David Evans

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David Evans
Longitudinal Models in Genetic Research
Queensland Institute of Medical
ResearchBrisbane, Australia
Twin Workshop Boulder 2002
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• These types of models are appropriate whenever
  one has repeated measures data
• short term trials of an experiment
• long term longitudinal studies
• When we have data from genetically informative
  individuals (e.g. MZ and DZ twins) it is possible
  to investigate the genetic and environmental
  influences affecting the trait over time.

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What sorts of questions?

• Are there changes in the magnitude of genetic and
  environmental effects over time?
• Do the same genetic and environmental influences
  operate throughout time?
• If there are no cohort effects then we can answer
  the first question using a cross-sectional study
  type design
• However, to answer the second question,
  longitudinal data is required

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Simplex Structure
From Fischbein (1977)
Weight1 Weight2 Weight3 Weight4
Weight5 Weight6 Weight1 1.000 Weight2
0.985 1.000 Weight3 0.968 0.981
1.000 Weight4 0.957 0.970
0.985 1.000 Weight5 0.932 0.940
0.964 0.975 1.000 Weight6
0.890 0.897 0.927 0.949
0.973 1.000
5
A1
Y2
Y1
Y3
Y4

• Factor models tend to fit this type of data
  poorly (Boomsma Molenaar, 1987)
• gt need a type of model which explicitly takes
  into account the longitudinal nature of the data

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Y - indicator variable ? - innovations ?
- latent variable ? - factor loadings e
- measurement error ß - transmission
coefficients
Phenotypic Simplex Model
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Measurement Model Yi ?i ?i ei Latent
Variable Model ?i ßi ?i-1 ?i
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ß2
ß3
ß4
?1
?2
?3
?4
?3
?4
?1
?2
Y2
Y1
Y3
Y4
e3
e2
e4
e1
? - Innovations are standardized to unit
variance ? - Factor loadings are estimated
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ß2
ß3
ß4
?1
?2
?3
?4
1
1
1
1
Y2
Y1
Y3
Y4
e3
e2
e4
e1
? -Variance of the innovations are estimated ? -
Factor loadings are constrained to unity
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?
ß2
ß3
ß4
?1
?2
?3
?4
1
1
1
1
Y2
Y1
Y3
Y4
e3
e2
e4
e1
CONSTRAINTS (1) var (e1) var (e4) (2) Need at
the VERY MINIMUM three measurement occasions
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Deriving the Expected Covariance Matrix
Path Analysis
Matrix Algebra
Covariance Algebra
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The Rules of Path Analysis
Adapted from Neale Cardon (1992)
(1) Trace backward along an arrow and then
forward, or simply forwards from one variable to
the other, but NEVER FORWARD AND THEN BACK (2)
The contribution of each chain traced between two
variables is the product of its path
coefficients (3) The expected covariance between
two variables is the sum of all legitimate routes
between the two variables (4) At any change in a
tracing route which is not a two way arrow
connecting different variables in the chain, the
expected variance of the variable at the point of
change is included in the product of path
coefficients
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The Rules of Path Analysis
Adapted from Neale Cardon (1992)
1
?1
?1
?2
Y2
Y1
cov (Y1, Y2) ?1 ?2
(2) The contribution of each chain traced between
two variables is the product of its path
coefficients
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The Rules of Path Analysis
Adapted from Neale Cardon (1992)
?2
?4
?1
?3
Y2
Y1
cov (Y1, Y2) ?1?2 ?3?4
(3) The expected covariance between two variables
is the sum of all legitimate routes between the
two variables
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The Rules of Path Analysis
Adapted from Neale Cardon (1992)
5
ß2
?2
?1
1
1
Y2
Y1
cov (Y1, Y2) 5ß2
(4) At any change in a tracing route which is not
a two way arrow connecting different variables in
the chain, the expected variance of the variable
at the point of change is included in the product
of path coefficients
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ß2
ß3
ß4
?1
?2
?3
?4
1
1
1
1
e3
Y2
e2
Y1
Y3
e1
e4
Y4
cov(y1, y2) ??? var(y1) ??? var(y2) ???
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ß2 var (?1)
cov(y1, y2)
(1) Trace backward along an arrow and then
forward, or simply forwards from one variable to
the other, but NEVER FORWARD AND THEN BACK
(4) At any change in a tracing route which is not
a two way arrow connecting different variables in
the chain, the expected variance of the variable
at the point of change is included in the product
of path coefficients
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cov(y1, y2) var(y1) var(y2)
ß2 var (?1)
var (?1) var (e1)
ß22 var (?1) var (?2) var (e2)
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Y1
Y2
Y3
Y4
Y1
var (?1) var (e1 ) ß2 var (?1) ß22 var (?1)
var (?2) var (e2 ) ß2 ß3 var (?1) ß3
var (?2) ß32(ß22 var (?1) var (?2))
var(?3) var (e3 ) ß2 ß3 ß4var (?1) ß3
ß4var (?2) ß4var (?3) ß42(ß32(ß22 var (?1)
var (?2)) var(?3)) var(?4) var (e4 )
Y2
Y3
Y4
Expected Phenotypic Covariance Matrix
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This can be expressed compactly in matrix algebra
form (I - B)-1 ? (I - B)-1 Te I is an
identity matrix B is the matrix of transmission
coefficients ? is the matrix of innovation
variances Te is the matrix of measurement
error variances
var(e1) 0 0 0 0
var(e2) 0 0 0 0
var(e3) 0 0 0
0 var(e4)
Te
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(1) Draw path model (2) Use path analysis to
derive the expected covariance matrix (3)
Decompose the expected covariance matrix into
simple matrices (4) Write out matrix
formulae (5) Implement in Mx
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Phenotypic Simplex Model MX Example Data taken
from Fischbein (1977) 66 Females had their
weight measured six times at 6 month intervals
from 11.5 years of age.
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Phenotypic Simplex Model Results
Time Latent Variable Variance Error. Total
ßn var(?n-1 ) var(?n )
Variance Variance 1
- - - 51.34 0.13 51.47 2 1.052
x 51.34 1.50 58.02 0.13 58.15 3 1
.032 x 58.02 2.07
63.52 0.13 63.66 4 1.062 x 63.52
1.86 72.69 0.13 72.82 5 0.972 x 72.69
3.27 71.50 0.13 71.64 6 0.942 x
71.50 3.27 66.72 0.13 66.86
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?a1
?a2
?a3
?a4
ßa2
ßa3
A1
A2
A3
?a1
?a2
?a3
e3
y2
e2
y1
y3
e1
?c3
?c2
?c1
ßc3
ßc2
C1
C2
C3
?c2
?c1
?e1
?c3
?e3
?e2
ße3
ße2
E1
E3
E2
?e2
?e1
?e3
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Measurement Model yi ?aiA i ?ciC i ?eiE i
ei Latent Variable Model Ai ßai Ai-1
?ai Ci ßci Ci-1 ?ci Ei ßei Ei-1
?ei
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?a1
?a2
?a3
?a4
ßa2
ßa3
A1
A2
A3
1
1
1
e3
y2
e2
y1
y3
e1
1
1
1
ßc3
ßc2
C1
C2
C3
?c2
?c1
1
?c3
1
1
ße3
ße2
E1
E3
E2
?e2
?e1
?e3
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Genetic Simplex Model MX Example
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Genetic Simplex Model Results
Time Genetic Variance Environmental
Variance Total var(?n ) ß var(?n-1
) var(?n ) ß var(?n-1 ) 1 4.792
22.98 1.822 3.30 26.28 2 1.122
1.052 x 22.98 26.72 0.562 0.922 x 3.30
3.09 29.81 3 1.502 1.042 x 26.72
31.40 0.982 1.052 x 3.09 4.39 35.79 4 1.232
1.022 x 31.40 34.07 0.952 0.852 x 4.39
4.08 38.15 5 1.392 1.022 x 34.07 37.57
0.812 0.852 x 4.08 3.55 41.12 6 1.392
0.972 x 37.57 37.40 1.002 1.012 x 3.55
4.62 42.02
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Useful References

• Boomsma D. I. Molenaar P. C. (1987). The
  genetic analysis of repeated measures. I. Simplex
  models. Behav Genet, 17(2), 111-23.
• Boomsma D. I., Martin, N. G. Molenaar P. C.
  (1989). Factor and simplex models for repeated
  measures application to two psychomotor measures
  of alcohol sensitivity in twins. Behav Genet,
  19(1), 79-96.