OpenMx
Frühling Rijsdijk
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
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
http://openmx.psyc.virginia.edu
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.
http://openmx.psyc.virginia.edu
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 2.10.1.
How do I install OpenMx?
• Open up an R session and copy the following line into the R
command line:
– source('http://openmx.psyc.virginia.edu/getOpenMx.R')
•
execute line
• A few lines of R output will scroll by and your OpenMx (and snow
and Matrix) library will be installed
Biometrical
Genetic
Theory
model
building
Observed Data
Twin Model
Summary
Statistics
System of
Linear
Equations
Covariance
Algebra
Path
Diagrams
Path Tracing
Rules
Predicted Var/Cov
of the Model
Observed Var/Cov
of the Data
SEM
Path Diagrams
for the Classical
ACE Twin Model
Model for MZ or DZ Pairs Reared Together
1
1 /.5
C
E
1
1
1
e
A
c
PTwin 1
C
A
a
1
1
1
a
E
c
PTwin 2
e
Predicted Var-Cov Matrices
Tw1
a 2  c 2  e 2
Cov MZ

2
2
Tw2
 a c
Tw1
Tw2

2
2
2
a c e 
2
a c
Tw1
 2
2
2
Tw1
a c e

Cov DZ

1 2
2
a c

Tw2
 2
2
Tw2

a c

2

2
2
2
a c e 

1
2
2
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
Variance / Covariance
L1
S1
V1tw1
r
Stand 2x2
L2
S2
V1tw2
Mean1 Mean2
1
r
r
1
Diag 2x2
S1
0
0
S2
Variance / Covariance
L1
S1
V1tw1
r
L2
S1
0
0
S2
*
1
r
r
1
*
S1
0
0
S2
S2
V1T1
V1tw2
Mean1 Mean2
V1T1
S12
V1T2
S1 * r * S2
V1T2
S1 * r * S2
S22
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
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")
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,
L1
name="expCovDZ" ),
mxData(dzData, type="raw"),
S1
mxFIMLObjective("expCovDZ", "expMeanDZ", selVars)),
mxAlgebra(MZ.objective + DZ.objective, name="-2sumLL"),
V1tw1
mxAlgebraObjective("-2sumLL"))
twinSatFit <- mxRun(twinSatModel)
r
L2
S2
V1tw2
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
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
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"),
C
A
E
mxAlgebra(e * t(e), name="E"),
1
1
# Algebra to compute total variances and SD
mxAlgebra( expression=A+C+E, name="V" ),
a
e
c
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
PTwin 1
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")
),
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
Extract Information
univTwinSatFit
all information in MxModel fitted object
univTwinSatFit@algebras
univTwinSatFit@algebras$’-2sumll’
list of algebras of container model
specific algebra
univTwinSatFit@submodels
univTwinSatFit@submodels$MZ
=univTwinSatFit$MZ
list of all items in all child models
list of all items in specific child model
univTwinSatFit$MZ@matrices
univTwinSatFit$MZ@matrices$CholMZ
=univTwinSatFit$MZ$CholMZ
list of all matrices in specific child model
specific matrix in specific child model
univTwinSatFit$MZ@algebras
univTwinSatFit$MZ@algebras$expCovMZ
=univTwinSatFit$MZ$expCovMZ
list of all algebras in specific child model
specific algebra in specific child model
univTwinSatFit$MZ$objective
univTwinSatFit$MZ$data
objective of specific child model
data of specific child model
R Matrix Operators
http://openmx.psyc.virginia.edu/wiki/matrix-operators-and-functions
For example:
Mx
Inverse
~
Transpose
‘
Row x col
*
Dot Prod
.
Kronecker Prod @
Vert adhesion _
Hor adhesion
|
OpenMx
solve()
t()
%*%
*
%x%
rbind()
cbind()
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