path analysis frühling rijsdijk sgdp centre institute of psychiatry king’s college london, uk
TRANSCRIPT
![Page 1: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/1.jpg)
Path Analysis
Frühling Rijsdijk
SGDP Centre Institute of Psychiatry
King’s College London, UK
![Page 2: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/2.jpg)
Twin Model Twin Data
HypothesisedSources ofVariation
Biometrical GeneticTheory
Predicted Var/Cov from Model
Structural Equation Modelling (Maximum Likelihood)
Path TracingRules
CovarianceAlgebra
SummaryStatistics
Matrix Algebra
ModelEquations
Path Diagrams
ObservedVariation
Data Preparation
Observed Var/Cov from Data
![Page 3: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/3.jpg)
Path Analysis• Path analysis was developed around 1918 by
Sewall Wright • Combines knowledge we have with regard to
causal relations with degree of observed correlations
• Guinea pigs: interrelationships of factors determining weight at birth and at weaning (33 days)
Wright, S. (1921). "Correlation and causation". J. Agricultural Research 20: 557–585
Birth weight Early gain Litter size Gestation period
Environmental conditionsHealth of damHeredity factors
![Page 4: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/4.jpg)
Path Diagram
![Page 5: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/5.jpg)
Path Analysis• Present linear relationships between variables by means of diagrams ;
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
![Page 6: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/6.jpg)
• In SEM 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 derivation of predicted variances and covariances of the
variables under the specified model
• Aims of this session: Derivation of predicted Var-Cov Matrices
using:
(1) Path Tracing & (2) Covariance Algebra
![Page 7: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/7.jpg)
Observed Variable
Latent Variable
Causal Path
Covariance Path
Path Diagram Conventions
![Page 8: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/8.jpg)
Twin 1
E C A1 1 1
Twin 2
A C E1 1 1
Model for an MZ PAIR
1
1
Note: a, c and e are the same cross twins
e ac a ec
![Page 9: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/9.jpg)
Twin 1
E C A1 1 1
Twin 2
A C E1 1 1
Model for a DZ PAIR
1
.5
Note: a, c and e are also the same cross groups
e ac a ec
![Page 10: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/10.jpg)
(1) 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 within the chain
• A legitimate chain is a path along arrows that
follow 3 rules:
![Page 11: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/11.jpg)
(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 standardized variables
CovBC : a*VA*b
NOT c*VD*d
VA
B C
D
A
e e
e
a b
c d
![Page 12: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/12.jpg)
(ii) Loops are not allowed, i.e. we can not
trace twice through the same variable
CovAB : a*VC*b
NOT
a* VC *c*e*d* VC
A B
C
D E
a b
c d
e
e
e e
VD VE
![Page 13: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/13.jpg)
(iii) 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)
C
D E
c d
e
e
VD VE CovCD : c*VD +
d*e NOT d*VE*e
![Page 14: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/14.jpg)
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
The Variance
![Page 15: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/15.jpg)
Twin 1
E
e c
C A1 1 1
Variance of Twin 1 AND Twin 2 (for MZ and DZ pairs)
a
![Page 16: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/16.jpg)
Twin 1
E
e c
C A1 1 1
Variance of Twin 1 AND Twin 2 (for MZ and DZ pairs)
a
![Page 17: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/17.jpg)
Twin 1
E
e c
C A1 1 1
Variance of Twin 1 AND Twin 2 (for MZ and DZ pairs)
a
![Page 18: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/18.jpg)
Twin 1
E
e c
C A1 1 1
Variance of Twin 1 AND Twin 2 (for MZ and DZ pairs)
a*1*a = a2
+a
![Page 19: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/19.jpg)
Twin 1
E
e c
C A1 1 1
Variance of Twin 1 AND Twin 2 (for MZ and DZ pairs)
a*1*a = a2
+c*1*c = c2
e*1*e = e2+
Total Variance = a2 + c2 + e2
a
![Page 20: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/20.jpg)
Twin 1
E C A1 1 1
Covariance Twin 1-2: MZ pairs
Twin 2
A C E1 1 1
1
1
e ac a ec
![Page 21: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/21.jpg)
Twin 1
E C A1 1 1
Covariance Twin 1-2: MZ pairs
Twin 2
A C E1 1 1
1
1
e ac a ec
![Page 22: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/22.jpg)
Twin 1
E C A1 1 1
Covariance Twin 1-2: MZ pairs
Total Covariance = a2 +
Twin 2
A C E1 1 1
1
1
e ac a ec
![Page 23: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/23.jpg)
Twin 1
E C A1 1 1
Covariance Twin 1-2: MZ pairs
Total Covariance = a2 + c2
Twin 2
A C E1 1 1
1
1
e ac a ec
![Page 24: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/24.jpg)
Twin 1
E C A1 1 1
Covariance Twin 1-2: DZ pairs
Twin 2
A C E1 1 1
.5
1
e ac a ec
![Page 25: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/25.jpg)
Twin 1
E C A1 1 1
Covariance Twin 1-2: MZ pairs
Twin 2
A C E1 1 1
.5
1
e ac a ec
![Page 26: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/26.jpg)
Twin 1
E C A1 1 1
Covariance Twin 1-2: DZ pairs
Total Covariance = .5a2 +
Twin 2
A C E1 1 1
.5
1
e ac a ec
![Page 27: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/27.jpg)
Twin 1
E C A1 1 1
Covariance Twin 1-2: DZ pairs
Total Covariance = .5a2 + c2
Twin 2
A C E1 1 1
.5
1
e ac a ec
![Page 28: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/28.jpg)
22222
22222
ecaca
caecaMZCov
Tw1 Tw2
Tw1
Tw2
22222
22222
2
12
1
ecaca
caecaDZCov
Tw1 Tw2
Tw1
Tw2
Predicted Var-Cov Matrices
![Page 29: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/29.jpg)
Twin 1
E D A1 1 1
Twin 2
A D E1 1 1
ADE Model
1(MZ) / 0.25 (DZ)
1/.5
e ad a ed
![Page 30: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/30.jpg)
22222
22222
edada
daedaMZCov
Tw1 Tw2
Tw1
Tw2
22222
22222
4
1
2
14
1
2
1
edada
daedaDZCov
Tw1 Tw2
Tw1
Tw2
Predicted Var-Cov Matrices
![Page 31: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/31.jpg)
ACE or ADE
Cov(mz) = a2 + c2 or a2 + d2
Cov(dz) = ½ a2 + c2 or ½ a2 + ¼ d2
VP = a2 + c2 + e2 or a2 + d2 + e2
3 unknown parameters (a, c, e or a, d, e), and only 3 distinctive predicted statistics:
Cov MZ, Cov DZ, Vp)
this model is just identified
![Page 32: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/32.jpg)
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 rmz = 0.60 rdz = 0.25
Effects of C and D are confounded
ADE
ACE
![Page 33: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/33.jpg)
Three Fundamental Covariance Algebra Rules
Cov (aX,bY) = ab Cov(X,Y)
Cov (X,Y+Z) = Cov (X,Y) + Cov (X,Z)
Var (X) = Cov(X,X)
(2) Covariance Algebra
![Page 34: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/34.jpg)
2
2
2
2
1
a
a
AVara
AACova
aAaACov
aAVarYVar
*
)(
),(
),(
)()(
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
Y
a
Y = aA
Example 1
A
1
![Page 35: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/35.jpg)
52
2
.*
,,
a
Cov(A,A)a
aA)Cov(aAZ)Cov(Y
Example 2
Y
a
Y = aA
A
Z
a
Z = aA
A11
.5
![Page 36: Path Analysis Frühling Rijsdijk SGDP Centre Institute of Psychiatry King’s College London, UK](https://reader035.vdocuments.net/reader035/viewer/2022062516/56649d755503460f94a56775/html5/thumbnails/36.jpg)
Summary• Path Tracing and Covariance Algebra have the
same aim: To work out the predicted variances and covariances of variables, given a specified model
• The Ultimate Goal:To fit predicted variances/covariances to observed variances/covariances of the data in
order to estimate the model parameters: regression coefficients, correlations