE model Script: Cholesky. Bivariate. R March 7, 2012 M. de Moor, Twin Workshop Boulder 59
Practical – exercise 2 • Fill in the table with fit statistics: -2 LL df ACE model chi 2 ∆df P-value - - - AE model • Question: – Is C significant? March 7, 2012 M. de Moor, Twin Workshop Boulder 60
Practical – exercise 3 • Now try to fill in the estimates for all paths in the path model (grey boxes): 1 1 A 2 FSIQ AP 1 1 E 2 M. de Moor, Twin Workshop Boulder March 7, 2012 61
- Slides: 61