Model building & assumptions. Matt Keller, Sarah Medland, Hermine Maes TC21 March 2008

Transcription

1 Model building & assumptions Matt Keller, Sarah Medland, Hermine Maes TC2 March 2008

2 cknowledgments John Jinks David Fulker Robert Cloninger John DeFries Lindon aves Nick Martin Dorret Boomsma Michael Neale

3 Behavior Genetic Methods For thirty years, BG studies have revolved around BUILDING and TSTING MODLS In particular, modeling means, variances, and covariances of genetically informative relatives Such models are based on our understanding of why relatives are similar to one another

4 Models mediate between theory and data

5 xample Model Building with MZ & DZ twins only Theory: variance due to &, & either D or C 3 unknowns > 3 independently informative equations needed to solve for V, V & VC (C) or V, V & VD (D) Mx arrives at much the same conclusion using a much different algorithm (ML). But for most BG models fitted today, no such easy close formed solutions exist. ML approach is better.

6 lgebra for V & VC assuming the C model VP = V + VC + V = rmz = V + VC rdz =.5V +VC - rmz = V rmz - rdz =.5V 2(rMZ - rdz) = V rmz - 2rDZ = V + VC - V - 2VC = - VC 2rDZ - rmz = VC or VP - V - V = VC

7 Practical I Solve for V & VD using the following equations: VP = V + VD + V = rmz = V + VD rdz =.5 V +.25VD

8 lgebra for V & VD assuming the D model VP = V + VD + V = rmz = V + VD rdz =.5 V +.25VD - rmz = V rmz - 4rDZ = V + VD - 2V - VD = -V 4rDZ - rmz = V rmz - 2rDZ = V + VD - V -.5VD =.5VD 2(rMZ - 2rDZ) = 2rMZ - 4rDZ = VD

9 In reality, models are constrained by data s ability to test particular theories

10 Model ssumptions ll models must make simplifying assumptions. It is no different in BG, e.g., with MZ & DZ twins reared together: C Model ssumes no D D Model ssumes no C

11 Testable ssumptions with twin data Normality of residuals No xc, x, Cx Interactions No C,, C Correlations qual nvironments ssumption No Sibling Interaction No Sex x (C) Interaction

12 Testable ssumptions with additional data Multivariate Measurement Invariance Longitudinal No ge x (C) Interaction Measurement rror vs Unique nvironment Other Relatives No ssortative Mating C&D, Cultural Transmission

13 Not Testable ssumptions No correlated errors No epistasis

14 Model ssumptions II Why must we make these simplifying assumptions?.g., why can t we estimate C & D at same time using twins only? Solve the following two equations for V, VC, and VD: rmz = V + VD + VC rdz = /2V + /4VD + VC

15 Classical Twin Design pproach When rmz < 2rDZ, we estimate VC and V if rmz>2rdz then VC negative or at 0 boundary V = 2(rMZ-rDZ) VC = 2rDZ-rMZ assumption: VD = 0 When rmz > 2rDZ, we estimate VD and V if rmz<2rdz then VD negative or at 0 boundary V = 4rDZ-rMZ VD = 2rMZ-4rDZ ssumption: VC = 0

16 Sensitivity nalysis What happens when our assumptions are wrong? Quantification of what happens when our models are wrong is a STRNGTH, not a weakness of model-based science!

17 Practical II Given rmz =.80 & rdz =.30 Solve algebraically for V and VD when VC is assumed to be 0 (the normal case): V = 4rDZ - rmz VD = 2rMZ - 4rDZ Solve algebraically for V and VD when VC is assumed to be.05 (a possibility after all) implies rmz-vc =.75 & rdz-vc =.25

18 Sensitivity nalysis Practical Given rmz =.80 & rdz =.30 Under normal assumptions (VC=0): V =.4 VD =.4 Under alternative assumptions (VC=.05): V =.25 = 4(.25) -.75 [4rDZ - rmz] VD =.5 = 2(.75) - 4(.25) [2rMZ - 4rDZ] had we run a twin-only model, we would have underestimated VD & VC and overestimated V

19 In reality, models determine study designs needed to test them

20 Genevolve Given a model, a study design is chosen (and assumptions made) Data are simulated under complex true world with Genevolve We can evaluate how biased results are given the chosen design/assumptions.

21 Simulation: True World V=.33 VC=. V=.36 VD=.2 VF=.07 COVF=.08 rsp=.2 cultural transmission G covariance spousal correlation

22 Model Design: rmz ssumptions: no C, D, assortment

23 Model vs True World True world C D F rsp T G Sources of Variance

24 C Model Design: rmz & rdz ssumptions: no D, assortment

25 C Model vs True World C True world C D F rsp T G Sources of Variance

26 D Model Design: rmz & rdz ssumptions: no C, assortment

27 D Model vs True World D True world C D F rsp T G Sources of Variance

28 CD Model

29 CD II Design: rmz & rdz & parents ssumptions: common C, no assortment lternative Designs: rmza & rdza (twins reared apart) rmz & rdz & half-sibs

30 rmza & rdza (twins reared apart) rmz= V + VC + VD rdz=.5v + VC +.25VD rmz= V + VD rdz=.5v +.25VD

31 rmz & rdz & halfsibs rhs rmz= V + VC + VD rdz=.5v + VC +.25VD rhs=.25v + VC

32 CD Model vs True World CD True world C D F rsp T G Sources of Variance

33 CF Model

34 CF II Partition c 2 in cultural transmission and non-parental shared environment Design: rmz & rdz & parents ssumptions: no assortment, D

35 CF Model vs True World CF True world C D F rsp T G Sources of Variance

36 CI Model

37 CI II ccount for assortative mating to correct estimates of a 2 and c 2 Design: rmz & rdz & parents ssumptions: non-parental C, no D

38 CI Model vs True World CI True world C D F rsp T G Sources of Variance

39 Twins & Parents Design C + Dominance OR Cultural Transmission Different mechanisms ssortative Mating Different mechanisms

40 Path Diagram Twins & Parents e P F e P M P F : Phenotype Father P M : Phenotype Mother e e P P 2 P : Phenotype Twin P 2 : Phenotype Twin 2

41 Genetic Transmission Model Genetic transmission Fixed at.5 Residual Genetic Variance Fixed at.5 quilibrium of variances across generations.5 a e e a P F P M a e e a P m P f

42 Genetic Transmission xpectations rsp= 0 rpo=.5a 2 rmz= a 2 rdz=.5a 2 Var= a 2 +e 2.5 a e e a P F P M a e e a P m P f

43 Dominance Model Dominance Common environment Non-parental ssortment D a e n P F c C d C c e a n P M D g a e P m D n c.25 C D n c e P f g a

44 Dominance xpectations rsp= d rpo=.5a 2 rmz= a 2 +c 2 +n 2 D a e n P F c C d C c e a n P M D rdz=.5a 2 +c n 2 Var= a 2 +c 2 +e 2 +n 2 g a e P m D n c.25 C D n c e P f g a

45 Common nvironment Model Common environment Same for all family members ssortment Function of common environment.5 C a e c c e a P F c c P M a e e a P m P f

46 Common nvironment xpectations rsp= c 2 rpo=.5a 2 +c 2 rmz= a 2 +c 2 rdz=.5a 2 +c 2 Var= a 2 +c 2 +e 2.5 C a e c c e a P F c c P M a e e a P m P f

47 Model ssumptions IIr But if we make simplifying assumptions, we can estimate C & D at same time using twins and parents? Solve the following three equations for V, VC, and VD: rmz = V + VD + VC rdz = /2V + /4VD + VC rpo = /2V + + VC

48 Social Homogamy Model ssortment Social Cultural Transmission From C to C Non-parental Shared nvironment Residual.5 C C C a e c c e a P F P M a e c c e a f d f r.5 P m P f

49 Social Homogamy xpectations rsp= dc 2 rpo= (f+df)c 2 +.5a 2 x= 2f 2 +2df 2 +r rmz= a 2 +xc 2 rdz=.5a 2 +xc 2 ; Var= a 2 +xc 2 +e 2.5 C C C a e c c e a P F P M a e c c e a f d f r.5 P m P f

50 Phenotypic ssortment Model ssortment Phenotypic Cultural Transmission From P to C Non-parental Shared nvironment Residual Genotype-nvironment Covariance g.5 C C C a e c c e a P F x P M a e c c e a P m P f s d x f f g r.5 s

51 Phenotypic ssortment xpectations rsp= w= pdp rgp= t= ga+sc rpo= (pf+wf)c+.5a(+pd)t rg= j= asc+csa rmz= ga2+xc2+j pa= y= g+.5(t(d+d)t) rdz=.5ya2+xc2+j + constraints g.5 C C C a e c c e a P F x P M a e c c e a s d x f f g r.5 s P m P f

52 Sex Differences s m s f q r u B v m x m C x f C v f u B r q e m a m b m c m c f a f e f P F i P M B p o m n C r c C.5 B.5 e m a m b m c m c f a f e f P m P f

53 T (Stealth/Cascade) Design xtended Twin Kinships (T model): twins, their parents siblings, spouses and children 88 unique sex-specific biological/social relationships

54 T Model (aves et al 999, Maes et al 999, 2006) Genetic dditive () Dominance (D) nvironment Specific () Shared/Common (C) Twin (T) Cultural transmission (w) G covariance (s) ssortative mating (i)

55 T vs True World T True world C D F rsp T G

56 nvironmental Factors rmz < >> specific environmental factors rsib = rdz = rpo >> shared environmental factors increase similarity between people living or having grown up in same home (first-degree relatives and MZs) rsib > rpo >> non-parental environmental factors in common for siblings, such as school environment, peers, friends rtwin > rsib >> special twin environment additional twin similarity due to greater sharing of aspects of environment rsib = rpo > /2 rmz >> cultural transmission

57 Genetic Factors rmz > first-degree relatives (rdz, rsib, rpo) > seconddegree relatives (grandparents, half-siblings, avuncular pairs) > more distant relatives such as cousins >> additive genetic factors rsib & rdz < /2 rmz (expectation: DZ=/4MZ) and zero correlations for other pairs of relatives >> dominance phenotypic cultural transmission + genetic transmission >> G covariance partner selection is based on phenotype >> non-random mating: source of similarity which may have both genetic and environmental implications

58 The 22th International Statistical Genetics Methods Workshop ugust - 5, 2008 Leuven, Belgium