1. Elizabeth Prom-Wormley & Hermine Maes
Univariate Twin Analysis- Saturated Models for Continuous and Categorical Data
September 2, 2014
Elizabeth Prom-Wormley & Hermine Maes

2. Overall Questions to be Answered
Does the data satisfy the assumptions of the classical twin study?
Does a trait of interest cluster among related individuals?

3. Family & Twin Study Designs
Family Studies
Classical Twin Studies
Adoption Studies
Extended Twin Studies

4. The Data Please open twinSatConECPW Fall2014.R
569 total MZ pairs
351 total DZ pairs
Please open twinSatConECPW Fall2014.R
Australian Twin Register
18-30 years old, males and females
Work from this session will focus on Body Mass Index (weight/height2) in females only
Sample size
MZF = 534 complete pairs (zyg = 1)
DZF = 328 complete pairs (zyg = 3)

5. A Quick Look at the Data

6. Classical Twin Studies Basic Background
The Classical Twin Study (CTS) uses MZ and DZ twins reared together
MZ twins share 100% of their genes
DZ twins share on average 50% of their genes
Expectation- Genetic factors are assumed to contribute to a phenotype when MZ twins are more similar than DZ twins

7. Classical Twin Study Assumptions
MZ twins are genetically identical
Equal Environments of MZ and DZ pairs

8. Basic Data Assumptions
MZ and DZ twins are sampled from the same population, therefore we expect :-
Equal means/variances in Twin 1 and Twin 2
Equal means/variances in MZ and DZ twins
Further assumptions would need to be tested if we introduce male twins and opposite sex twin pairs

9. “Old Fashioned” Data CheckingMZ DZ T1 T2
mean
21.35
21.34
21.45
21.46
variance
0.73
0.79
0.77
0.82
covariance(T1-T2)
0.59
0.25
Nice, but how can we actually be sure that these means and variances are truly the same?

10. Univariate Analysis A Roadmap
1- Use the data to test basic assumptions (equal means & variances for twin 1/twin 2 and MZ/DZ pairs)
Saturated Model
2- Estimate contributions of genetic and environmental effects on the total variance of a phenotype
ACE or ADE Models
3- Test ACE (ADE) submodels to identify and report significant genetic and environmental contributions
AE or CE or E Only Models 10

11. Saturated Twin Model

12. Saturated Code Deconstructed
meanMZ <- mxMatrix( type="Full", nrow=1, ncol=ntv, free=TRUE, values=meVals, labels=c("mMZ1","mMZ2"), name=”meanMZ" )
meanDZ <- mxMatrix( type="Full", nrow=1, ncol=ntv, free=TRUE, values=meVals, labels=c("mDZ1","mDZ2"), name=”meanDZ" )
mMZ1
mMZ2
mDZ1
mDZ2
mean MZ = 1 x 2 matrix
mean DZ = 1 x 2 matrix

13. Saturated Code Deconstructed
covMZ <- mxMatrix( type="Symm", nrow=ntv, ncol=ntv, free=TRUE, values=cvVals, lbound=lbVals, labels=c("vMZ1","cMZ21","vMZ2"), name=”covMZ" )
covDZ <- mxMatrix( type="Symm", nrow=ntv, ncol=ntv, free=TRUE, values=cvVals, lbound=lbVals, labels=c("vDZ1","cDZ21","vDZ2"), name=”covDZ" )
vMZ1
cMZ21
vMZ2
covMZ = 2 x 2 matrix T1 T2
vDZ1
cDZ21
vDZ2
covDZ = 2 x 2 matrix

14. Time to Play... Continue with the File twinSatConECPW Fall2014.R

15. Estimated Values T1 T2 Saturated Model mean MZ DZ cov
10 Total Parameters Estimated
mMZ1, mMZ2, vMZ1,vMZ2,cMZ21
mDZ1, mDZ2, vDZ1,vDZ2,cDZ21
Standardize covariance matrices for twin pair correlations (covMZ & covDZ)

16. Estimated Values T1 T2 Saturated Model mean MZ 21.34 21.35 DZ 21.45
21.46
cov
0.73
0.77
0.59
0.79
0.24
0.82
10 Total Parameters Estimated
mMZ1, mMZ2, vMZ1,vMZ2,cMZ21
mDZ1, mDZ2, vDZ1,vDZ2,cDZ21
Standardize covariance matrices for twin pair correlations (covMZ & covDZ)

17. Fitting Nested Models Saturated Model
likelihood of data without any constraints
fitting as many means and (co)variances as possible
Equality of means & variances by twin order
test if mean of twin 1 = mean of twin 2
test if variance of twin 1 = variance of twin 2
Equality of means & variances by zygosity
test if mean of MZ = mean of DZ
test if variance of MZ = variance of DZ

18. Estimated Values T1 T2 Equate Means & Variances across Twin Order meanMZ DZ
cov
Equate Means Variances across Twin Order & Zygosity

19. Estimates T1 T2 Equate Means & Variances across Twin Order mean MZ
21.35 DZ
21.45
cov
0.76
0.79
0.59
0.24
Equate Means Variances across Twin Order & Zygosity
21.39
0.78
0.61
0.23

20. Stats Model ep -2ll df AIC diff -2ll diff p Saturated 10 4055.93 1767
521.93
mT1=mT2 8
4056
1769
518
0.07 2
0.97
mT1=mT2 &
varT1=VarT2 6
1771
516.94
3.01 4
0.56
Zyg
MZ=DZ
1773
517.45
7.52
0.28
No significant differences between saturated model
and models where means/variances/covariances
are equal by zygosity and between twins

21. Working with Binary and Ordinal Data
Elizabeth Prom-Wormley and Hermine Maes
Special Thanks to Sarah Medland

22. Transitioning from Continuous Logic to Categorical Logic
Ordinal data has 1 less degree of freedom compared to continuous data
MZcov, DZcov, Prevalence
No information on the variance
Thinking about our ACE/ADE model
4 parameters being estimated
A/ C/ E/ mean
ACE/ADE model is unidentified without adding a constraint

23. Two Approaches to the Liability Threshold Model
Traditional
Maps data to a standard normal distribution
Total variance constrained to be 1
Alternate
Fixes an alternate parameter (usually E)
Estimates the remaining parameters

24. Please open BinaryWarmUp.R
Time to Look at the Data!
Please open BinaryWarmUp.R

25. Observed Binary BMI is Imperfect Measure of Underlying Continuous Distribution
Mean (bmiB2) = 0.39
SD (bmiB2) = 0.49
Prevalence “low” BMI = 60.6%
We are interested in the liability
of risk for being in the
“high” BMI category

26. It’s Helpful to Rescale
Raw Data (Unstandardized)
mean=0.49, SD=0.39
-Data not mapped to a standard normal
-No easy conversion to %
-Difficult to compare between groups
Since the scaling is now arbitrary
Standard Normal (Standardized)
mean=0, SD=1
Area under the curve
between two z-values is interpreted
as a probability or percentage

27. Binary Review Threshold calculated using the
cumulative normal distribution (CND)
-We used frequencies and
inverse CND to do our
own estimation of the threshold
qnorm(0.816) = 0.90
- Threshold is the Z Value that
corresponds with the proportion
of the population
having “low BMI”

28. Moving to Ordinal Data!

29. Getting a Feel for the Data Open twinSatOrd.R
Calculate the frequencies of the 5 BMI categories for the second twins of the MZ pairs
CrossTable(mzDataOrdF$bmi2)

30. Estimating MZ Twin 2 Thresholds by Hand
T1 = qnorm(0.124)
T1 =
T2 = qnorm( )
T2 =
T3 = qnorm( )
T3 =0.388
T4 = qnorm( )
T4 =
Estimate Twin Pair Correlations for the Liabilities Too!

31. Translating Back to the SEM Approach in OpenMx

32. Handling Ordinal Data in OpenMx
1- Determine the 1st threshold
2- Determine displacements between 1st threshold and subsequent thresholds
3- Add the 1st threshold and the displacement to obtain the subsequent thresholds 32

33. Ordinal Saturated Code Deconstructed Defining Threshold Matrices
covMZ
covDZ
μ MZT1
μ MZT2
μ DZT1
μ DZT2 1 1
LT1
LT2
LT1
LT2
Variance
Constraint 1 1 1 1
Threshold
Model
t1MZ1
t1MZ2
t2MZ1
t2MZ2
t3MZ1
t3MZ2
t4MZ1
t4MZ2
t1DZ1
t1DZ2
t2DZ1
t2DZ2
t3DZ1
t3DZ2
t4DZ1
t4DZ2
threM <- mxMatrix( type="Full", nrow=nth, ncol=ntv, free=TRUE, values=thVal, lbound=thLB, labels=thLabMZ, name="ThreMZ" )
threD <- mxMatrix( type="Full", nrow=nth, ncol=ntv, free=TRUE, values=thVal, lbound=thLB, labels=thLabDZ, name="ThreDZ" )

34. Ordinal Saturated Code Deconstructed Defining Threshold Matrices- ThreMZ
Tw1
Tw2
-1.89
-1.16
0.81
0.79
0.73
0.76
0.55
1- Determine the 1st threshold
2- Determine displacements
between1st thresholds and
subsequent thresholds

35. Double Check- Moving from Frequencies to Displacements
Frequency BMI T2
Cumulative Frequency
Z Value
Displacement
0.124
-1.16
0.236
0.360
-0.37
0.79
0.291
0.651
0.39
0.76
0.175
0.826
0.94
0.55 1 -

36. Ordinal Saturated Code Deconstructed Estimating Expected Threshold Matrices
threMZ <- mxAlgebra( expression= Inc %*% ThreMZ, name="expThreMZ" )
Inc <- mxMatrix( type="Lower", nrow=nth, ncol=nth, free=FALSE, values=1, name="Inc" )
-1.19
-1.16
0.81
0.79
0.73
0.76
0.55
-1.19
-1.16
-0.38
-0.37
0.34
0.39
0.89
0.93 1
% * % =
3- Add the 1st threshold and the displacement
to obtain the subsequent thresholds

37. Ordinal Saturated Code Deconstructed Estimating Correlations & Fixing Variance
corMZ <- mxMatrix( type="Stand", nrow=ntv, ncol=ntv, free=TRUE, values=corVals, lbound=lbrVal, ubound=ubrVal, labels="rMZ", name="expCorMZ" )
corDZ <- mxMatrix( type="Stand", nrow=ntv, ncol=ntv, free=TRUE, values=corVals, lbound=lbrVal, ubound=ubrVal, labels="rDZ", name="expCorDZ" )

38. How Many Parameters in this Ordinal Model?
MZ correlation- rMZ
DZ correlation- rDZ
Thresholds
t1MZ1,t2MZ1,t3MZ1,t4MZ1
t1MZ2,t2MZ2,t3MZ2,t4MZ2
t1DZ1,t2DZ1,t3DZ1,t4DZ1
t1DZ2,t2DZ2,t3DZ2,t4DZ2

39. Problem Set 1 Open twinSatOrdA.R and twinSatOrd.R
What do these scripts do?
Looking at the scripts only: How are they similar? How are they different?
What do these differences in the scripts reflect regarding conceptual differences in the two models?
Run either script and double check against your previously hand-calculated values of thresholds. Report your results. If you can’t get it to match up, don’t panic…do .
Run twinSatOrd.R
Is testing an ACE model with the usual model assumptions justified? Why or why not?

40. Univariate Analysis with Ordinal Data A Roadmap
1- Use the data to test basic assumptions inherent to standard ACE (ADE) models
Saturated Model
2- Estimate contributions of genetic and environmental effects on the liability of a trait
ADE or ACE Models
3- Test ADE (ACE) submodels to identify and report significant genetic and environmental contributions
AE or E Only Models

41. Questions?