Multivariate Twin Model Independent Pathway Model Common Path model Conor Dolan Biological psychology VU Amsterdam Dirk Pelt Biological psychology VU Amsterdam Boulder 2022 Practical Intro Multiv IPM CPM Dolan amp Pelt ID: 1014856
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1. Intro to the Day 5 practical: Multivariate Twin Model ....Independent Pathway Model .... Common Path modelConor DolanBiological psychology, VU,AmsterdamDirk PeltBiological psychology, VU,AmsterdamBoulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}1
2. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}2DZ twin 1DZ twin 2UNIVARIATE TWIN MODEL ...biceps skinfold .... ADE model rmz> 2*rdz (path diagram of the DZ group) The twin design is a means to decompose phenotypic variance (ADE model): sbic2 = sAb2 + sDb2 + sEb2
3. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}3bivariate ADE model: SPh = SA + SD + SESPhbictribicsbic2sbic,tritrisbic,tristri2Univariate model (question? answer?): decomposition of phenotypic variance sbic2 = sAb2 + sDb2 + sEb2Bivariate model (question? answer?): decomposition of phenotypic variances (sbic2 stri2) and covariance (sbic,tri)sAbAtsDbDtbicepsEbDbAb111tricepsEtDtAt111sEbEtsAb2sDb2sEb2sAt2sDt2sEt2Bivariate model for biceps and triceps:
4. 4SAbictribicsAb2sAb,AttrisAb,AtsAt2SPh = SA + SD + SESDbictribicsDb2sDb,DttrisDb,DtsDt2SEbictribicsEb2sEb,EttrisEb,EtsEt2SPhbictribicsAb2+sDb2+sEb2sAb,At+sDb,Dt+sEb,EttrisAb,At+sDb,Dt+sEb,EtsAt2+sDt2+sEt2SPhbictribicsbic2sbic,tritrisbic,tristri2 = + + =Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}variances in red; covariance in blue
5. 5Bivariate ADE path diagram for twin data. DZ: RA=½ RD=¼. MZ: RA=1 RD=1 Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}SPh (2 x 2) Covariance Matrix - 2 phenotypesSPhMZ (4 x 4) Covariance Matrix - 2 phenotypes X 2 twinsSPhDZ (4 x 4) Covariance Matrix - 2 phenotypes X 2 twinsSA + SD + SESA + SDSA + SDSA + SD + SESA + SD + SE½SA + ¼SD½SA + ¼SDSA + SD + SESPhMZSPhDZ
6. sAbAtsDbDtbicepsEbDbAb111tricepsEtDtAt111sEbEtsAb2sDb2sEb2sAt2sDt2sEt2bicepsEbDbAbtricepsEtDtAtrEbEtrAbAtrDbDt111111ebdbabetdtatSAbicepstricepsbicepssAb2 sAb,AttricepssAb,AtsAt2SAbicepstricepsbicepsab2 ab*rAbAt*attricepsab*rAbAt*atat2 Standard deviations: ab and at and correlation: rAbAtcovariance (A): ab*rAbAt*at variance components (used to fit) vs. path coefficients (used to show)6Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}covariance (A):sAbAt
7. sAbAtsDbDtbicepsEbDbAb111tricepsEtDtAt111sEbEtsAb2sDb2sEb2sAt2sDt2sEt2bicepsEbDbAbtricepsEtDtAtrEbEtrAbAtrDbDt111111ebdbabetdtatvariance components (used to fit) vs. path coefficients (used to show)7Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}trace up from bicepts to Ab (1)trace from Ab to At (covariance : sAbAt )trace down from At to triceps (1) .....Additive genetic contribution to covariance: 1*sAbAt*1trace up from bicepts to Ab (ab)trace from Ab to At (correlation : rAbAt)trace down from At to triceps (at) .....Additive genetic contribution to covariance: ab*rAbA*attracing - - -tracing ....
8. The generalization from p=1 (univariate) to p=2 (bivariate) to p=4 (4-variate).Biceps and Triceps (purple ), Subscapular (blue ), Suprailiacal (orange ).8Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}SA + SD + SESA + SDSA + SDSA + SD + SESA + SD + SE½SA + ¼SD½SA + ¼SDSA + SD + SESPhMZSPhDZ
9. 9Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}SPh (4 x 4) Covariance Matrix - 4 phenotypesSPhMZ, SPhDZ (8 x 8) Covariance Matrix - 4 phenotypes X 2 twinsA orange; D green; E blue
10. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}10SPh =SA + SD + SEEach symmetric matrix has p*(p+1)/2 = 4*5/2 = 10 elements, i.e., 4 variances, 6 covariances (counting covariances once!)SA + SD + SE½SA + ¼SD½SA + ¼SDSA + SD + SEA model for the phenotypic 2p x 2p DZ phenotypic covariance matrixSPhDZThe pxp covariance matrices themselves SA, SD, & SE can be modelled"A model (for SA... SD... SE ) in a model (for SPhMZ ... SPhDZ)"
11. Single common factor model - a possible model for SAGenetic theory of the common factor model1) there are genes with common additive genetic effects on all 4 skinfold phenotypes (pleiotropic genes)2) there are genes with phenotype specific additive genetic effects - unique influences on the 4 phenotype are a source of variance, but not covarianceSubscript c in Ac stands for “common” because Ac genes are common to the 4 phenotypes (Ac genes are pleiotropic)AcbicepstricepssubscsupriAbicAsiAsAtri11Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}SA is part of model SPh
12. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}12The 4x4 covariance matrices SA, SD, & SE, each a common factor model SA has 4*5/2 = 10 elements, but modelled using 8 parameters SA = LA LAt + TALA (4x1) with 4 factor loadings (path coefficients)TA (diagonal 4x4) with 4 residual variances Neuroticism items data
13. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}13Acn1n2n3n4D1D4D3D2DcEcE1A1A2E2A3E3A4E4Practical: applied to 4 neuroticism items (n1, n2, n3, n4)The independent pathway model
14. neu scoreneuroticismADEADEWhat we really want:A statement about the latent variable neuroticismWhat we get: A statement about the proxy ofthe latent variable neuroticismA proxy: Neu scorei = n1i + n2i + n3i + n4i 0.378.612d2a2e2Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}14Psychometrics: the statistical science of measuring latent phenotypes
15. NeuroticismADEn1e1n2n3e3e2n4e4The common pathway modelA, D (C), Edecomposition of latent variableThe common factor model... a measurement model relates items to latentvariableBoulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}15common factor model(applied to items)what we want
16. Aipmn1D1n2n3n41sA2DipmEipmE1A11Neuroticismn1n2n3n4ADE111sA2sE2sD2D2E2A2D3E3A3D4E4A4sD2sE2f4f3f2e41e3e2a2a4a3D1E1A1D2E2A2D3E3A3D4E4A41d2d4d34+4+4 + 4+4+4 = 24 parameters3+3+4+4+4 = 18 parametersBoulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}16
17. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}17Practical Part 1: 4-variate model fitted to the skinfold dataADE modelPractical Part 2:4-variate models fitted to the 4 neuroticism itemsADE vs AE modelAE Independent Pathway modelAE Common Pathway model cut & paste, & answer the questions.
18. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}18
19. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}19Wrap-up of the practical: Multivariate Twin Model ....Independent Pathway Model .... Common Path modelKeep your eye on the substantive hypotheses IPM pleiotropywhat is the story?
20. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}20Keep your eye on the substantive hypotheses CPMPsychometric interpretation latent phenotype as measured by items ... a strong motivation for CPM ... but a fairly parsimonious idealphenotype latent variable of interestPhenotypes: items depend directly and causally onthe latent variable NMeasurement Error (acts like E)Also: CPM is a mediation model: N mediates effects of A and E on the items n1 n2 n3 & n4.
21. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}21What we found out about skinfold measures1) the skinfold measures are highly correlated phenotypically2) Model is a ADE model3) A + D make a large contribution to the phenotypic variances4) The A, D and E correlations are all large.5) The D correlations are equal to 1 (dominance effects 100% common to all measures)
22. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}22What we found out about neuroticism items1) Model is an AE model (no C & no D)2) The IPM fitted OK3) The CPM fitted well, as we expect based on psychometric theory (why did it fit well?)4) The narrow-sense heritability of the latent variable neuroticism is .4655) The narrow-sense heritability of the neuroticism proxy (sumscore) is .3786) The factor loading are between .64 and .77.6) The item specific residuals are mainly due to E (item specific A variance is low)
23. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}23
24. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}24rA1A2 = .98rD1D2Ph1E1D1A1√sE12√sD12√sA12Ph2E2D2A2rE1E2111111√sE22√sD22√sA22cov(Ph1, Ph2) = sA1 rA1A2 sA2 + sD1 rD1D2 sD2 + sE1 rE1E2 sE2Given rA1A2 = .98, does that mean that A is contributing greatly to cov(Ph1, Ph2)?cov(Ph1, Ph2) = sA1*.98*sA2 + sD1 rD1D2 sD2 + sE1 rE1E2 sE2
25. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}25rA1A2rC1C2Ph1E1C1A1√sE12√sC12= .03√sA12Ph2E2C2A2rE1E2111111√sE22√sC22√sA22cov(Ph1, Ph2) = sA1 rA1A2 sA2 + sC1 rC1C2 sC2 + sE1 rE1E2 sE2Given sC1 is very small (standardized: .03), why is it hard to estimate rC1C2 reliably?cov(Ph1, Ph2) = sA1 rA1A2 sA2 + .03*rC1C2*.35 + sE1 rE1E2 sE2
26. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}26Suppose ADE model, sA2>0, sD2>0 and sE2>0, and suppose sample size is very large (power not an issue)Can phenotype item n4 be characterized by an AE model in the CPM?Can phenotype item n4 be characterized by an AE model in the IPM?
27. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}27
28. Boulder 2022 Practical Intro Multiv / IPM / CPM {Dolan & Pelt}28RCR
29. GHB 2021 – lecture 729RCR Case 1:Phenotype: criminal behavior (e.g., an objective measure using police contact: misdemeanors, felonies, convictions)Reseach question2: do genetic factors contribute to individual differences in criminal behavior?do environmental factors contribute to individual differences in criminal behavior?Question: Does this have any bearing on responsibility / culpability? Should it be of interest to a judge or an attorney?
30. GHB 2021 – lecture 730RCR Case 2:Phenotype: victim of a crime e.g., have you ever been a victim of a mugging? Measured yes (1) or no (2), with underlying liabilityProposition: heritability of phenotype h2 > 0 Q: Is h2>0 plausible? what does this mean?Q: if true, how would culpability fit in? Are you in part to blame if you are mugged? Should it be interest to a judge or an attorney?