1. OpenMx
Frühling Rijsdijk

2. OpenMx Models
Steven Boker1 Michael Neale2 Hermine Maes2 Paras Mehta3 Michael Wilde4 Timothy Brick1 Jerey Spies1 Michael Spiegel1 Ryne Estabrook1 Sarah Kenny4 John Fox5 Timothy Bates6
1University of Virginia;
2Virginia Commonwealth University;
3University of Houston;
4University of Chicago, Argonne National Labs;
5McMasters University; 6University of Edinburgh

3. Open Mx is
OpenMx is free open source software for fitting Structural Equation Models (SEM) to observed data.
OpenMx offers the features you would expect in an SEM software package, but OpenMx works in ways that will make your modeling jobs easier and will allow you to do things that other SEM packages don't.
Integration with R
OpenMx works as an integral part of the R statistical software system. You have available the full power of the R statistical software system for data manipulation, graphics, simulation, and report generation

4. Open Mx is 1. A free, full-featured, open source SEM package.
2. Runs on Windows, Mac OS-X, and Linux.
3. Runs inside the R statistical programming environment.
OpenMx features:
1. A new approach to model specification.
2. Allows both path-style and matrix-style scripting.
3. Web-based forums, tutorials, and a wiki.

5. Install OpenMx What computers run OpenMx?
You can run OpenMx on computers using Windows XP, Windows Vista, Mac (Intel or PPC) OS-X 10.5 or later, and varieties of Linux (32 and 64 bit).
What do I need to do first?
In order to install OpenMx, you will need R version 2.9.x or
How do I install OpenMx?
Open up an R session and copy the following line into the R command line:
source('
execute line
A few lines of R output will scroll by and your OpenMx (and snow and Matrix) library will be installed

6. model building SEM Biometrical Genetic Theory Twin Model Covariance
Observed Data
model building
Twin Model
Summary
Statistics
System of
Linear
Equations
Path
Diagrams
Covariance
Algebra
Path Tracing
Rules
Predicted Var/Cov
of the Model
Observed Var/Cov
of the Data
SEM

7. Path Diagrams for the Classical ACE Twin Model

8. Model for MZ or DZ Pairs Reared Together1
1 /.5 E C A A C E 1 1 1 1 1 1 e c a a c e
PTwin 1
PTwin 2

9. Predicted Var-Cov Matrices
Tw1
Tw2
Tw1
Tw2
Tw1
Tw2
Tw1
Tw2

10. Saturated Twin Model To get Twin correlations (MZ and DZ)
Testing e.g. equality in means and
variances across Twin 1 and 2 and
MZ and DZ groups

11. Variance / Covariance Stand 2x2 r Diag 2x2 L1 L2 S1 S2 V1tw1 V1tw2
Mean1
Mean2

12. Variance / Covariance r * * V1tw1 V1tw2 L1 L2 Mean1 Mean2 S1 S2
V1T V1T2
V1T1
S12
S1 * r * S2
V1T2
S1 * r * S2
S22

13. Means and Variances Multinormal Probability Density Function:
-|2πΣ |-n/2 e -.5((xi - μ) Σ-1 (xi - μ)’)
make use of all available data
get unbiased estimates if missing data are missing at random
Use Maximum Likelihood to estimate
free Parameters:
2 means, 2 variances, 1 covariance

14. ACEuniv.R require(OpenMx) source("GenEpiHelperFunctions.R")#
# Prepare Data
NCdata <- read.table ('N-CortisolNA.csv', header=T, sep=',')
names (NCdata)
str(NCdata)
nv <- 1
ntv <- nv*2
Vars <-('ncomp')
selVars <- c('ncomp1','ncomp2')
summary(NCdata)
mzData <- subset(NCdata, zyg==1, selVars)
dzData <- subset(NCdata, zyg==2, selVars)
summary(mzData)
summary(dzData)
colMeans(mzData,na.rm=TRUE)
cov(mzData,use="complete")
colMeans(dzData,na.rm=TRUE)
cov(dzData,use="complete")

15. Specify and Run Saturated Model with RawData and Matrix-style Input
twinSatModel <- mxModel("twinSat",
mxModel("MZ",
mxMatrix(type="Full", nrow=1, ncol=ntv, free=T, values=c(30,30), name="expMeanMZ"),
mxMatrix(type="Diag", nrow=ntv, ncol=ntv, free=TRUE, values=8, lbound=.001, name="expSDMZ" ),
mxMatrix(type="Stand", nrow=ntv, ncol=ntv, free=TRUE, values=.8, lbound=-.99, ubound=.99, name="expCorMZ" ),
mxAlgebra( expression= expSDMZ %*% expCorMZ %*% expSDMZ, name="expCovMZ" ),
mxData(mzData, type="raw"),
mxFIMLObjective("expCovMZ", "expMeanMZ", selVars)),
mxModel("DZ",
mxMatrix("Full", 1, 2, T, 30, name="expMeanDZ"),
mxMatrix(type="Diag", nrow=ntv, ncol=ntv, free=TRUE, values=8, lbound=.001, name="expSDDZ" ),
mxMatrix(type="Stand", nrow=ntv, ncol=ntv, free=TRUE, values=.6, lbound=-.99, ubound=.99, name="expCorDZ" ),
mxAlgebra( expression= expSDDZ %*% expCorDZ %*% expSDDZ, name="expCovDZ" ),
mxData(dzData, type="raw"),
mxFIMLObjective("expCovDZ", "expMeanDZ", selVars)),
mxAlgebra(MZ.objective + DZ.objective, name="-2sumLL"),
mxAlgebraObjective("-2sumLL"))
twinSatFit <- mxRun(twinSatModel) r L1 L2 S1 S2
V1tw1
V1tw2
Mean1
Mean2

16. TESTS ASSUMPTION OF EQUAL VARIANCES#
# Specify and Run Saturated SubModel 1 equating Variances ACROSS TWINS #
twinSatModelSub1 <- twinSatModel
twinSatModelSub1$MZ$expSDMZ <- mxMatrix("Diag", 2, 2, T, 8, "sdMZ", name="expSDMZ")
twinSatModelSub1$DZ$expSDDZ <- mxMatrix("Diag", 2, 2, T, 8, "sdDZ", name="expSDDZ")
twinSatFitSub1 <- mxRun(twinSatModelSub1)
#Check!!
ExpSDMZ <- mxEval(MZ.expSDMZ, twinSatFitSub1)
ExpSDMZ
ExpSDDZ <- mxEval(DZ.expSDDZ, twinSatFitSub1)
ExpSDDZ

17. TESTS ASSUMPTION OF EQUAL VARIANCES#
# Specify and Run Saturated SubModel 1 equating Variances ACROSS TWINS and ZYG #
twinSatModelSub1 <- twinSatModel
twinSatModelSub1$MZ$expSDMZ <- mxMatrix("Diag", 2, 2, T, 8, "sd", name="expSDMZ")
twinSatModelSub1$DZ$expSDDZ <- mxMatrix("Diag", 2, 2, T, 8, "sd", name="expSDDZ")
twinSatFitSub1 <- mxRun(twinSatModelSub1)
#Check!!
ExpSDMZ <- mxEval(MZ.expSDMZ, twinSatFitSub1)
ExpSDMZ
ExpSDDZ <- mxEval(DZ.expSDDZ, twinSatFitSub1)
ExpSDDZ

18. SPECIFY AND RUN ACE MODEL WITH RAWDATA AND MATRIX STYLE INPUT
univACEModel <- mxModel("univACE",
mxModel("ACE",
# Matrix for expected means vector for MZ and DZ twins
mxMatrix("Full", 1, 2, T, 20, "mean", name="expMean"),
# Matrices a11, c11, and e11 to store the a, c, and e path coefficients
mxMatrix("Full", nrow=1, ncol=1, free=TRUE, values=3, label="a11", name="a"),
mxMatrix("Full", nrow=1, ncol=1, free=TRUE, values=3, label="c11", name="c"),
mxMatrix("Full", nrow=1, ncol=1, free=TRUE, values=3, label="e11", name="e"),
# Matrices A, C, and E to compute variance components
mxAlgebra(a * t(a), name="A"),
mxAlgebra(c * t(c), name="C"),
mxAlgebra(e * t(e), name="E"),
# Algebra to compute total variances and SD
mxAlgebra( expression=A+C+E, name="V" ),
mxMatrix( type="Iden", nrow=nv, ncol=nv, name="I"),
mxAlgebra( expression=solve(sqrt(I*V)), name="sd"),
# Algebra to model expected variance/covariance matrix in MZ
mxAlgebra(rbind (cbind(A+C+E , A+C),
cbind(A+C , A+C+E)), name="expCovMZ"),
# Algebra to model expected variance/covariance matrix in DZ
mxAlgebra(rbind (cbind(A+C+E , .5%x%A+C),
cbind(.5%x%A+C , A+C+E)), name="expCovDZ") ),
PTwin 1 E C A 1 e a c

19. mxData(mzData, type="raw"),
mxModel("MZ",
mxData(mzData, type="raw"),
mxFIMLObjective("ACE.expCovMZ", "ACE.expMean",selVars) ),
mxModel("DZ",
mxData(dzData, type="raw"),
mxFIMLObjective("ACE.expCovDZ", "ACE.expMean",selVars)
mxAlgebra(MZ.objective + DZ.objective, name="-2sumLL"),
mxAlgebraObjective("-2sumLL")
) # end of model univACE
univACEFit <- mxRun(univACEModel) #
# Generate ACE Model Output
LL_ACE <- mxEval(objective, univACEFit)
LL_ACE

20. Extract Information all information in MxModel fitted object
list of algebras of container model
specific algebra
list of all items in all child models
list of all items in specific child model
list of all matrices in specific child model
specific matrix in specific child model
list of all algebras in specific child model
specific algebra in specific child model
objective of specific child model
data of specific child model
univTwinSatFit
=univTwinSatFit$MZ
=univTwinSatFit$MZ$CholMZ
=univTwinSatFit$MZ$expCovMZ
univTwinSatFit$MZ$objective
univTwinSatFit$MZ$data

21. R Matrix Operators For example: Mx OpenMx Inverse ~ solve()
For example:
Mx OpenMx
Inverse ~ solve()
Transpose ‘ t()
Row x col * %*%
Dot Prod . *
Kronecker %x%
Vert adhesion _ rbind()
Hor adhesion | cbind()

22. GenEpiHelperFunctions.R
Function "parameterSpecifations()" prints labels of a MxMatrix with square brackets surrounding free parameters; returns a matrix of strings
Function "expectedMeansCovariances()" prints expected means and expected covariance matrices for all submodels
Function "formatOutputMatrices()" prints matrix with specified labels and number of decimals
Function "formatMatrix()" returns a matrix with specified dimnames and of decimal places
Function "tableFitStatistics()" prints fit statistics with labels for Full Model and list of Nested Models