1. Institute of Psychiatry King’s College London, UK
Path Analysis
Frühling Rijsdijk
SGDP Centre
Institute of Psychiatry
King’s College London, UK

2. Structural Equation Modelling
Twin Model
Twin Data
Assumptions
Data Preparation
Biometrical
Genetic
Theory
Hypothesised
Sources of
Variation
Observed
Variation
Summary
Statistics
Matrix
Algebra
Model
Equations
Path
Diagrams
Covariance
Algebra
This slide represents an overview of the topics and lectures that will be covered this week. It is intended to serve as a visual aid to understand where different lectures fit in.
The diagram consists of 2 ‘arms’: Twin Model and Twin Data. In the Model side are addressed the hypothesised sources of variation (based on BG theory) and how the model is expressed in both equation form and in path diagrams: by means of cov algebra and/or path tracing we derive the predicted Var/Cov of the Model. The Data arm is concerned with all data preparation processes in order to derive the observed var/Cov of the data. The link between both arms represents structural equation model fitting: the estimation of the parameters of the tested model; estimation method; deciding on the model fit > decision process etc.
Path Tracing
Rules
Predicted Var/Cov
from Model
Observed Var/Cov
from Data
Structural Equation Modelling
(Maximum Likelihood)

3. Path Analysis Developed by the geneticist Sewall Wright (1920)
Now widely applied to problems in genetics and the behavioural sciences.

4. Path Analysis
This technique allows us to present linear relationships between
variables in diagrams and to derive predictions for the variances
and covariances of the variables under the specified model.
The relationships can also be represented as structural
equations and covariance matrices
All three forms are mathematically complete, it is possible to
translate from one to the other.
Structural equation modelling (SEM) represents a unified platform
for path analytic and variance components models.

5. A system of linear model equations or
In (twin) models, expected relationships between observed variables are expressed by:
A system of linear model equations or
Path diagrams which allow the model to be represented in schematic form
Both allow us to derive predictions for the
variances and covariances of the variables
under the specified model

6. Aims of this Session Derivation of Predicted Var-Cov
matrices of a model using:
(1) Path Tracing
(2) Covariance Algebra

7. Path Diagram Conventions
Observed Variable
Latent Variable
In path diagrams we use square boxes to represent observed or measured variables and circles to represent unobserved or latent variables. The relationship among variables is represented by two kinds of arrows: a straight, single-headed arrow represents a causal relationship, and a curved two-headed arrow represents a covariance or correlational path.
Causal Path
Covariance Path

8. Path Diagrams for the Classical Twin Model

9. Model for an MZ PAIR e c a a c e1 1 E C A A C E 1 1 1 1 1 1 e c a a c e
Twin 1
Twin 2
This path diagram represents an ACE twin model for MZ pairs. The model represents specific hypotheses about (causal) relationships of latent genetic and environmental factors (single-headed arrows) on the observed variables and also about the correlations (curved arrows) between these latent factors between individuals in a pair.
The meaning of ‘causal’ is the assumption that change in the variable at the tail of the arrow will result in change in the variable at the head of the arrow, with all other variables in the diagram held constant. The causal relationships represented by straight arrows are assumed to be linear.
Variables that do not receive causal input from any one variable in the diagram are referred to as independent, source or predictor variables. Variables that do, are referred to as dependent variables. In general, only independent variables are connected by double-headed arrows.
Because the latent variables have no scale, the double-headed arrow from the factor to itself is fixed to 1 in order to give the explained variance of A, C and E the same scale as the observed variable (e.g. height).
The path coefficients a, c and e are the regression coefficient which tells us to what extent a change in the variable at the tail of the arrow is transmitted to the variable at the head of the arrow.
Model for an MZ PAIR
Note: a, c and e are the same cross twins

10. Model for a DZ PAIR e c a a c e1 .5 E C A A C E 1 1 1 1 1 1 e c a a c e
Twin 1
Twin 2
Model for a DZ PAIR
Note: a, c and e are also the same cross groups

11. Path Tracing
The covariance between any two variables is the sum of all legitimate chains connecting the variables
The numerical value of a chain is the product of all traced path coefficients in it
A legitimate chain
is a path along arrows that follow 3 rules:
The numerical value of each chain is the product of the path coefficients of all constituent arrows: both causal effects and correlations.

12. (i) Trace backward, then forward, or simply forward
from one variable to another.
NEVER forward then backward!
Include double-headed arrows from the independent
variables to itself. These variances will be
1 for latent variables
Loops are not allowed, i.e. we can not trace twice through
the same variable
The first rule may seem less arbitrary if you realise that it allows variables to be connected by common cause, but not by common consequence.
(iii) There is a maximum of one curved arrow per path.
So, the double-headed arrow from the independent
variable to itself is included, unless the chain includes
another double-headed arrow (e.g. a correlation path)

13. The Variance Since the variance of a variable is
the covariance of the variable with
itself, the expected variance will be
the sum of all paths from the variable
to itself, which follow Wright’s rules

14. Variance of Twin 1 AND Twin 2
(for MZ and DZ pairs) E C A 1 1 1 e c a
Twin 1

15. Variance of Twin 1 AND Twin 2
(for MZ and DZ pairs) E C A 1 1 1 e c a
Twin 1

16. Variance of Twin 1 AND Twin 2
(for MZ and DZ pairs) E C A 1 1 1 e c a
Twin 1

17. Variance of Twin 1 AND Twin 2
(for MZ and DZ pairs)
a*1*a = a2 E C A 1 1 1 + e c a
Twin 1

18. Variance of Twin 1 AND Twin 2
(for MZ and DZ pairs)
a*1*a = a2 E C A 1 1 1 +
c*1*c = c2 e c a +
e*1*e = e2
Twin 1
Total Variance = a2 + c2 + e2

19. Covariance Twin 1-2: MZ pairs

20. Covariance Twin 1-2: MZ pairs

21. Covariance Twin 1-2: MZ pairs
Total Covariance = a2 +

22. Covariance Twin 1-2: MZ pairs
Total Covariance = a2 + c2

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

24. ADE Model e d a a d e 1(MZ) / 0.25 (DZ) E D A A D E Twin 1 Twin 2 1/.5
Why ¼? This will hopefully have been explained in previous session.
Given the 4 possible offspring genotypes Amam, Afaf, Afam, afAm (m=mother/f=father) two siblings (or DZ twin pairs) will have 16 possible genotype combinations. If we tabulate these 16, you can see that we can identify 3 distinct combinations:
When sibs share 2 alleles IBD (from same parent), left diagonal, 4 cells; P2= 4/16 = 1/4
When sibs share 0 alleles IBD, right diagonal, 4 cells; P0 = 4/16 = ¼
When sibs share 1 alleles IBD, rest of the cells, 8 cells; P1 = 8/16 = 1/2
But why is P2 used for the Dominance Genetic Variance contribution to correlations across individuals?
Since D is the interaction effect between alleles at the same locus, these interaction effects can only be correlated across relatives if the genotypes are exactly the same, i.e. IBD 2. This is ¼ in sibs and DZ twins and 1 in MZ twins. Since the prob of IBD=2 is 0 in all other relations, Dominance does not contribute to the genetic covariance (Covg = Ra * Va + P2*Vd).
Twin 1
Twin 2

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

26. this model is just identified
ACE or ADE
Cov(mz) = a2 + c2 or a2 + d2
Cov(dz) = ½ a2 + c2 or ½ a2 + ¼ d2
VP = a2 + c2 + e or a2 + d2 + e2
3 unknown parameters (a, c, e or a, d, e), and only 3 distinct predicted statistics:
Cov MZ, Cov DZ, Vp)
this model is just identified
In the classical twin model (twin data only) we have enough information to estimate at the most 3 model parameters since we have only 3 distinct predicted statistics. However not any combination of A C D or E is identified: we always need E in the model since it includes measurement error. C and D cannot be tested at the same time with twin only data: we can have an ACE or ADE model.

27. Effects of C and D are confounded
The twin correlations indicate which of the two
components is more likely to be present:
Cor(mz) = a2 + c2 or a2 + d2
Cor(dz) = ½ a2 + c2 or ½ a2 + ¼ d2
If a2 =.40, c2 =.20
rmz = 0.60
rdz = 0.40
If a2 =.40, d2 =.20
rdz = 0.25
ACE
ADE

28. ADCE: classical twin design + adoption data
Cov(mz) = a2 + d2 + c2
Cov(dz) = ½ a2 + ¼ d2 + c2
Cov(adopSibs) = c2
VP = a2 + d2 + c2 + e2
4 unknown parameters (a, c, d, e), and 4 distinct predicted statistics:
Cov MZ, Cov DZ, Cov adopSibs, Vp)
this model is just identified
In the classical twin model (twin data only) we have enough information to estimate at the most 3 model parameters since we have only 3 distinct predicted statistics. However not any combination of A C D or E is identified: we always need E in the model since it includes measurement error. C and D cannot be tested at the same time with twin only data: we can have an ACE or ADE model.

29. Path Tracing Rules are based on Covariance Algebra

30. Three Fundamental Covariance Algebra Rules
Var (X) = Cov(X,X)
Cov (aX,bY) = ab Cov(X,Y)
Cov (X,Y+Z) = Cov (X,Y) + Cov (X,Z)

31. Example 1 1 A a Y
Y = aA
The variance of a dependent variable (Y) caused by independent
variable A, is the squared regression coefficient multiplied
by the variance of the independent variable

32. Example 2 .5 1 1 A A a a Y Z
Y = aA
Z = aA

33. Summary Path Tracing and Covariance Algebra have the same aim :
to work out the predicted Variances
and Covariances of variables,
given a specified model