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 isOpenMx 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 OpenMxWhat 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

BiometricalGeneticTheory

Predicted Var/Cov of the Model

Path TracingRules

System of Linear

Equations

Path Diagrams

Observed Var/Cov of the Data

CovarianceAlgebra

Twin Model

SEM

model building

Observed Data

Summary Statistics

Path Diagramsfor the Classical ACE Twin Model

PTwin 1

E C A1 1 1

PTwin 2

A C E1 1 1

Model for MZ or DZ Pairs Reared Together

1

1 /.5

e ac a ec

22222

22222

ecaca

caecaMZCov

Tw1 Tw2

Tw1

Tw2

22222

22222

2

12

1

ecaca

caecaDZCov

Tw1 Tw2

Tw1

Tw2

Predicted Var-Cov Matrices

Saturated Twin Model

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

Variance / Covariance

1

1

r

r

V1tw1 V1tw2

L1 L2

Mean1 Mean2

S1 S2

Stand 2x2r

Diag 2x2

2

1

0

0

S

S

Variance / Covariance

1

1

r

r

2

1

0

0

S

S

V1tw1 V1tw2

L1 L2

Mean1 Mean2

S1 S2

r2

1

0

0

S

S**

V1T1 V1T2

V1T1

V1T2

S12

S22

S1 * r * S2

S1 * r * S2

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.Rrequire(OpenMx)source("GenEpiHelperFunctions.R")# -----------------------------------------------------------------------# Prepare Data# -----------------------------------------------------------------------NCdata <- read.table ('N-CortisolNA.csv', header=T, sep=',')names (NCdata)str(NCdata)nv <- 1ntv <- nv*2Vars <-('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, name="expCovDZ" ),mxData(dzData, type="raw"), mxFIMLObjective("expCovDZ", "expMeanDZ", selVars)),

mxAlgebra(MZ.objective + DZ.objective, name="-2sumLL"), mxAlgebraObjective("-2sumLL"))

twinSatFit <- mxRun(twinSatModel)

V1tw1 V1tw2

L1 L2

Mean1 Mean2

S1 S2

r

# ------------------------------------------------------------------------------------------------------------------# Specify and Run Saturated SubModel 1 equating Variances ACROSS TWINS# ----------------------------------------------------------------------------------------------------------------

twinSatModelSub1 <- twinSatModeltwinSatModelSub1$MZ$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)ExpSDMZExpSDDZ <- mxEval(DZ.expSDDZ, twinSatFitSub1)ExpSDDZ

TESTS ASSUMPTION OF EQUAL VARIANCES

# ------------------------------------------------------------------------------------------------------------------# Specify and Run Saturated SubModel 1 equating Variances ACROSS TWINS and ZYG# ----------------------------------------------------------------------------------------------------------------

twinSatModelSub1 <- twinSatModeltwinSatModelSub1$MZ$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)ExpSDMZExpSDDZ <- mxEval(DZ.expSDDZ, twinSatFitSub1)ExpSDDZ

TESTS ASSUMPTION OF EQUAL VARIANCES

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 coefficientsmxMatrix("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 componentsmxAlgebra(a * t(a), name="A"),mxAlgebra(c * t(c), name="C"),mxAlgebra(e * t(e), name="E"), # Algebra to compute total variances and SDmxAlgebra( 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 MZmxAlgebra(rbind (cbind(A+C+E , A+C),

cbind(A+C , A+C+E)), name="expCovMZ"), # Algebra to model expected variance/covariance matrix in DZmxAlgebra(rbind (cbind(A+C+E , .5%x%A+C),

cbind(.5%x%A+C , A+C+E)), name="expCovDZ")),

SPECIFY AND RUN ACE MODEL WITH RAWDATA AND MATRIX STYLE INPUT

PTwin 1

E C A

1 1

e ac

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

univTwinSatFit@algebrasunivTwinSatFit@algebras$’-2sumll’

univTwinSatFit@submodelsunivTwinSatFit@submodels$MZ=univTwinSatFit$MZ

univTwinSatFit$MZ@matricesunivTwinSatFit$MZ@matrices$CholMZ=univTwinSatFit$MZ$CholMZ

univTwinSatFit$MZ@algebrasunivTwinSatFit$MZ@algebras$expCovMZ=univTwinSatFit$MZ$expCovMZ

univTwinSatFit$MZ$objectiveunivTwinSatFit$MZ$data

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

R Matrix Operators

http://openmx.psyc.virginia.edu/wiki/matrix-operators-and-functions

For example:Mx OpenMx

Inverse ~ solve() Transpose ‘ t()Row x col * %*%Dot Prod . *Kronecker Prod @ %x%Vert adhesion _ rbind()Hor adhesion | 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