1. Structural Equation Modeling
HGEN502, 2008
Danielle M. Dick
Hermine H. Maes
I’ve added notes to each of the slides. I’m sure you could have done a fine job without them
But at least, you see how I was thinking about linking various slides.
Of course, ignore and make changes as you like.
Sorry you’re getting this at the last minute. I actually spent considerable time thinking about
how to best present this material, so that the different pieces fit into the overall puzzle.M
Not sure if this accomplishes that, but I’m sure we’ll find out from the students one way or another.
Thanks again for stepping in!
I looked at your presentation for the last class (next Monday) and that was exactly what I was thinking.
I can add an example for cultural transmission / assortative mating.
I also have a presentation with various path models for different types of questions,
if any of the slides were of any use. We can ‘talk’ about that later this week. Do you use skype?
Hope the lecture goes well! Best, Hermine.

2. Role of model mediating between theory and data
Tim used this in his intro presentation, and Lindon referred to it indirectly.
I think it provides a good’ model’ to connect the various pieces together.
It is also in the first chapter of the book which they received as background reading.
The talk sets out to go over the various steps of the figure.
Notice the little figures in the bottom right corner of each of the slide,
highlighting the part that’s being discussed.

3. Theory Phenotype = Genotype+Environment (VP=VG+VE)
Genetic factors (VG):
Additive (VA)
Dominance (VD)
Environmental factors (VE):
Common / Shared (VC)
Specific / Unique (VE)
Assumptions: additivity & independence of genetic and environmental effects

4. Model Building Write equations Draw path diagrams or
Trace backwards, change direction at a 2-headed arrow, then trace forwards (implies that we can never trace through two-headed arrows in the same chain).
The expected covariance between two variables, or the expected variance of a variable, is computed by multiplying together all the coefficients in a chain, and then summing over all possible chains.
How do we translate our theory in a testable mathematical model?
Some people prefer equations.
Other find diagrams more intuitive (a picture is worth a thousand words (or a minimum of three equations..).
Segway (not sure how you spell this) into path analysis.

5. Model VP = VA + VC + VE ACE Model
‘Full’ Model: requires various different types of relative to uniquely solve for each of the unknowns.
Thus let’s start with ACE.

6. Study Design Uniquely quantify sources of variation of interest
Subjects available in large enough samples to obtain stable estimates of sources of variation, and representative of general population
Classical Twin Study (MZ & DZ twins reared together): Separation of genetic and shared environmental factors
Equal environment assumption; Random mating; NO: GE correlation, G x E interaction, sex limitation, G x age interaction
What study designs are being used and what questions can they answer?
Depending on access to twin or adoption registries, researchers might have a preference or bias.
It’s important to know advantages and disadvantages of various designs.

7. Twin Correlations Observed Statistics: 1. MZ correlation
2. DZ correlation
3. Variance
Unknown Parameters:
A: additive genetic
C: shared environment
E: unique environment
Twin Correlations

8. Data
Collect phenotypic data from large sample of relatives/ twins (N > )
Power session on Friday
Calculate / Estimate means & variances
Estimate covariances/ correlations between different types of relatives
Fit ‘the’ model
Data are essential to evaluate theory & models based on theory.

9. Model Fitting Formula’s Analysis of Variance (Jinks & Fulker, 1970)
Falconer: h2b= 2(rMZ - rDZ): broad sense heritability
Holzinger: H= (rMZ - rDZ)(1- rDZ)
Analysis of Variance (Jinks & Fulker, 1970)
Structural Equation Modeling (SEM)
Includes Regression Analysis & Factor Analysis & most Linear Models
Evaluate series of equations
Solve for unknowns iteratively
Path Analysis: tool to derive expectations
Early recognitions of use of MZ & DZ twins to answer the nature/nurture question.
Galton (Lindon nicely described the historical background.
Falconer: maybe explain difference between broad and narrow sense heritability.
Many more attempts at coming up with a simple way to estimate the role of genes.
These approaches are still around, although using SEM model fitting has become the standard.

10. SEM Advantages
Systematic approach to hypothesis testing and parameter estimation
Evaluate goodness-of-fit of model - closeness of observed & expected values & Compare fit under alternative models
Obtain maximum likelihood estimates & Evaluate significance of parameters- effect size & sample size
Both continuous and categorical variables
Extendable: multiple variables; covariates; extended pedigrees; selected samples
Make use of all available data
Key here is combination of hypothesis testing and parameter estimation. (principles introduced by Lindon)
Not only calculate how big h^2 is, but whether genetic factors contribute significantly to the model.

11. Model (Estimates) Goodness-of-fit: likelihood
ML estimates and confidence intervals
We typically use a statistical program to obtain the likelihood of a set of data given a particular model.
Thus the likelihood is a measure of goodness-of-fit.
Remember the curve of likelihood for different values of the parameter(s),and finding the maximum
(sometimes minimum if we evaluate -2 log likelihood) see Wednesday.
Estimates of free parameters (A, C and E) are ML estimates. Confidence intervals can be obtained.

12. Revision Test alternative models Test significance of parameters ACE
ADE
Test significance of parameters
AE: test significance of C
CE: test significance of A
E: test significance of A&C, familial resemblance
If the model does not fit the data, we have to revise the model.
We may also want to test alternative models.
With twin data of twins reared together, C & D are confounded
(DZ correlation increased relative to MZ correlations for C)
(DZ correlation decreased relative to MZ correlation for D,
remember expectation of .25 for correlation of dominance factors in sibs/DZ twins
(from Lindon’s biometrical genetics lecture)
One can drop (fix to zero) parameters to test their significance.

13. Theory
Conclusions regarding the best fitting model taking into account model fit and parsimony (which sources of variance are significant?)
Conclusions regarding the contributions: magnitude of effects of genetic versus environmental factors (how big is their effect?)
After fitting a series of models, we can use statistical principles to pick the ‘best fitting’ model.
Introduce parsimony (Occam’s razor): come up with simplest explanation (model) for the data
Quantify the contributions of A, C, and E,
depending on which sources contribute significantly to the variance of phenotype of interest.
Concrete example of Model Fitting and Likelihood estimation in context of twin data (ACE model)
Wednesday! Please download excel spreadsheet and bring computers to class so that you can
actively follow along.
Mike Neale recently wrote a nice chapter for the Handbook of Behavioral Genetics, which covers
most of the topics covered in this module (Tim’s, Lindon’s and my lectures) in a similar order.
Note that there is significant overlap with the chapters provided by Lindon.