longitudinal modeling nathan gillespie & dorret boomsma \\nathan\2008\longitudinal...
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Longitudinal Modeling
Nathan Gillespie & Dorret Boomsma
\\nathan\2008\Longitudinalneuro_f_chol.mxneuro_f_simplex.mxjepq6.dat
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Why conduct longitudinal analyses?
1. Can improve power by using multiple observations from the same individual
- Cross twin cross trait correlation
2. Can examine and estimate time-dependent genetic and environmental effects
- Changing magnitude of genetic & environmental influence across time
- Same versus different genes across development
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Methods for Longitudinal Data Analysis
1. Cholesky Models
2. Simplex Models
Eaves et al., 1986
Boomsma & Molenaar, 1987
3. Growth Curve Models
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Aims
1. Revisit the Mx trivariate Cholesky
2. Take a look at simplex or auto-regression models
- Explain some of the underlying theory of this form of longitudinal modelling
- Run through an Mx script
3. Compare the Cholesky and simplex models
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Longitudinal modeling of adolescent personality
Introduce longitudinal modeling in the context of personality change
Eaves, Eysenck & Martin (1989)
- Adult personality - High of genetic continuity over time
- Effect stronger in Neuroticism vs Extraversion
Genetic continuity in adolescents?
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X2 = 57.462, df = 45, p = .101, AIC = -32.538, RMSEA = .011
Personality Data
Twin Mole and Twin MAPS projects
Assess the genetic / environmental etiology of Melanocytic Naevi (common moles) in twins aged 12 & 14 years + Cognition at 16
81 items JEPQ: Psychoticism (P), Extraversion (E), and Neuroticism (N) - Lie (L)
Twin Mole Twin MAPS 12 yrs 14 yrs 16 yrs
P E N L P E N L P E N L
Male 603 604 606 605 465 466 466 467 416 412 416 415
Female 605 602 607 609 471 470 473 473 442 438 442 442
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X2 = 57.462, df = 45, p = .101, AIC = -32.538, RMSEA = .011
Personality DataRaw continuous data methods
Assumptions of mean and variance homogeneity by twin order and zygosity (and necessarily by sex and age)
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1. Cholesky Model
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Cholesky Model
Advantages:- Logical: organized such that all factors are constrained to
impact later, but not earlier time points
- Requires few assumptions, can predict any pattern of change
Disadvantages:- Not falsifiable
- Does not make predictions about what will happen in the future (as yet unmeasured time points)
- Only feasible for limited number of measurements
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Cholesky Model
Questions you can address:- Magnitude of genetic/environmental influence at
each time
- Extent to which genetic/environmental influences overlap across time
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Run ACE Cholesky Model
neuro_f_chol.mx
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Model Fit for Female Neuroticism
Model LL df 2LL df p BIC
ACE Cholesky 10424.88 1803 - - - 2352.134
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Cholesky Model Fitting Results
Phenotypic correlations
Proportions of variance
ACE
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Cholesky Model Fitting Results
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2. Simplex Models
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Simplex or autoregressive modelsSimplex designs model changes in true scores (y[t]) over time by fitting auto-regressive or Markovian chains
Each true score is predicted to be causally related to the immediately preceding latent true score in a linear fashion
y[t]n = [t]n y[t-1]n + [t]n
[t] = linear regression of latent factor (y[t]) on the previous latent factor (y[t-1]), [t] = new input, change or innovation at time [t], uncorrelated with y[t-1]
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Simplex Model
Latent True Scores
Innovations
Errors
Observed phenotypes
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ACE Simplex Model script
neuro_f_simplex.mx
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Model Fitting Results
Model LL df 2LL df p BICACE Cholesky 10424.88 1803 - - - 2352.13Full Simplex
AE Simplex
CE Simplex
E Simplex
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Model Fitting Results
Model LL df 2LL df p BICACE Cholesky 10424.88 1803 - - - 2352.13
Full Simplex 10425.113 1805 0.23 2 0.89 2349.08
AE Simplex 10425.339 1810 0.23 5 0.99 2341.26
CE Simplex 10432.45 1810 7.34 5 0.20 2344.81
E Simplex 10473.31 1815 48.20 10 <0.001 2357.31
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Test for non-significant parametersRun confidence intervals on all parameters
! AE Simplex StructureGet neuro_f.mxsDr 12 13 ! C transmission coefficientsDR 4 5 6 ! C innovationsIN X 1 3 3ENd
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Matrix Element Int. Estimate Lower Upper Lfail Ufail X 1 3 3 95.0 0.8752 -1.7949 1.7949 0 2 0 2
! AE Simplex Structure, Remove final genetic innovationGet neuro_f.mxsDr 12 13 ! C transmission coefficientsDR 4 5 6 ! C innovationsDR 3ENd
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Model Fitting Results
Model LL df 2LL df p BIC
ACE Cholesky 10424.88 1803 - - - 2352.13
Full Simplex 10425.113 1805 0.23 2 0.89 2349.08
AE Simplex 10425.339 1810 0.23 5 0.99 2341.26
Dropζa3 10425.70 1811 0.58 6 1.00 2339.85
CE Simplex 10432.45 1810 7.34 5 0.20 2344.81
E Simplex 10473.31 1815 48.20 10 <0.001 2357.31
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Best Fitting Model for Female Neuroticism
Degree of genetic continuityAge specific genetic effectsGenetic innovation at 14 years
Is it related to developmental or hormonal changes during puberty and psychosexual development?
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Additional Longitudinal ModelsDual Change Score (DCS) Model for Ordinal Data
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Additional Longitudinal ModelsBivariate Dual Change Score (DCS) Model for Ordinal Data
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Age 16Drinking
Age 17Drinking
Age 18.5Drinking
A1
C1 C2 C3
.58(.51 - .66)
.70(.62 - .78)
.71(.63 - .79)
.61(.53 - .67)
.22(.10 - .32)
.39(.29 - .45)
.37(.16 - .47)
Best-fitting model for drinking
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Methods for Longitudinal Analysis
Cholesky ModelsSimplex ModelsGrowth Curve Models
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Simplex Model
(Boomsma & Molenaar, 1987)
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Genetic Simplex Model
A A AA
BAC2 BAC6BAC4BAC3
E E EE
n21 n32 n43
p21 p32 p43
x11 x22 x33 x44
z11 z22 z33 z44
1 111
1 111
u11u22 u33 u44
x and z = genetic and nonshared environmental innovations respectivelyn and p = genetic and nonshared environmental transmission respectivelyu = error variances
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Simplex Model
Advantages: Makes restrictive predictions about covariance
pattern Falsifiable
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Today’s example Grant et al., 1999, Behavior Genetics, 29, 463-472. Australian alcohol challenge data, collected between 1979 and
1981 Mean age = 23.5 years
Subjects drank 0.75 g/kg alcohol at a steady rate over a 20-minute period. Blood Alcohol Concentration (BAC) was assessed at 6 points after consumption:
Minutes post-
consump.
Mean BAC
# of individuals with data
MZM (43 prs)
DZM (37 prs)
BAC 1 56 89.0 83 72
BAC 2 68 88.9 83 74
BAC 3 83 88.8 84 71
BAC 4 123 80.9 86 74
BAC 5 143 76.6 52 56
BAC 6 182 67.7 83 74
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A simplex correlation pattern…
BAC 2
BAC 3
BAC 4
BAC 6
BAC 2
1.00
BAC 3
0.90 1.00
BAC 4
0.69 0.84 1.00
BAC 6
0.58 0.77 0.93 1.00
Sample correlations (the DZM twin A quadrant of an intraclass correlation matrix)
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Practical - Simplex Model
A A AA
BAC2 BAC6BAC4BAC3
E E EE
n21 n32 n43
p21 p32 p43
x11 x22 x33 x44
z11 z22 z33 z44
1 111
1 111
u11u22 u33 u44
x and z = genetic and nonshared environmental innovations respectivelyn and p = genetic and nonshared environmental transmission respectivelyu = error variances
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Practical - Simplex Model
A A AA
BAC2 BAC6BAC4BAC3
E E EE
n21 n32 n43
p21 p32 p43
x11 x22 x33 x44
z11 z22 z33 z44
1 111
1 111
u11u22 u33 u44
x and z = genetic and nonshared environmental innovations respectivelyn and p = genetic and nonshared environmental transmission respectivelyu = error variances
1 0 0 00 2 0 00 0 3 00 0 0 4
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Practical - Simplex Model
A A AA
BAC2 BAC6BAC4BAC3
E E EE
?
1 111
1 111
x and z = genetic and nonshared environmental innovations respectivelyn and p = genetic and nonshared environmental transmission respectivelyu = error variances
? ? ?
? ? ? ?
? ? ? ?
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Full Genetic Simplex Model
A A AA
BAC2 BAC6BAC4BAC3
E E EE
0.9892 0.5332 1.3123
0.7001 0.5027 0.4130
12.0279 6.5234 .7036 4.2649
9.4624 5.0094 10.3650 5.1284
-0.1840
1 111
1 111
Basic_simplex.mxo -2*LL=4620.028, 23 est. parameters, 606 df
-0.1840 -0.1840 -0.1840
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Sub-Models
1) Is the error variance on individual variable assessments significant?
2) Is the genetic innovations on BAC6 significant? BAC4? BAC2?
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Sub-Models
1) Is the error variance on individual variable assessments significant?
- drop 200
2) Is the genetic innovations on BAC6 significant? BAC4? BAC2? - drop 4, 3, 2
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Simplex Model
Advantages: Makes restrictive predictions about covariance
pattern Falsifiable
Disadvantages: Makes restrictive predictions about covariance
pattern (future depends on current state only) Number of parameters increases with number of
measurements
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Methods for Longitudinal Analysis
Cholesky ModelsSimplex ModelsGrowth Curve Models
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Latent Growth Curve Model(shown here as linear)
• Mean Level of the Trait (Intercept) • Rate of Change In Trait (Slope)
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Latent Growth Curve Model(shown here as linear)
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Genetically InformativeLatent Growth Curve Model
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Genetically Informative Latent Growth Curve Model
→Like a bivariate model!
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Growth Model Questions
What is the contribution of genetic/environmental factors to the variation of α (intercept) and (slope)?
Same or different genes influencing α (intercept) and (slope)?
Same or different environments influencing α (intercept) and (slope)?
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Practical
Mx latent growth curve example (script from http://www.psy.vu.nl/mxbib/)
Submodels to test:1. No covariance between slope and intercept2. No genetic effect on intercept3. No genetic effect on slope4. No common environmental effect on intercept5. No common environmental effect on slope6. Best fitting model? (i.e., ACE, AE, CE, E?)
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Slope
A
AC CE E
1.0 (0.5)1.0
Intercept
Time 1 Time 2 Time 3 Time 4
1.01.0 1.0
1.00
1 23
x11x21
x22
F11
F21F31
F41
L
L11
L
L11
L
L11
L
L11
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Practical
Mx latent growth curve example (script from http://www.psy.vu.nl/mxbib/)
Submodels to test:1. No covariance between slope and intercept – signif
decrease in fit2. No genetic effect on intercept – signif decrease in
fit3. No genetic effect on slope – signif decrease in fit4. No common environmental effect on intercept -- ns5. No common environmental effect on slope -- ns 6. Best fitting model? (i.e., ACE, AE, CE, E?) -- AE
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Growth Curve Model
Advantages: Very efficient: number of parameters does not
increase with number of measurements Provides prediction about behavior beyond measured
timepoints
Disadvantages: Note regarding slope parameters Can be computationally intense Assumptions to reduce computational burden
Linearity, no genetic effects on residuals, equal variance among residuals at differing timepoints
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Latent Growth Curve ModelingAdditional Considerations
Standard approach assumes data are collected at identical set of fixed ages for all individuals (e.g., start at age 12, yearly assessments)
Age heterogeneity and unequal spacing of measurements can be handled using definition variables Mehta & West, 2000, Psychological Methods
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S1 I1 S2 I2
As1Cs1
Es1 Ai1Ci1
Ei1 As2Cs2 Es2
Ai2Ci2
Ei2
lgdri171 lgdri181 lgdri162 lgdri172 lgdri182
T1 T2 T3 T1 T2 T3
ArCr Er
ArCr
ErAr
Cr
Er Ar
CrEr
ArCr
ErAr Cr
Er
M
Age
lgdri161
Latent Growth Curve Model with Measured Variable
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Extensions of Growth Curve Models
Incorporation of measured variables (genotype, environment)
Nonlinear growth Neale, MC & McArdle, JJ (2000). A structured
latent growth curves for twin data. Twin Research, 3, 165-177. SlopeIntercept
Time 1 Time 2 Time 3 Time 4
1.01.0 1.0
1.0 01 2 4
L L L L
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Latent Growth Curve Modeling
McArdle, JJ (1986). Latent variable growth within behavior genetic models. Behavior Genetics, 16, 163-200.
Baker, LA et al. (1992). Biometrical analysis of individual growth curves. Behavior Genetics, 22, 253-264.
McArdle, JJ et al. (1998). A contemporary method for developmental -genetic analyses of age changes in intellectual abilities. Developmental Neuropsychology, 14, 69-114.
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Summary of Longitudinal Models
Cholesky Model Few assumptions, predict any pattern of correlations Not falsifiable Limited measurements
Simplex Model Falsifiable Limited measurements
Growth Curve Model G, E influences on initial level, rate of change Unlimited measurements Computationally intensive, assumptions
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