Path Analysis, SEM and the Classical Twin Model

Transcription

1 Path Analysis, SEM and the Classical Twin Model Michael Neale & Frühling Rijsdijk 2 Virginia Institute for Psychiatric and Behavioral Genetics Virginia Commonwealth University 2 MRC SGDP Centre, Institute of Psychiatry, Psychology & Neuroscience, King s College London

2 Aims of Session. Introduction to SEM 2. Path Coefficient ACE Model 3. Variance Components ACE Model 4. ADE Model 5. RAM Algebra

3 Path Analysis and SEM Sewall Wright (92) PNAS, 6, Causal and correlational relationships between variables in path diagrams One-to-one mathematical equivalence with simple matrix algebra expression Structural equation modelling (SEM) is a unified platform for path analysis, regression, factor and variance components models

4 Path Analysis and SEM OpenMx Software R package Open Source Since 990

5 Path Diagram Conventions Observed Variables Latent Variables Causal Paths Covariance Paths

6 Tracing Rules of Path Analysis Find All Distinct Chains between Variables: a Go backwards along zero or more single-headed arrows b Change direction at one and only one Double-headed arrow c Trace forwards along zero or more Single-headed arrows 2 Multiply path coefficients in a chain 3 Sum the results of step 2. For covariance of a variable with itself (Variance), chains are distinct if they have different paths or a different order

7 Chain Examples I c c c c c From To From To b d b b b a e a a a From To To From From To a*b*c*d*e a*b*c a*b*c c a*b*c*b*a Thou shalt not pass through adjacent arrowheads

8 Chain Examples II e c f F F 2 a d From To a*e*a + d*f*d + a*c*d + d*c*a Variance: Chains in Different Order Count

9 Path Diagrams for the Classical Twin Model Part : Path Coefficients

10 Path Model for an MZ Pair E C A A 2 C 2 E 2 e c a a c e T T 2 Latent variables A C and E have variance, and cause phenotype T via path coefficients a, c and e. Same model for T2. Cov(A,A2)=

11 Path Model for a DZ Pair 0.5 E C A A 2 C 2 E 2 e c a a c e T T 2 Latent variables A C and E have variance, and cause phenotype T via regression paths a, c and e. Same model for T2. Cov(A,A2) =.5

12 Variance of Twin AND Twin 2 (for MZ and DZ pairs) What Chains? E C A e c a T Total Variance = a 2 + c 2 + e 2

13 Variance of Twin AND Twin 2 (for MZ and DZ pairs) a* = E C A e c a T Total Variance = a 2 + c 2 + e 2

14 Variance of Twin AND Twin 2 (for MZ and DZ pairs) a* E C A e c a T Total Variance = a 2 + c 2 + e 2

15 Variance of Twin AND Twin 2 (for MZ and DZ pairs) a**a = a 2 E C A e c a T Total Variance = a 2 +

16 Variance of Twin AND Twin 2 (for MZ and DZ pairs) a**a = a 2 E C A + c**c = c 2 e c a + T Total Variance = a 2 + c 2 +

17 Variance of Twin AND Twin 2 (for MZ and DZ pairs) a**a = a 2 E C A + c**c = c 2 e c a + e**e = e 2 T Total Variance = a 2 + c 2 + e 2

18 Covariance of Twin AND Twin 2 (for DZ pairs) 0.5 a*.5*a =.5a 2 E C A A 2 C 2 E 2 + e c a a c e T T 2 Covariance =.5a 2 +

19 Covariance of Twin AND Twin 2 (for DZ pairs) 0.5 a*.5*a =.5a 2 E e C A A 2 C 2 c a a c e E 2 + c* *c = c 2 T T 2 Total Covariance =.5a 2 + c 2

20 Predicted Variance-Covariance Cov MZ Matrices ACE Path Model Tw Tw a 2 +c 2 +e 2 a 2 +c 2 a 2 +c 2 a 2 +c 2 +e 2 Cov DZ Tw Tw a 2 +c 2 +e 2 ½a 2 +c 2 ½a 2 +c 2 a 2 +c 2 +e 2

21 Path Diagrams for the Classical Twin Model Part 2: Variance Components

22 Variance Component Model: MZ VC VA VA VA VA VA VC VE E C A A 2 C 2 E 2 T T 2 Latent variables A C and E have variances VA, VC and VE, and cause phenotype T via regression paths. Same model for T2

23 Variance Component Model: DZ VC.25VA VA VA VA VA VC VE E C A A 2 C 2 E 2 T T 2 Latent variables A C and E have variances VA, VC and VE, and cause phenotype T via regression paths. Same model for T2

24 Predicted Variance-Covariance Cov MZ Matrices ACE VC Model Tw Tw VA+VC+VE VA+VC VA+VC VA+VC+VE Tw Cov DZ Tw VA+VC+VE.5VA+VC.5VA+VC VA+VC+VE

25 What s the Difference? Path: Implicit Boundary Constraint Estimate a but a 2 never negative Variance Component: Unbounded Estimates VA, VC and VE can be positive or negative Variance Component may fit better No bias from implicit boundary Negative Variances? Model wrong?

26 ADE Path Coefficient Model DZ pairs E D A A 2 D 2 E 2 e d a a d e T T 2 MZ Covariance = a 2 + d 2 DZ Covariance =.5a d 2 Total Variance = a 2 + d 2 + e 2

27 Predicted Var-Cov Matrices Cov MZ Tw ADE Model Tw a 2 +d 2 +e 2 a 2 +d 2 a 2 +d 2 a 2 +d 2 +e 2 Cov DZ Tw Tw a 2 +d 2 +e 2 ½a 2 +¼d 2 ½a 2 +¼d 2 a 2 +d 2 +e 2

28 ADE Variance Component Model DZ pairs.25vd.5va VE VD VA VA VD VE E D A A 2 D 2 E 2 T T 2 MZ Covariance = VA + VD DZ Covariance =.5VA +.25VD Total Variance = VA + VD + VE

29 Predicted Variance-Covariance Cov MZ Matrices ADE VC Model Tw Tw VA+VD+VE VA+VD VA+VD VA+VD+VE Tw Cov DZ Tw VA+VD+VE.5VA+.25VD.5VA+.25VD VA+VD+VE

30 One-to-one Translation to Matrices RAM Algebra: Standardized Univariate Regression Ve e y b x Asymmetric Arrows Symmetric Slings Funky Filter A = S = F = x y e x y e x y x y e b V e apple

31 One-to-one Translation to Matrices Completely General RAM Algebra Expected Covariance Matrix F * (I - A) - * S * (I - A) - * F Thank you: Jack McArdle & Steve Boker Workshop Faculty & Students & NIH Also see: for path model drawing software