Multivariate Analysis. Hermine Maes TC19 March 2006

Transcription

1 Multivariate Analysis Hermine Maes TC9 March 2006

2 Files to Copy to your Computer Faculty/hmaes/tc9/maes/multivariate *.rec *.dat *.mx Multivariate.ppt

3 Multivariate Questions I Bivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between two traits? Multivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between more than two traits?

4 Phenotypic Cholesky F2 F3 F4 P f F2 F3 F4 P2 f 2 f f 2 f 3 f 4 P3 f 3 P4 f 4 P t P2 t P3 t P4 t

5 Phenotypic Cholesky F2 F2 F3 F4 P f F2 F3 0 F4 P2 f 2 f 22 f 22 f 32 f 42 P3 f 3 f 32 P4 f 4 f 42 P t P2 t P3 t P4 t

6 Phenotypic Cholesky F2 F3 F4 P f F2 F3 0 0 F4 P2 0 f 33 f 43 P3 f 2 f 22 f 3 f 33 P4 f 4 f 32 f 42 f 43 P t P2 t P3 t P4 t

7 Phenotypic Cholesky F2 F3 F4 P f F2 F3 F P2 f 2 f 22 0 f 33 0 f 44 P3 f 3 f 32 0 f f P4 f 4 f 42 P t P2 t P3 t P4 t

8 Cholesky Decomposition F2 F3 F4 P f f f 2 f 3 f 4 P2 P3 f 2 f f 3 f 32 f 33 0 f f * 0 0 f 22 f 32 f 42 0 f 33 f 43 P4 f 4 f f 44 F * F'

9 Saturated Model Use Cholesky decomposition to estimate covariance matrix Fully saturated Model: Cov P = F*F F: Lower nvar nvar

10 Phenotypic Single Factor P P2 f f 2 * f f 2 f 3 f 4 P3 f 3 f f 2 f 3 f 4 P4 f 4 F * F' P t P2 t P3 t P4 t

11 Residual Variances P P2 P3 E E2 E3 E4 e e e 33 0 e 44 * e e e 33 0 e 44 f f 2 f 3 f 4 P E * E' P t P2 t P3 t P4 t e e 22 e 33 e 44 E E2 E3 E4

12 Factor Analysis Explain covariance by limited number of factors Exploratory / Confirmatory Model: Cov P = F*F + E*E F: Full nvar nfac E: Diag nvar nvar Model: Cov P = F*I*F + E*E

13 Twin Data? f f 2 f 3 f 4 f f 2 f 3 f 4 P t P2 t P3 t P4 t P t2 P2 t2 P3 t2 P4 t2 e e 22 e 33 e 44 e e 22 e 33 e 44 E E2 E3 E4 E E2 E3 E4

14 Genetic Single Factor.0 / 0.5 A A a a 2 a 3 a 4 a a 2 a 3 a 4 P t P2 t P3 t P4 t P t2 P2 t2 P3 t2 P4 t2 e e 22 e 33 e 44 e e 22 e 33 e 44 E E2 E3 E4 E E2 E3 E4

15 Common Environmental Single Factor.0 / A C A C c c 2 c 3 c 4 c c 2 c 3 c 4 P t P2 t P3 t P4 t P t2 P2 t2 P3 t2 P4 t2 e e 22 e 33 e 44 e e 22 e 33 e 44 E E2 E3 E4 E E2 E3 E4

16 Specific Environmental Single Factor.0 / A C E A C E e e 2 e 3 e 4 e e 2 e 3 e 4 P t P2 t P3 t P4 t P t2 P2 t2 P3 t2 P4 t2 e e 22 e 33 e 44 e e 22 e 33 e 44 E E2 E3 E4 E E2 E3 E4

17 Single [Common] Factor X: genetic Full 4 x Full nvar x nfac P P2 P3 P4 A a a 2 * a a 2 a 3 a 4 a 3 a 4 X * X' Y: shared environmental Z: specific environmental

18 Residuals partitioned in ACE.0 / A C E A C E P t P2 t P3 t P4 t P t2 P2 t2 P3 t2 P4 t2 a c A C e e 22 e 33 e 44 E E2 E3 E4 e e 22 e 33 e 44 E E2 E3 E4

19 Residual Factors T: genetic U: shared environmental V: specific environmental Diag 4 x 4 Diag nvar x nvar E E2 E3 E4 P e P2 0 e P3 0 0 e 33 0 P e 44 V * * e e e e 44 V'

20 Independent Pathway Model.0 / A C C C [X] [Y] [Z] E C A C C C E C P t P2 t P3 t P4 t P t2 P2 t2 P3 t2 P4 t2 [U] C A E A2 E2 A3 E3 A4 E4 [T] [V] A E A2 E2 A3 E3 A4 E4.0 / / / / 0.5

21 IP Independent pathways Biometric model Different covariance structure for A, C and E

22 Independent Pathway I G: Define matrices Calculation Begin Matrices; X full nvar nfac Free! common factor genetic path coefficients Y full nvar nfac Free! common factor shared environment paths Z full nvar nfac Free! common factor unique environment paths T diag nvar nvar Free! variable specific genetic paths U diag nvar nvar Free! variable specific shared env paths V diag nvar nvar Free! variable specific residual paths M full nvar Free! means End Matrices; Start Begin Algebra; A= X*X' + T*T';! additive genetic variance components C= Y*Y' + U*U';! shared environment variance components E= Z*Z' + V*V';! nonshared environment variance components End Algebra; End indpath.mx

23 Independent Pathway II G2: MZ twins #include iqnlmz.dat Begin Matrices = Group ; Means M M ; Covariance A+C+E A+C _ A+C A+C+E ; Option Rsiduals End G3: DZ twins #include iqnldz.dat Begin Matrices= Group ; H full End Matrices; Matrix H.5 Means M M ; Covariance A+C+E H@A+C _ H@A+C A+C+E ; Option Rsiduals End

24 Independent Pathway III G4: Calculate Standardised Solution Calculation Matrices = Group I Iden nvar nvar End Matrices; Begin Algebra; R=A+C+E;! total variance S=(\sqrt(I.R))~;! diagonal matrix of standard deviations P=S*X_ S*Y_ S*Z;! standardized estimates for common factors Q=S*T_ S*U_ S*V;! standardized estimates for spec factors End Algebra; Labels Row P a a2 a3 a4 a5 a6 c c2 c3 c4 c5 c6 e e2 e3 e4 e5 e6 Labels Col P var var2 var3 var4 var5 var6 Labels Row Q as as2 as3 as4 as5 as6 cs cs2 cs3 cs4 cs5 cs6 es es2 es3 es4 es5 es6 Labels Col Q var var2 var3 var4 var5 var6 Options NDecimals=4 End

25 Practical Example Dataset: NL-IQ Study 6 WAIS-III IQ subtests var = onvolledige tekeningen / picture completion var2 = woordenschat / vocabulary var3 = paren associeren / digit span var4 = incidenteel leren / incidental learning var5 = overeenkomsten / similarities var6 = blokpatronen / block design N MZF: 27, DZF: 70

26 Dat Files iqnlmz.dat Data NInputvars=8 Missing=-.00 Rectangular File=iqnl.rec Labels famid zygos age_t sex_t var_t var2_t var3_t var4_t var5_t var6_t age_t2 sex_t2 var_t2 var2_t2 var3_t2 var4_t2 var5_t2 var6_t2 Select if zygos < 3 ;!select mz's Select var_t var2_t var3_t var4_t var5_t var6_t var_t2 var2_t2 var3_t2 var4_t2 var5_t2 var6_t2 ; iqnldz.dat... Select if zygos > 2 ;!select dz's...

27 Goodness-of-Fit Statistics -2LL df 2 df p AIC 2 df p Saturated (iqnlsat2.mx) Cholesky (cholesky.mx) Independent Pathway Common Pathway

28 MATRIX M This is a FULL matrix of order by MATRIX X This is a LOWER TRIANGULAR matrix of order 6 by MATRIX Y This is a LOWER TRIANGULAR matrix of order 6 by MATRIX Z This is a LOWER TRIANGULAR matrix of order 6 by

29 Goodness-of-Fit Statistics -2LL df 2 df p AIC 2 df p Saturated Cholesky Independent Pathway Common Pathway

30 MATRIX M This is a FULL matrix of order by MATRIX P This is a computed FULL matrix of order 8 by 6 [=S*X_S*Y_S*Z] VAR VAR2 VAR3 VAR4 VAR5 VAR6 A A A A A A C C C C C C E E E E E E

31 Independent Pathway Model.0 / A C C C [X] [Y] [Z] E C A C C C E C P t P2 t P3 t P4 t P t2 P2 t2 P3 t2 P4 t2 [U] C A E A2 E2 A3 E3 A4 E4 [T] [V] A E A2 E2 A3 E3 A4 E4.0 / / / / 0.5

32 Path Diagram to Matrices Variance Component a 2 c 2 e 2 Common Factors [X] 6 x [Y] 6 x [Z] 6 x Residual Factors [T] 6 x 6 [U] 6 x 6 [V] 6 x 6 #define nvar 6 #define nfac

33 MATRIX M This is a FULL matrix of order by MATRIX T This is a DIAGONAL matrix of order 6 by MATRIX U This is a DIAGONAL matrix of order 6 by MATRIX V This is a DIAGONAL matrix of order 6 by MATRIX X This is a FULL matrix of order 6 by MATRIX Y This is a FULL matrix of order 6 by MATRIX Z This is a FULL matrix of order 6 by

34 Goodness-of-Fit Statistics -2LL df 2 df p AIC 2 df p Saturated Cholesky Independent Pathway Common Pathway

35 MATRIX M This is a FULL matrix of order by MATRIX P 8 by [=S*X_S*Y_S*Z] VAR A A A A A A C C C C C C E E E E E5 0.3 E MATRIX Q This is a computed FULL matrix of order 8 by 6 [=S*T_S*U_S*V] VAR VAR2 VAR3 VAR4 VAR5 VAR6 AS AS AS AS AS AS CS CS CS CS CS CS ES ES ES ES ES ES

36 Exercise I. Drop common factor for shared environment 2. Drop common factor for specific environment 3. Add second genetic common factor

37 Phenotypic Single Factor P P2 f f 2 * f f 2 f 3 f 4 P3 f 3 f f 2 f 3 f 4 P4 f 4 F * F' P t P2 t P3 t P4 t

38 Latent Phenotype A C E a c e f f 2 f 3 f 4 P t P2 t P3 t P4 t

39 Twin Data.0 / A C E A C E a c e a c e f f 2 f 3 f 4 f f 2 f 3 f 4 P t P2 t P3 t P4 t P t P2 t P3 t P4 t

40 Factor on Latent Phenotype P P2 P3 P4 = f f 2 * a * a * f f 2 f 3 f 4 f 3 f 4 F * X * X' * F & ( X * X' ) F'

41 Common Pathway Model.0 / A C [X] a [Y] c e [Z] E A a C c e E [F] f f 2 f 3 f 4 f f 2 f 3 f 4 P t P2 t P3 t P4 t P t P2 t P3 t P4 t [U] C A E A2 E2 A3 E3 A4 E4 [T] [V] A E A2 E2 A3 E3 A4 E4.0 / / / / 0.5

42 Common Pathway Model I G: Define matrices Calculation Begin Matrices; X full nfac nfac Free! latent factor genetic path coefficient Y full nfac nfac Free! latent factor shared environment path Z full nfac nfac Free! latent factor unique environment path T diag nvar nvar Free! variable specific genetic paths U diag nvar nvar Free! variable specific shared env paths V diag nvar nvar Free! variable specific residual paths F full nvar nfac Free! loadings of variables on latent factor I Iden 2 2 M full nvar Free! means End Matrices; Start.. Begin Algebra; A= F&(X*X') + T*T';! genetic variance components C= F&(Y*Y') + U*U';! shared environment variance components E= F&(Z*Z') + V*V';! nonshared environment variance components L= X*X' + Y*Y' + Z*Z';! variance of latent factor End Algebra; End

43 Common Pathway II G4: Constrain variance of latent factor to Constraint Begin Matrices; L computed =L I unit End Matrices; Constraint L = I ; End G5: Calculate Standardised Solution Calculation Matrices = Group D Iden nvar nvar End Matrices; Begin Algebra; R=A+C+E;! total variance S=(\sqrt(D.R))~;! diagonal matrix of standard deviations P=S*F;! standardized estimates for loadings on F Q=S*T_ S*U_ S*V;! standardized estimates for specific factors End Algebra; Options NDecimals=4 End

44 CP Common pathway Psychometric model Same covariance structure for A, C and E

45 Common Pathway Model.0 / A C [X] a [Y] c e [Z] E A a C c e E [F] f f 2 f 3 f 4 f f 2 f 3 f 4 P t P2 t P3 t P4 t P t P2 t P3 t P4 t [U] C A E A2 E2 A3 E3 A4 E4 [T] [V] A E A2 E2 A3 E3 A4 E4.0 / / / / 0.5

46 Path Diagram to Matrices Variance Component a 2 c 2 e 2 Common Factor [X] x [Y] x [Z] x [F] 6 x Residual Factors [T] 6 x 6 [U] 6 x 6 [V] 6 x 6 #define nvar 6 #define nfac

47 MATRIX M This is a FULL matrix of order by MATRIX T This is a DIAGONAL matrix of order 6 by MATRIX U This is a DIAGONAL matrix of order 6 by MATRIX X This is a FULL matrix of order by MATRIX Y This is a FULL matrix of order by 2 MATRIX Z This is a FULL matrix of order by 3 MATRIX V This is a DIAGONAL matrix of order 6 by

48 Goodness-of-Fit Statistics -2LL df 2 df p AIC 2 df p Saturated Cholesky Independent Pathway Common Pathway

49 MATRIX M This is a FULL matrix of order by MATRIX P 6 by [=S*F] F F F F F MATRIX Q This is a computed FULL matrix of order 8 by 6 [=S*T_S*U_S*V] AS AS AS AS AS AS CS CS CS CS CS CS ES ES ES ES ES ES

50 WAIS-III IQ Verbal IQ var2 = woordenschat / vocabulary var3 = paren associeren / digit span var5 = overeenkomsten / similarities Performance IQ var = onvolledige tekeningen / picture completion var4 = incidenteel leren / incidental learning var6 = blokpatronen / block design

51 Exercise II. Change to 2 Latent Phenotypes corresponding to Verbal IQ and Performance IQ

52 Summary Independent Pathway Model Biometric Factor Model Loadings differ for genetic and environmental common factors Common Pathway Model Psychometric Factor Model Loadings equal for genetic and environmental common factor

53 Pathway Model [X] A [Y] C [Z] E A2.0 / 0.5 C2 E2.0 A3 C3 E3 A C E a a 2 a 22 a 3 a 32 a 33 a c e F2 F3 F2 F3 [F] f f 2 f 3 f 4 f 5 f 6 f f f 2 f 3 f 4 f 5 f 6 P t P2 t P3 t P4 t P5 t P6 t P t2 P2 t2 P3 t2 P4 t2 P5 t2 P6 t2 A E C [T] [V] [U] E2 E3 E4 E5 E6 E.0 / A C

54 Two Common Pathway Model [X] A [Y] C [Z] E A2.0 / 0.5 C2 E2.0 A3 C3 E3 A C E a a 2 a 22 a c e F2 F3 F2 F3 [F] f f 2 f 3 f 4 f 5 f f f 2 f 3 f 4 f 5 f 6 P t P2 t P3 t P4 t P5 t P6 t P t2 P2 t2 P3 t2 P4 t2 P5 t2 P6 t2 A E C [T] [V] [U] E2 E3 E4 E5 E6 E.0 / A C

55 Two Independent CP Model [X] A [Y] C [Z] E A2.0 / 0.5 C2 E2.0 A3 C3 E3 A C E a a 22 a c e F2 F3 F2 F3 [F] f f 2 f 3 f 4 f 5 f f f 2 f 3 f 4 f 5 f 6 P t P2 t P3 t P4 t P5 t P6 t P t2 P2 t2 P3 t2 P4 t2 P5 t2 P6 t2 A E C [T] [V] [U] E2 E3 E4 E5 E6 E.0 / A C

56 Two Reduced Indep CP Model [X] A [Y] C [Z] E A2.0 / 0.5 C2 E2.0 A3 C3 E3 A C E a a 22 a c e F2 F3 F2 F3 [F] f 2 f 3 f 2 f 5 f 42 f 62 f f 2 f 3 f 4 f 5 f 6 P t P2 t P3 t P4 t P5 t P6 t P t2 P2 t2 P3 t2 P4 t2 P5 t2 P6 t2 E E2 E3 E4 E5 E6 E A C A C [T] [V] [U].0 / 0.5.0

57 Common Pathway Model [X] A [Y] C [Z] E A2.0 / 0.5 C2 E2.0 A3 C3 E3 A C E a a c e F2 F3 F2 F3 [F] f f 2 f 3 f 4 f 5 f 6 f f 2 f 3 f 4 f 5 f 6 P t P2 t P3 t P4 t P5 t P6 t P t2 P2 t2 P3 t2 P4 t2 P5 t2 P6 t2 E E2 E3 E4 E5 E6 E A C A C [T] [V] [U].0 / 0.5.0

58 Independent Pathway Model [X] A [Y] C [Z] E A2.0 / 0.5 C2 E2.0 A3 C3 E3 A C E F2 F3 F2 F3 [F] f f 2 f 3 f 4 f 5 f 6 f f f 2 f 3 f 4 f 5 f 6 P t P2 t P3 t P4 t P5 t P6 t P t2 P2 t2 P3 t2 P4 t2 P5 t2 P6 t2 A E C [T] [V] [U] E2 E3 E4 E5 E6 E.0 / A C