David M. Evans

Sarah E. Medland

Developmental Models in Genetic Research

Wellcome Trust Centre for Human Genetics Oxford

United Kingdom

Twin Workshop Boulder 2004

Queensland Institute of Medical Research Brisbane Australia

• These type 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.

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

“Simplex” Structure

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

From Fischbein (1977)

• “Factor” models tend to fit this type of data poorly (Boomsma & Molenaar, 1987)

• => need a type of model which explicitly takes into account the longitudinal nature of the data

Y1Y2 Y3 Y4

A1

Phenotypic Simplex Model

η1 η2 η3

β2

ζ1

β3

λ1 λ2λ3

Y1 ε3Y2 Y3ε2ε1 ε4 Y4

η4

λ4

β4

ζ2 ζ3 ζ4

Y - “indicator variable” ζ - “innovations”

η - “latent variable” λ - “factor loadings”

ε - “measurement error” β - “transmission coefficients”

η1 η2 η3

β2

ζ1

β3

λ1 λ2λ3

Y1 ε3Y2 Y3ε2ε1 ε4 Y4

η4

λ4

β4

ζ2 ζ3 ζ4

Measurement Model: Yi = λi ηi + εi

Latent Variable Model: ηi = βi ηi-1 + ζi

η1 η2 η3

β2

ζ1

β3

λ1 λ2λ3

Y1

ε3

Y2 Y3

ε2ε1 ε4

Y4

η4

λ4

β4

ζ2 ζ3 ζ4

ζ - Innovations are standardized to unit variance

λ - Factor loadings are estimated

1 11 1

η1 η2 η3

β2

ζ1

β3

1 1 1

Y1

ε3

Y2 Y3

ε2ε1 ε4

Y4

η4

1

β4

ζ2 ζ3 ζ4

ζ -Variance of the innovations are estimated

λ - Factor loadings are constrained to unity

? ?? ?

η1 η2 η3

β2

ζ1

β3

1 1 1

Y1

ε3

Y2 Y3

ε2ε1 ε4

Y4

η4

1

β4

ζ2 ζ3 ζ4

CONSTRAINTS

(1) var (ε1) = var (ε4)

(2) Need at the VERY MINIMUM three measurement occasions

? ?? ?

Deriving the Expected Covariance Matrix

Path Analysis

Matrix Algebra

Covariance Algebra

(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

The Rules of Path Analysis

Adapted from Neale & Cardon (1992)

(2) The contribution of each chain traced between two variables is the product of its path coefficients

The Rules of Path Analysis

Adapted from Neale & Cardon (1992)

η1

Y1Y2

λ1 λ2

1

cov (Y1, Y2) = λ1 λ2

(3) The expected covariance between two variables is the sum of all legitimate routes between the two variables

The Rules of Path Analysis

Adapted from Neale & Cardon (1992)

Y1Y2

λ2 λ4

η1

1

cov (Y1, Y2) = λ1λ2 + λ3λ4

η2

1

λ1 λ3

η1 η2

β2

ζ1

1 1

Y1Y2

(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

The Rules of Path Analysis

Adapted from Neale & Cardon (1992)

cov (Y1, Y2) = β2var(ζ1)

η1 η2 η3

β2

ζ1

β3

1 1 1

Y1 ε3Y2 Y3ε2ε1 ε4 Y4

η4

1

β4

ζ2 ζ3 ζ4

cov(y1, y2) = ???

var(y1) = ???

var(y2) = ???

η1 η2 η3

β2

ζ1

β3

1 1 1

Y1 ε3Y2 Y3ε2ε1 ε4 Y4

η4

1

β4

ζ2 ζ3 ζ4

β2 var (ζ1)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

η1 η2 η3

β2

ζ1

β3

1 1 1

Y1 ε3Y2 Y3ε2ε1 ε4 Y4

η4

1

β4

ζ2 ζ3 ζ4

β2 var (ζ1)cov(y1, y2) =

var(y1) =

var(y2) =β2

2 var (ζ1) + var (ζ2) + var (ε2)

var (ζ1) + var (ε1)

var (ζ1) + var (ε1 )

β2 var (ζ1) β22 var (ζ1) + var (ζ2)

+ var (ε2 )

β2 β3 var (ζ1) β3 var (ζ2) β32(β2

2 var (ζ1) + var (ζ2))

+ var(ζ3) + var (ε3 )

β2 β3 β4var (ζ1) β3 β4var (ζ2) β4var (ζ3) β42(β3

2(β22 var (ζ1) + var (ζ2))

+var(ζ3)) + var(ζ4) + var (ε4 )

Y1 Y2 Y3 Y4

Y4

Y3

Y2

Y1

Expected Phenotypic Covariance Matrix

This can be expressed compactly in matrix algebra form:

(I - B)-1 * Ψ * (I - B)-1 ’ + Θε

I is an identity matrix

B is the matrix of transmission coefficients

Ψ is the matrix of innovation variances

Θε is the matrix of measurement error variances

var(ζ1) 0 0 0

0 var(ζ2) 0 0

0 0 var(ζ3) 0

0 0 0 var(ζ4)

Ψ =

var(ε1) 0 0 0

0 var(ε2) 0 0

0 0 var(ε3) 0

0 0 0 var(ε4)

Θε =

0 0 0 0

β2 0 0 0

0 β3 0 0

0 0 β4 0

B =

(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

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.

Time Latent Variable Variance Error. Total

βn var(ηn-1 ) var(ζn ) 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

Phenotypic Simplex Model: Results

A1 A2 A3

βa2

ζa1

βa3

λa1 λa2 λa3

C1 C2 C3

E1 E3E2

y1 ε3y2 y3

ζc2 ζc3

ζe1

ζc1

ζe3ζe2

ε2ε1

βc2 βc3

βe2βe3

λc1λc3λc2

λe2λe3

λe1

A4

βa4

λa4

C4

E4

ε4 y4

ζc4

ζe4

βc4

βe4

λc4

λe4

ζa2 ζa3 ζa4

Measurement Model: yi = λaiA i + λciC i+ λeiE i + εi

Latent Variable Model: Ai = βai Ai-1 + ζai

Ci = βci Ci-1 + ζci

Ei = βei Ei-1 + ζei

A1 A2 A3

βa2

ζa1

βa3

1 1 1

C1 C2 C3

E1 E3E2

y1 ε3y2 y3

ζc2 ζc3

ζe1

ζc1

ζe3ζe2

ε2ε1

βc2 βc3

βe2βe3

1 11

1 11

A4

βa4

1

C4

E4

ε4 y4

ζc4

ζe4

βc4

βe4

1

1

ζa2 ζa3 ζa4

Genetic Simplex Model: MX Example

• Equate measurement error across all time points

• Drop the measurement error structure from the model– Where will the measurement error go?

• Can you drop the common environmental structure from the model?

Time Genetic Variance Environmental Variance Total

var(ζn ) β var(ζn-1 ) var(ζn ) β var(ζn-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 ? + ? x 37.57 = ? ? + ? x 3.55 = ? ?

Genetic Simplex Model: Results

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.

Time Genetic Variance Environmental Variance Total

var(ζn ) β var(ζn-1 ) var(ζn ) β var(ζn-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

Genetic Simplex Model: Results