The Model Building Process Part I: Checking Model Assumptions
Model building & assumptions
description
Transcript of Model building & assumptions
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Model building & assumptionsMatt Keller, Sarah Medland, Hermine Maes
TC21
March 2008
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Acknowledgments John Jinks David Fulker Robert Cloninger John DeFries Lindon Eaves Nick Martin Dorret Boomsma Michael Neale
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Behavior Genetic Methods
For thirty years, BG studies have revolved around BUILDING and TESTING MODELS
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
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Models mediate between theory and data
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Example Model Building with MZ & DZ twins only Theory: variance due to E & A, & either D or C 3 unknowns > 3 independently informative
equations needed to solve for VE, VA & VC (ACE) or VE, VA & VD (ADE)
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.
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Algebra for VA & VC assuming the ACE model VP = VA + VC + VE = 1 rMZ = VA + VC rDZ = .5VA +VC
1 - rMZ = VE rMZ - rDZ = .5VA
2(rMZ - rDZ) = VA rMZ - 2rDZ = VA + VC - VA - 2VC = - VC
2rDZ - rMZ = VC or VP - VE - VA = VC
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Practical I
Solve for VA & VD using the following equations:VP = VA + VD + VE = 1rMZ = VA + VDrDZ = .5 VA +.25VD
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Algebra for VA & VD assuming the ADE model VP = VA + VD + VE = 1 rMZ = VA + VD rDZ = .5 VA +.25VD
1 - rMZ = VE rMZ - 4rDZ = VA + VD - 2VA - VD = -VA
4rDZ - rMZ = VA rMZ - 2rDZ = VA + VD - VA - .5VD = .5VD
2(rMZ - 2rDZ) = 2rMZ - 4rDZ = VD
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In reality, models are constrained by data’s ability to test particular theories
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Model Assumptions
All models must make simplifying assumptions. It is no different in BG, e.g., with MZ & DZ twins reared together:
ACE ModelAssumes no D
ADE ModelAssumes no C
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Testable Assumptions with twin data Normality of residuals No AxC, AxE, CxE Interactions No AC, AE, CE Correlations Equal Environments Assumption No Sibling Interaction No Sex x A(CE) Interaction
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Testable Assumptions with additional data Multivariate
Measurement Invariance Longitudinal
No Age x A(CE) Interaction Measurement Error vs Unique Environment
Other RelativesNo Assortative MatingC&D, Cultural Transmission
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Not Testable Assumptions
No correlated errors No epistasis
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Model Assumptions II
Why must we make these simplifying assumptions? E.g., why can’t we estimate C & D at same time using twins only?
Solve the following two equations for VA, VC, and VD:rMZ = VA + VD + VCrDZ = 1/2VA + 1/4VD + VC
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Classical Twin Design Approach
When rMZ < 2rDZ, we estimate VC and VA if rMZ>2rDZ then VC negative or at 0 boundary VA = 2(rMZ-rDZ) VC = 2rDZ-rMZ assumption: VD = 0
When rMZ > 2rDZ, we estimate VD and VA if rMZ<2rDZ then VD negative or at 0 boundary VA = 4rDZ-rMZ VD = 2rMZ-4rDZ Assumption: VC = 0
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Sensitivity Analysis
What happens when our assumptions are wrong?
Quantification of what happens when our models are wrong is a STRENGTH, not a weakness of model-based science!
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Practical II
Given rMZ = .80 & rDZ = .30 Solve algebraically for VA and VD when
VC is assumed to be 0 (the normal case):VA = 4rDZ - rMZVD = 2rMZ - 4rDZ
Solve algebraically for VA and VD when VC is assumed to be .05 (a possibility after all) implies rMZ-VC = .75 & rDZ-VC = .25
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Sensitivity Analysis Practical
Given rMZ = .80 & rDZ = .30 Under normal assumptions (VC=0):
VA = .4 VD = .4
Under alternative assumptions (VC=.05): VA = .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 VA
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In reality, models determine study designs needed to test them
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GeneEvolve
Given a model, a study design is chosen (and assumptions made)
Data are simulated under complex true world with GeneEvolve
We can evaluate how biased results are given the chosen design/assumptions.
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Simulation: True World
VA=.33 VC=.1 VE=.36 VD=.2 VF=.07 cultural transmission COVAF=.08 GE covariance rSP=.2 spousal correlation
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AE Model
Design: rMZ Assumptions: no C, D, assortment
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AE Model vs True World
1.01
.24
.00 .00 .00 .00 .00 .00
.33.26
.10.20
.07
.20.10 .08
.00
.20
.40
.60
.80
1.00
1.20
A E C D F rSP T GE
Sources of Variance
AE True world
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ACE Model
Design: rMZ & rDZ Assumptions: no D, assortment
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ACE Model vs True World
.65
.24.32
.00 .00 .00 .00 .00
.33.26
.10
.20
.07
.20
.10 .08.00
.10
.20
.30
.40
.50
.60
.70
.80
A E C D F rSP T GE
Sources of Variance
ACE True world
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ADE Model
Design: rMZ & rDZ Assumptions: no C, assortment
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ADE Model vs True World
.95
.23
.00 .00 .00 .00 .00 .00
.33.26
.10.20
.07
.20.10 .08
.00
.20
.40
.60
.80
1.00
1.20
A E C D F rSP T GE
Sources of Variance
ADE True world
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ACED Model
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ACED II
Design: rMZ & rDZ & parents Assumptions: common C, no assortment
Alternative Designs: rMZa & rDZa (twins reared apart)rMZ & rDZ & half-sibs
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rMZa & rDZa (twins reared apart) rMZ= VA + VC + VD rDZ= .5VA + VC + .25VD rMZA= VA + VD rDZA= .5VA + .25VD
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rMZ & rDZ & halfsibs rHS
rMZ= VA + VC + VD rDZ= .5VA + VC + .25VD rHS= .25VA + VC
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ACED Model vs True World
.40
.24
.31.27
.00 .00 .00 .00
.33
.26
.10
.20
.07
.20
.10 .08
.00
.10
.20
.30
.40
.50
.60
A E C D F rSP T GE
Sources of Variance
ACED True world
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ACEF Model
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ACEF II
Partition c2 in cultural transmission and non-parental shared environment
Design: rMZ & rDZ & parents Assumptions: no assortment, D
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ACEF Model vs True World
.65
.24.29
.00 .02 .00 .00 .00
.33.26
.10
.20
.07
.20
.10 .08.00
.10
.20
.30
.40
.50
.60
.70
.80
A E C D F rSP T GE
Sources of Variance
ACEF True world
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ACEI Model
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ACEI II
Account for assortative mating to correct estimates of a2 and c2
Design: rMZ & rDZ & parents Assumptions: non-parental C, no D
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ACEI Model vs True World
.84
.23
.12
.00 .00
.20
.00 .00
.33.26
.10.20
.07
.20.10 .08
.00
.10
.20
.30
.40
.50
.60
.70
.80
.901.00
A E C D F rSP T GE
Sources of Variance
ACEI True world
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Twins & Parents Design
ACE + Dominance OR Cultural Transmission
Different mechanisms Assortative Mating
Different mechanisms
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Path Diagram Twins & Parents
PM
P1 P2
PF
E
e
1
E
e
1
E
e
1
E
e
1
PF : Phenotype Father
PM : Phenotype Mother
P1 : Phenotype Twin 1
P2 : Phenotype Twin 2
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Genetic Transmission Model
Genetic transmission Fixed at .5
Residual Genetic Variance Fixed at .5 Equilibrium of
variances across generations
PM
Pm Pf
A
A A
A
a a
a a
.5 .5.5.5PF
1
.5
1
.5
E
e
1
E
e
1
E
e
1
E
e
1
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Genetic Transmission Expectations
rSP= 0 rPO= .5a2
rMZ= a2
rDZ= .5a2
Var= a2+e2
PM
Pm Pf
A
A A
A
a a
a a
.5 .5.5.5PF
1
.5
1
.5
E
e
1
E
e
1
E
e
1
E
e
1
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Dominance Model
PM
Pm Pf
A A
C
C
A
a a
a a
c
c c
.5 .5.5.5PF
1
1
g
11
g
E
e
1
E
e
1
E
e
1
E
e
1
d
Dn
1
D
n
D
n
1
Dn
1
1
.25
C1
c
A
Dominance Common environment
Non-parental
Assortment
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DominanceExpectations
rSP= d rPO= .5a2
rMZ= a2+c2+n2
rDZ= .5a2+c2+.25n2
Var= a2+c2+e2+n2
PM
Pm Pf
A A
C
C
A
a a
a a
c
c c
.5 .5.5.5PF
1
1
g
11
g
E
e
1
E
e
1
E
e
1
E
e
1
d
Dn
1
D
n
D
n
1
Dn
1
1
.25
C1
c
A
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Common Environment Model
Common environment Same for all family
members
Assortment Function of common
environment
PM
Pm Pf
A
A A
A C
a a
a
c c
c c
.5 .5.5.5PF
1
.5
11
.5
E
e
1
E
e
1
E
e
1
E1
ae
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Common Environment Expectations
rSP= c2
rPO= .5a2 +c2
rMZ= a2+c2
rDZ= .5a2+c2
Var= a2+c2+e2
PM
Pm Pf
A
A A
A C
a a
a
c c
c c
.5 .5.5.5PF
1
.5
11
.5
E
e
1
E
e
1
E
e
1
E1
ae
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Model Assumptions 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 VA, VC, and VD:rMZ = VA + VD + VCrDZ = 1/2VA + 1/4VD + VCrPO = 1/2VA + + VC
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Social Homogamy Model
Assortment Social
Cultural Transmission From C to C
Non-parental Shared Environment Residual
PM
Pm Pf
A
A A
C
C
A C
a a
a a
c c
c c
.5 .5.5.5PF
1
r .5
111
.5
E
e
1
E
e
1
E
e
1
E
e
1
f f
d
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Social Homogamy Expectations
rSP= dc2
rPO= (f+df)c2+.5a2
x= 2f2+2df2+r rMZ= a2+xc2
rDZ= .5a2+xc2; Var= a2+xc2+e2
PM
Pm Pf
A
A A
C
C
A C
a a
a a
c c
c c
.5 .5.5.5PF
1
r .5
111
.5
E
e
1
E
e
1
E
e
1
E
e
1
f f
d
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Phenotypic Assortment Model
Assortment Phenotypic
Cultural Transmission From P to C
Non-parental Shared Environment Residual
Genotype-Environment Covariance
PM
Pm Pf
A
A A
C
C
A C
a a
a a
c c
c c
.5 .5.5.5PF
g
r .5
gxx
.5
E
e
1
E
e
1
E
e
1
E
e
1
f f
d
s s
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Phenotypic Assortment Expectations
rSP= w= pdp rGP= t= ga+sc
rPO= (pf+wf)c+.5a(1+pd)t rGE= j= asc+csa
rMZ= ga2+xc2+j pa= y= g+.5(t(d+d)t)
rDZ= .5ya2+xc2+j + constraints
PM
Pm Pf
A
A A
C
C
A C
a a
a a
c c
c c
.5 .5.5.5PF
g
r .5
gxx
.5
E
e
1
E
e
1
E
e
1
E
e
1
f f
d
s s
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Sex Differences
Pm Pf
A
A A
CA C
am af
am af
cm cf
.5 .5.5.5
q
.5
qxfxm
.5
B
bm
uB
u
B.5
B
bm
.5
i
sm sf
EE1 1
efem
E1
ef
E1
em
C
cm cf
1
p m
C1
o n
rvfvmr
.5 .5.5.5PF PM
rc
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ET (Stealth/Cascade) Design
Extended Twin Kinships (ET model):
twins, their parents
siblings, spouses
and children
88 unique sex-specific biological/social relationships
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ET Model (Eaves et al 1999, Maes et al 1999, 2006)
Genetic Additive (A) Dominance (D)
Environment Specific (E) Shared/Common (C) Twin (T) Cultural transmission (w)
GE covariance (s) Assortative mating (i)
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ET vs True World
.25 .24
.08
.26
.09
.18
.09
.02
.33
.26
.10
.20
.07
.20
.10.08
.00
.05
.10
.15
.20
.25
.30
.35
A E C D F rSP T GE
ET True world
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Environmental Factors
rMZ < 1 >> 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 > 1/2 rMZ >> cultural transmission
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Genetic Factors
rMZ > first-degree relatives (rDZ, rSIB, rPO) > second-degree relatives (grandparents, half-siblings, avuncular pairs) > more distant relatives such as cousins >> additive genetic factors
rSIB & rDZ < 1/2 rMZ (expectation: DZ=1/4MZ) and zero correlations for other pairs of relatives >> dominance
phenotypic cultural transmission + genetic transmission >> GE covariance
partner selection is based on phenotype >> non-random mating: source of similarity which may have both genetic and environmental implications
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The 22th International Statistical Genetics Methods Workshop
August 11 - 15, 2008
Leuven, Belgium