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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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Path analysis

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.

-

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.


Path analysis

ADDITIVE

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


Path analysis

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


Path analysis

CONVENTIONS OF PATH ANALYSIS

Squares or rectangles denote observed variables.

Circles or ellipses denote latent (unmeasured) variables.

Upper-case letters are used to denote variables.

Lower-case letters (or numeric values) are used to denote covariances or path

coefficients.

Single-headed 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


Path analysis

Double-headed arrows ( ) are used to represent a covariance between two

variables, which may arise through common causes not represented in the

model. Double-headed arrows ( ) may also be used to represent the variance

of an exogenous variable or (in some conventions) the residual variance in an

endogenous variable.

Double-headed arrows may

be used to link any variables with one or more

not

single-headed arrows pointing to them -- these variables are called endogenous

variables. Other variables are exogenous variables.

Single-headed arrows may be drawn from exogenous to endogenous variables

(A P) or from endogenous variables to other endogenous variables

(P

P

).

1

2


Path analysis

Omission covariance between two of a two-headed arrow between two exogenous variables implies the assumption that the covariance of those variables is zero (e.g., no genotype-environment 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.


Path analysis

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 genotype-environment

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


Tracing rules of path analysis

  • (Simplified rules assuming no ‘feedback loops’, i.e., no reciprocal causation -

  • reciprocal sibling environmental influences - etc.)

  • (1) Trace backwards, change direction at a 2-headed arrow, then trace

  • forwards.

    • implies that we can never trace through two 2-headed arrows in

    • the same chain.

  • (2) 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.

TRACING RULES OF PATH ANALYSIS


Path analysis

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


Path analysis

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


Path analysis exercises
PATH ANALYSIS: EXERCISES no reciprocal causation -

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?


Path analysis

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.


Path analysis

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?


Path analysis

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


Path analysis

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 adopted-away offspring?

Exercise 8. What 10 assumptions are implied by Figure 3?


Path analysis

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


Path analysis

Exercise 9. no reciprocal causation -Using Figure 4, what is the expected covariance between

biological uncle and nephew?


Path analysis

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)


Path analysis

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”)?


Path analysis

Figure 6. no reciprocal causation -DIRECTION-OF-CAUSATION 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


Path analysis

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?


Path analysis

(3) What are the corresponding expected DZ covariances when no reciprocal causation -

(a) CONDUCT PROBLEMS ALCOHOL DEPENDENCE

(b) ALCOHOL DEPENDENCE CONDUCT PROBLEMS


Path analysis

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, j-th element of

the matrix

X

(where i denotes

R

ow, and j denotes

C

olumn) will be equal to the

covariance of the i-th, j-th exogenous variables.

will be a

matrix.

X

square


Path analysis

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


Path analysis

Let no reciprocal causation -

denote the matrix of paths from the exogenous variables to the endogenous

W

variables. The i, j-th element of the matrix

will be equal to the path

from

the j-th

W

exogenous variable to the i-th 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


Path analysis

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


Path analysis

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


Path analysis

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


Path analysis

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).


Path analysis

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


Path analysis

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


Path analysis

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?


Path analysis

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?


Path analysis

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).