1. Univariate Model Fitting
Sarah Medland
QIMR – openMx workshop
Brisbane 16/08/10

2. Univariate Twin Models
= using the twin design to assess the aetiology of one trait (univariate)
Path Diagrams
Basic ACE R Script

3. 1. Path Diagrams for Univariate Models

4. Basic Twin Model - MZ
Twin 1 Trait A C E c e a
Twin 2 Trait
1.0 1

5. Basic Twin Model - DZ
Twin 1 Trait A C E c e a
Twin 2 Trait
0.5 1
1.0

6. Basic Twin Model Twin 1 Trait A C E c e a Twin 2 Trait 1 rDZ = 0.5
rMZ = 1.0;
rDZ = 1.0
rDZ = 0.5 1

7. The Variance
Since the variance of a variable is the covariance of the variable with itself, the expected variance will be the sum of all paths from the variable to itself, which follow Wright’s rules

8. Variance for Twin 1 - A 1 1
a*1*a = a2 1 A C E c a e
Twin 1 Trait

9. a*1*a = a2 c*1*c = c2 Variance for Twin 1 - C c a e 1 1 1 A C E
Twin 1 Trait

10. a*1*a = a2 c*1*c = c2 e*1*e = e2 Variance for Twin 1 - E c a e 1 1 1 A
Twin 1 Trait

11. Total Variance for Twin 1
a*1*a = a2
c*1*c = c2
e*1*e = e2
Twin 1 Trait A C E c e a 1
Total Variance = a2 + c2 + e2

12. a2 + c2 Covariance - MZ 1.0 1.0 1 1 1 1 1 1 A C E E C A c e c a e a
Twin 1 Trait
Twin 2 Trait

13. 0.5a2 + c2 Covariance - DZ 0.5 1.0 1 1 1 1 1 1 A C E E C A c e c a e a
Twin 1 Trait
Twin 2 Trait

14. Variance-Covariance MatricesMZ
Twin 1
Twin 2
a2 + c2 + e2
a2 + c2

15. Variance-Covariance MatricesDZ
Twin 1
Twin 2
a2 + c2 + e2
0.5a2 + c2

16. Variance-Covariance MatricesMZ
Twin 1
Twin 2
a2 + c2 + e2
a2 + c2 DZ
0.5a2 + c2

17. Why isn’t e2 included in the covariance?
Because, e2 refers to environmental influences UNIQUE to each twin. Therefore, this cannot explain why there is similarity between twins.
Why is a2 only .5 for DZs but not MZs?
Because DZ twins share on average half of their genes, whereas MZs share all of their genes.

18. 2. Basic openMx ACE Script

19. Overview
OpenMx script
Running the script
Describing the output

20. Do some algebra to get the variances
ACE model
Specify the
matrices you need
To build the model
# Fit ACE Model with RawData and Matrices Input # univACEModel <- mxModel("univACE", mxModel("ACE", # Matrices a, c, and e to store a, c, and e path coefficients mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=10, label="a11", name="a" ), mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=10, label="c11", name="c" ), mxMatrix( type="Lower", nrow=nv, ncol=nv, free=TRUE, values=10, label="e11", name="e" ), # Matrices A, C, and E compute variance components mxAlgebra( expression=a %*% t(a), name="A" ), mxAlgebra( expression=c %*% t(c), name="C" ), mxAlgebra( expression=e %*% t(e), name="E" ), # Algebra to compute total variances and standard deviations (diagonal only) mxAlgebra( expression=A+C+E, name="V" ),
Twin 1 Trait A C E c e a 1
Do some algebra to get the variances

21. Start values?
a2 = additive genetic variance (A) c2 = Shared E variance (C) e2 = Non-shared E variance (E) Sum is modelled to be expected Total Variance Start Values for a, c, e:  (Total Variance / 3)
Twin 1 Trait A C E c e a 1

22. Standardize parameter estimates
# Algebra to compute total variances and standard deviations (diagonal only) mxAlgebra( expression=A+C+E, name="V" ), mxMatrix( type="Iden", nrow=nv, ncol=nv, name="I"), mxAlgebra( expression=solve(sqrt(I*V)), name="iSD"), # Algebra to compute standardized path estimares and variance components mxAlgebra( expression=a%*%iSD, name="sta"), mxAlgebra( expression=c%*%iSD, name="stc"), mxAlgebra( expression=e%*%iSD, name="ste"), mxAlgebra( expression=A/V, name="h2"), mxAlgebra( expression=C/V, name="c2"), mxAlgebra( expression=E/V, name="e2"), a *
1/SD =
a/SD
Twin 1 Trait A a 1
The regression coefficient a is standardized by: (a * SD(A)) / SD(Trait)
where SD(Trait) is the standard deviation of the dependent variable, and SD(A) is the standard deviation of the predictor, the latent factor ‘A’ (=1) V 1 * = V SD
inv =
1/SD

23. Standardize parameter estimates
# Algebra to compute total variances and standard deviations (diagonal only) mxAlgebra( expression=A+C+E, name="V" ), mxMatrix( type="Iden", nrow=nv, ncol=nv, name="I"), mxAlgebra( expression=solve(sqrt(I*V)), name="iSD"), # Algebra to compute standardized path estimares and variance components mxAlgebra( expression=a%*%iSD, name="sta"), mxAlgebra( expression=c%*%iSD, name="stc"), mxAlgebra( expression=e%*%iSD, name="ste"), mxAlgebra( expression=A/V, name="h2"), mxAlgebra( expression=C/V, name="c2"), mxAlgebra( expression=E/V, name="e2"),
Twin 1 Trait A
sta 1
The heritability ‘h2’ is the proportion of the total variance due to A (additive genetic effects; = A / V. Note: this will be “sta” squared.
The standardized variance components for C and E are: C / V; E / V N
“sta” 2 V A / =
“h2”

24. Standardising data
V = A + C + E A/V = 73/233 = .31 V = [73] + [90] + [70] C/V = 90/233 = .39 V = [233] E/V = 70/233 = .30 a = 8.55 c = 9.49 e = 8.35 SD = sqrt(V) = sta = 8.55 / = 0.56 squared = .31 stc = 9.49 / = 0.62 squared = .39 ste = 8.35 / = 0.55 squared = .30

25. # Algebra for expected variance/covariance matrix in MZ
mxAlgebra( expression= rbind ( cbind(A+C+E , A+C),
cbind(A+C , A+C+E)), name="expCovMZ" ),
# Algebra for expected variance/covariance matrix in DZ
mxAlgebra( expression= rbind ( cbind(A+C+E , 0.5%x%A+C),
cbind(0.5%x%A+C , A+C+E)), name="expCovDZ" ) MZ
Twin 1
Twin 2
a2 + c2 + e2
a2 + c2 DZ
0.5a2 + c2
Twin 1 Trait A C E c e a
Twin 2 Trait
1/ 0.5 1
1.0

26. mxModel("MZ", mxData( observed=mzData, type="raw" ),
mxFIMLObjective( covariance="ACE.expCovMZ", means="ACE.expMean", dimnames=selVars ) ),
mxModel("DZ",
mxData( observed=dzData, type="raw" ),
mxFIMLObjective( covariance="ACE.expCovDZ", means="ACE.expMean", dimnames=selVars )
mxAlgebra( expression=MZ.objective + DZ.objective, name="m2ACEsumll" ),
mxAlgebraObjective("m2ACEsumll"),
mxCI(c('ACE.A', 'ACE.C', 'ACE.E')) )
univACEFit <- mxRun(univACEModel, intervals=T)
univACESumm <- summary(univACEFit)
univACESumm

27. You can fit sub-models to test the significance of your parameters
# Fit AE model # univAEModel <- mxModel(univACEFit, name="univAE", mxModel(univACEFit$ACE, mxMatrix( type="Lower", nrow=1, ncol=1, free=FALSE, values=0, label="c11", name="c" ) ) ) univAEFit <- mxRun(univAEModel) univAESumm <- summary(univAEFit) univAESumm
Twin 1 Trait A E
Twin 2 Trait
You can fit sub-models to test the significance of your parameters
-you simply drop the parameter and see if the model fit is significantly worse than full model

28. Sub-models The E parameter can never not be dropped
# Fit CE model # univCEModel <- mxModel(univACEFit, name="univCE", mxModel(univACEFit$ACE, mxMatrix( type="Lower", nrow=1, ncol=1, free=FALSE, values=0, label="a11", name="a" ) ) ) univCEFit <- mxRun(univCEModel) univCESumm <- summary(univCEFit) univCESumm
Twin 1 Trait C E
Twin 2 Trait
Twin 1 Trait E
Twin 2 Trait
# Fit E model #
univEModel <- mxModel(univAEFit, name="univE",
mxModel(univAEFit$ACE,
mxMatrix( type="Lower", nrow=1, ncol=1, free=FALSE, values=0, label="a11", name="a" ) ) )
univEFit <- mxRun(univEModel)
univESumm <- summary(univEFit)
univESumm
The E parameter can never not be dropped
because it includes measurement error

29. OpenMx Output

30. univACESumm free parameters: name matrix row col Estimate
1 a11 ACE.a
2 c11 ACE.c
3 e11 ACE.e
4 mean ACE.Mean
observed statistics: 2198
estimated parameters: 4
degrees of freedom: 2194
-2 log likelihood:
saturated -2 log likelihood: NA
number of observations: 1110
chi-square: NA
p: NA
AIC (Mx):
BIC (Mx):
adjusted BIC:
RMSEA: NA

31. tableFitStatistics models compared to saturated model
Name ep -2LL df AIC diffLL diffdf p
M1 : univTwinSat
M2 : univACE
M3 : univAE
M4 : univCE
M5 : univE
Smaller -2LL means better fit. -2LL of sub-model is always higher (worse fit). The question is: is it significantly worse.
Chi-sq test: dif in -2LL is chi-square distributed. Evaluate sig of chi-sq test.
A non-sig p-value means that the model is consistent with the data.

32. Nested.fit models compared to ACE model
Name ep -2LL df diffLL diffdf p
univACE NA NA NA
univAE
univCE
univE
Smaller -2LL means better fit. -2LL of sub-model is always higher (worse fit). The question is: is it significantly worse.
Chi-sq test: dif in -2LL is chi-square distributed. Evaluate sig of chi-sq test. Critical Chi-sq value for 1 DF = 3.84
A non-sig p-value means that the dropped parameter(s) are non-significant.

33. Estimates ACE model
> univACEFit$ACE.h2 [,1] [1,] > univACEFit$ACE.c2 [1,] > univACEFit$ACE.e2 [1,]