PATH ANALYSIS. (see Chapters 5,6 of Neale & Cardon). . allows us to represent. linear. models for the relationships between. variables in diagrammatic form. e.g. a genetic model;. a factor model;. a regression model. .
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(see Chapters 5,6 of Neale & Cardon)

allows us to represent
linear
models for the relationships between
variables in diagrammatic form
e.g.
a genetic model;
a factor model;
a regression model.

makes it easy to derive expectations for the variances and covariances
of variables in terms of the parameters of the proposed linear model.

permits easy translation into a matrix formulation as used by programs
such as MX.
DOMINANCE
GENETIC
UNIQUE
SHARED
GENETIC
e.g.
EFFECT
ENVIRONMENT
ENVIRONMENT
EFFECT
E
A
C
D
1
1
1
1
e
a
c
d
P
PHENOTYPE
P = eE + aA + cC + dD
Figure 1. A PATH MODEL FOR MZ TWIN PAIRS
1.0
1.0 (MZT) or 0.0 (MZA)
1.0
1
EXOGENOUS
D
D
C
C
E
A
E
A
1
1
1
1
1
1
1
T1
T2
T1
T2
T1
T1
T2
T2
VARIABLES
e
a
c
d
e
a
c
d
ENDOGENOUS
P
P
T1
T2
VARIABLES
TWIN 1
TWIN 2
Squares or rectangles denote observed variables.
Circles or ellipses denote latent (unmeasured) variables.
Uppercase letters are used to denote variables.
Lowercase letters (or numeric values) are used to denote covariances or path
coefficients.
Singleheaded arrows or paths ( ) are used to represent hypothesized
causal relationships between variables under a particular model  where the
variable at the tail of the arrow is hypothesized to have a direct causal influence
on the variable at the head of the arrow (e.g., A
P
).
T1
T1
Doubleheaded arrows ( ) are used to represent a covariance between two
variables, which may arise through common causes not represented in the
model. Doubleheaded arrows ( ) may also be used to represent the variance
of an exogenous variable or (in some conventions) the residual variance in an
endogenous variable.
Doubleheaded arrows may
be used to link any variables with one or more
not
singleheaded arrows pointing to them  these variables are called endogenous
variables. Other variables are exogenous variables.
Singleheaded arrows may be drawn from exogenous to endogenous variables
(A P) or from endogenous variables to other endogenous variables
(P
P
).
1
2
Omission covariance between two of a twoheaded arrow between two exogenous variables implies the assumption that the covariance of those variables is zero (e.g., no genotypeenvironment correlation).
Omission of a direct path from an exogenous (or endogenous) variable to an endogenous variable implies that there is no direct causal effect of the former on the latter variable.
ASSUMPTIONS OF OUR SIMPLE PATH MODEL FOR MZ TWIN PAIRS covariance between two
:
(i)
linear additive effects  no multiplicative (genotype x environment)
interactions;
(ii)
E, A, C, D are mutually uncorrelated  no genotypeenvironment
covariance;
(iii)
path coefficients e, a, c, d are the same for first, second twins;
(iv)
no reciprocal sibling environmental influences; i.e., no direct paths
P
P
1
2
e.g., for the MZ twin pair covariance we have: no reciprocal causation 
CONTRIBUTION
a 1 a
a2
c 1 c
d 1 d
CovMZ = a2 + c2 + d2
P1 AT1 AT2 P2
P1 CT1 CT2 P2 c2
P1 DT1 DT2 P2 d2
e.g., for the total phenotypic variance for MZ twins: no reciprocal causation 
CONTRIBUTION
e 1 e
e2
a 1 a
c 1 c
d 1 d
VarMZ = e2 + a2 + c2 + d2
P1 ET1 ET1 P1
P1 AT1 AT1 P1 a2
P1 CT1 CT1 P1 c2
P1 DT1 DT1 P1 d2
Exercise 1.Using Figure 1, what is the expected covariance between MZ
twin pairs reared apart (MZAs)?
Exercise 2. Using Figure 1, what is the expected variance of twins from
MZ pairs reared apart?
Figure 2. no reciprocal causation DZ TWIN PAIRS/FULL SIBLINGS
0.25
1 (DZT/FST) or 0 (DZA/FSA)
0.5
(1+x)
1
1
1
1
1
1
1
1
D
E
A
C
E
A
C
D
d
e
h
c
e
h
c
d
P
P
TWIN 1
TWIN 2
x = 0 under random mating.
Exercise 3. no reciprocal causation Using Figure 2, and assuming random mating (x=0), what is
the expected covariance between DZ twin pairs reared together?
Exercise 4. Using Figure 2, what is the expected variance of DZ twins or
full siblings reared together?
Exercise 5. Using Figure 2, what is the expected covariance of DZ twins
or full sibling pairs reared apart?
Figure 3. no reciprocal causation BIOLOGICAL PARENT  OFFSPRING
1
1
1
1
1
1
1
1
E
C
A
D
E
C
A
D
d
a
c
e
a
e
c
d
0.5
0.5
0.5
0.5
P
P
MOTHER
FATHER
1
0.5
R
R
0.5
A
0.25
A
1
1
E
C
A
D
D
A
C
E
1
1
1
1
1
1
e
c
h
d
d
h
c
e
P
P
CHILD 2
CHILD 1
Exercise 6. no reciprocal causation Using Figure 3, what is the expected covariance of parent and
offspring?
Exercise 7. Using Figure 3, what is the expected covariance of biological
parent and adoptedaway offspring?
Exercise 8. What 10 assumptions are implied by Figure 3?
Figure 4. no reciprocal causation BIOLOGICAL UNCLE / AUNT  NEPHEW / NIECE
1
0.5
0.25
E
C
A
D
E
C
A
D
E
C
A
D
e
c
a
d
e
c
a
d
e
c
a
d
0.5
P
0.5
FATHER
P
P
AUNT/UNCLE
MOTHER
R
0.5
1
A
E
C
A
D
e
c
a
d
P
NEPHEW/NIECE
Exercise 9. no reciprocal causation Using Figure 4, what is the expected covariance between
biological uncle and nephew?
Figure 5. no reciprocal causation OFFSPRING OF MZ TWIN PAIRS
1
1
1
1
1
1
1
1
1
1
1
E
D
C
A
E
D
C
A
A
C
D
E
A
C
D
E
1
1
1
1
1
1
1
1
e
d
c
a
e
d
c
a
a
c
d
e
a
c
d
e
½
½
½
½
PSPOUSE1
PSPOUSE2
PTWIN1
PTWIN2
0.5
0.5
R
R
A
A
1
1
1
E
D
A
C
E
D
A
C
1
1
1
e
d
a
c
e
d
a
c
POFFSPRING (Twin 2)
POFFSPRING (Twin 1)
Exercise 10. no reciprocal causation Using Figure 5, what is the expected covariance between
MZ uncle/aunt and nephew/niece, i.e. between an MZ twin and the child of his or her cotwin?
Exercise 11. What is the expected covariance between first cousins related through MZ twins (“MZ half sibs”)?
Figure 6. no reciprocal causation DIRECTIONOFCAUSATION MODEL: MZ PAIRS
1
E
C
E
C
1
1
1
1
e
c
e
c
CONDUCT
PROBLEMS
TWIN 1
CONDUCT
PROBLEMS
TWIN 2
i
i
1
E
A
E
A
1
1
1
1
e
a
e
a
ALCOHOL
DEPENDENCE
TWIN 1
ALCOHOL
DEPENDENCE
TWIN 2
Exercise 12. no reciprocal causation Using Figure 6,
(1) What is the expected MZ covariance for
(a) conduct problems;
(b) alcohol dependence;
(c) conduct problems in one twin and alcohol dependence in
the cotwin?
(2) Redraw Figure 6 so that the direction of causation is now
ALCOHOL DEPENDENCE CONDUCT PROBLEMS
What are the expected MZ covariances now for
(a) conduct problems;
(b) alcohol dependence;
(c) conduct problems in one twin and alcohol dependence in
the cotwin?
(3) What are the corresponding expected DZ covariances when no reciprocal causation 
(a) CONDUCT PROBLEMS ALCOHOL DEPENDENCE
(b) ALCOHOL DEPENDENCE CONDUCT PROBLEMS
PART TWO: no reciprocal causation 
PATH ANALYSIS: FROM PATH DIAGRAMS TO MATRICES
It is straightforward to convert from a path diagram to a matrix representation of a
path model, and vice versa. Here we illustrate a computationally efficient approach 
though other alternatives exist (see e.g. the MX manual).
Consider first a model with no feedback loops for example Figure 1 for MZ twin pairs
reared together, where all endogenous variables are directly observable, and there are
no paths from endogenous variables to other endogenous variables.
Let
X
denote the covariance matrix for the exogenous variables. The i, jth element of
the matrix
X
(where i denotes
R
ow, and j denotes
C
olumn) will be equal to the
covariance of the ith, jth exogenous variables.
will be a
matrix.
X
square
1.0 no reciprocal causation 
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
1.0
X
=
MZ
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
1.0
Let no reciprocal causation 
denote the matrix of paths from the exogenous variables to the endogenous
W
variables. The i, jth element of the matrix
will be equal to the path
from
the jth
W
exogenous variable to the ith endogenous variable.
will be a
matrix,
rectangular
W
with number of rows equal to the number of endogenous variables, number of
columns equal to the number of exogenous variables.
e
a
c
d
0
0
0
0
=
W
0
0
0
0
e
a
c
d
For this simplified case, with no paths from endogenous variables to other endogenous
variables, and all endogenous variables directly observable, we can derive the expected
covariance matrix as a product of matrices,
SMZ = WXMZ W
e variables to other endogenous
a
c
d
0
a
c
d
=
WX
0
a
c
d
e
a
c
d
e
a
c
d
0
a
c
d
e
0
=
WXW
0
a
c
d
e
a
c
d
a
0
c
0
d
0
0
e
0
a
0
c
0
d
2
2
2
2
2
2
2
e
+
a
+
c
+
d
a
+ c
+
d
=
S MZ
2
2
2
2
2
2
2
a
+
c
+
d
e
+ a
+ c
+ d
Exercise 13. variables to other endogenous Using Figure 2, write down the elements of the covariance matrix of the exogenous variables for DZ pairs, XDZ.
SDZ = WXDZ W
Derive
Figure 7 represents a more complicated model, with paths from endogenous variables to other endogenous variables  in this example, representing the reciprocal environmental influences of the twins upon each other (or perhaps a rater contrast effect, if twin data are ratings by the same parent).
Figure 7. from endogenous variables to other endogenous variables  in this example, representing the reciprocal environmental influences of the twins upon each other (or perhaps a rater contrast effect, if twin data are ratings by the same parent).MODEL WITH RECIPROCAL SIBLING EFFECTS
0.25
1 (MZ) or
(DZ/FS)
1 (MZT/DZT/FST) or 0 (MZA/DZA/FSA)
0.5
1 (MZ) or
(DZ/FS)
D
E
A
C
E
A
C
D
d
e
a
c
e
a
c
d
0 (MZA/DZA/FSA) or i (MZT/DZT/FST)
P
P
TWIN 1
TWIN 2
0 (MZA/DZA/FSA) or i (MZT/DZT/FST)
i  reciprocal sibling interaction effect
It may be shown from endogenous variables to other endogenous variables  in this example, representing the reciprocal environmental influences of the twins upon each other (or perhaps a rater contrast effect, if twin data are ratings by the same parent).
, again assuming that all endogenous
(see Neale and Cardon, C6)
variables are directly observable, that
SMZ = ZWXMZ W Z
1
where Z = (I  Y)
SDZ = ZWXDZ W Z
and
Here
is the identity matrix, and
denotes the inverse.
1
I
Exercise 14. from endogenous variables to other endogenous variables  in this example, representing the reciprocal environmental influences of the twins upon each other (or perhaps a rater contrast effect, if twin data are ratings by the same parent).What are the expected covariance matrices for MZ and DZ twin pairs reared together, and for MZ and DZ twin pairs reared apart, under the model of Figure 7?
Exercise 15. from endogenous variables to other endogenous variables  in this example, representing the reciprocal environmental influences of the twins upon each other (or perhaps a rater contrast effect, if twin data are ratings by the same parent).Redraw Figure 6 for the case of reciprocal causation, i.e. a path i from ALCOHOL DEPENDENCE to CONDUCT PROBLEMS, in addition to the path i from CONDUCT PROBLEMS to ALCOHOL DEPENDENCE. What is the expected 4x4 covariance matrix for MZ pairs for this case?
THE END from endogenous variables to other endogenous variables  in this example, representing the reciprocal environmental influences of the twins upon each other (or perhaps a rater contrast effect, if twin data are ratings by the same parent).