1. 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

2. 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.

3. 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

4. “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)

5. “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 Y1Y1 Y2Y2 Y3Y3 Y4Y4 A1A1

6. Phenotypic Simplex Model η1η1 η2η2 η3η3 β2β2 ζ1ζ1 β3β3 λ1λ1 λ2λ2 λ3λ3 Y1Y1 ε3ε3 Y2Y2 Y3Y3 ε2ε2 ε1ε1 ε4ε4 Y4Y4 η4η4 λ4λ4 β4β4 ζ2ζ2 ζ3ζ3 ζ4ζ4 Y - “indicator variable” ζ - “innovations” η - “latent variable” λ - “factor loadings” ε - “measurement error” β - “transmission coefficients”

7. η1η1 η2η2 η3η3 β2β2 ζ1ζ1 β3β3 λ1λ1 λ2λ2 λ3λ3 Y1Y1 ε3ε3 Y2Y2 Y3Y3 ε2ε2 ε1ε1 ε4ε4 Y4Y4 η4η4 λ4λ4 β4β4 ζ2ζ2 ζ3ζ3 ζ4ζ4 Measurement Model:Y i = λ i η i + ε i Latent Variable Model:η i = β i η i-1 + ζ i

8. η1η1 η2η2 η3η3 β2β2 ζ1ζ1 β3β3 λ1λ1 λ2λ2 λ3λ3 Y1Y1 ε3ε3 Y2Y2 Y3Y3 ε2ε2 ε1ε1 ε4ε4 Y4Y4 η4η4 λ4λ4 β4β4 ζ2ζ2 ζ3ζ3 ζ4ζ4 ζ - Innovations are standardized to unit variance λ - Factor loadings are estimated 1 1 1 1

9. η1η1 η2η2 η3η3 β2β2 ζ1ζ1 β3β3 1 1 1 Y1Y1 ε3ε3 Y2Y2 Y3Y3 ε2ε2 ε1ε1 ε4ε4 Y4Y4 η4η4 1 β4β4 ζ2ζ2 ζ3ζ3 ζ4ζ4 ζ -Variance of the innovations are estimated λ - Factor loadings are constrained to unity ? ? ? ?

10. η1η1 η2η2 η3η3 β2β2 ζ1ζ1 β3β3 1 1 1 Y1Y1 ε3ε3 Y2Y2 Y3Y3 ε2ε2 ε1ε1 ε4ε4 Y4Y4 η4η4 1 β4β4 ζ2ζ2 ζ3ζ3 ζ4ζ4 CONSTRAINTS (1) var (ε 1 ) = var (ε 4 ) (2) Need at the VERY MINIMUM three measurement occasions ? ? ? ?

11. Deriving the Expected Covariance Matrix Path Analysis Matrix Algebra Covariance Algebra

12. (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)

13. (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) η1η1 Y1Y1 Y2Y2 λ1λ1 λ2λ2 1 cov (Y 1, Y 2 ) = λ 1 λ 2

14. (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) Y1Y1 Y2Y2 λ2λ2 λ4λ4 η1η1 1 cov (Y 1, Y 2 ) = λ 1 λ 2 + λ 3 λ 4 η2η2 1 λ1λ1 λ3λ3

15. η1η1 η2η2 β2β2 ζ1ζ1 1 1 Y1Y1 Y2Y2 (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 (Y 1, Y 2 ) = β 2 var(ζ 1 )

16. η1η1 η2η2 η3η3 β2β2 ζ1ζ1 β3β3 1 1 1 Y1Y1 ε3ε3 Y2Y2 Y3Y3 ε2ε2 ε1ε1 ε4ε4 Y4Y4 η4η4 1 β4β4 ζ2ζ2 ζ3ζ3 ζ4ζ4 cov(y 1, y 2 ) = ??? var(y 1 ) = ??? var(y 2 ) = ???

17. η1η1 η2η2 η3η3 β2β2 ζ1ζ1 β3β3 1 1 1 Y1Y1 ε3ε3 Y2Y2 Y3Y3 ε2ε2 ε1ε1 ε4ε4 Y4Y4 η4η4 1 β4β4 ζ2ζ2 ζ3ζ3 ζ4ζ4 β 2 var (ζ 1 ) cov(y 1, y 2 ) = (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

18. η1η1 η2η2 η3η3 β2β2 ζ1ζ1 β3β3 1 1 1 Y1Y1 ε3ε3 Y2Y2 Y3Y3 ε2ε2 ε1ε1 ε4ε4 Y4Y4 η4η4 1 β4β4 ζ2ζ2 ζ3ζ3 ζ4ζ4 β 2 var (ζ 1 ) cov(y 1, y 2 ) = var(y 1 ) = var(y 2 ) = β 2 2 var (ζ 1 ) + var (ζ 2 ) + var (ε 2 ) var (ζ 1 ) + var (ε 1 )

19. β 2 var (ζ 1 ) β 2 2 var (ζ 1 ) + var (ζ 2 ) + var (ε 2 ) β 2 β 3 var (ζ 1 ) β 3 var (ζ 2 ) β 3 2 (β 2 2 var (ζ 1 ) + var (ζ 2 )) + var(ζ 3 ) + var (ε 3 ) β 2 β 3 β 4 var (ζ 1 ) β 3 β 4 var (ζ 2 ) β 4 var (ζ 3 )β 4 2 (β 3 2 (β 2 2 var (ζ 1 ) + var (ζ 2 )) +var(ζ 3 )) + var(ζ 4 ) + var (ε 4 ) Y1Y1 Y2Y2 Y3Y3 Y4Y4 Y4Y4 Y3Y3 Y2Y2 Y1Y1 Expected Phenotypic Covariance Matrix

20. This can be expressed compactly in matrix algebra form: (I - B) -1 * Ψ * (I - B) -1 ’ + Θ ε I is an identity matrix B is the matrix of transmission coefficients Ψ is the matrix of innovation variances Θ ε is the matrix of measurement error variances var(ζ 1 ) 0 0 0 0 var(ζ 2 ) 0 0 0 0 var(ζ 3 ) 0 0 0 0 var(ζ 4 ) Ψ = var(ε 1 ) 0 0 0 0 var(ε 2 ) 0 0 0 0 var(ε 3 ) 0 0 0 0 var(ε 4 ) Θε =Θε = 0 0 β 2 0 0 0 0 β 3 0 0 0 0 β 4 0 B =

21. (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

22. 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.

23. TimeLatent Variable VarianceError.Total β n var(η n-1 ) var(ζ n ) Variance Variance 1 - - -51.340.1351.47 21.05 2 x 51.34 + 1.50 =58.020.1358.15 31.03 2 x 58.02 + 2.07 =63.520.1363.66 41.06 2 x 63.52 + 1.86 =72.690.1372.82 50.97 2 x 72.69 + 3.27 =71.500.1371.64 60.94 2 x 71.50 + 3.27 =66.720.1366.86 Phenotypic Simplex Model: Results

24. A1A1 A2A2 A3A3 β a2 ζ a1 β a3 λ a1 λ a2 λ a3 C1C1 C2C2 C3C3 E1E1 E3E3 E2E2 y1y1 ε3ε3 y2y2 y3y3 ζ c2 ζ c3 ζ e1 ζ c1 ζ e3 ζ e2 ε2ε2 ε1ε1 β c2 β c3 β e2 β e3 λ c1 λ c3 λ c2 λ e2 λ e3 λ e1 A4A4 β a4 λ a4 C4C4 E4E4 ε4ε4 y4y4 ζ c4 ζ e4 β c4 β e4 λ c4 λ e4 ζ a2 ζ a3 ζ a4

25. Measurement Model:y i = λ ai A i + λ ci C i + λ ei E i + ε i Latent Variable Model:A i = β ai A i-1 + ζ ai C i = β ci C i-1 + ζ ci E i = β ei E i-1 + ζ ei

26. A1A1 A2A2 A3A3 β a2 ζ a1 β a3 1 1 1 C1C1 C2C2 C3C3 E1E1 E3E3 E2E2 y1y1 ε3ε3 y2y2 y3y3 ζ c2 ζ c3 ζ e1 ζ c1 ζ e3 ζ e2 ε2ε2 ε1ε1 β c2 β c3 β e2 β e3 1 1 1 1 1 1 A4A4 β a4 1 C4C4 E4E4 ε4ε4 y4y4 ζ c4 ζ e4 β c4 β e4 1 1 ζ a2 ζ a3 ζ a4

27. Genetic Simplex Model: MX Example

28. 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?

29. Time Genetic Variance Environmental VarianceTotal var(ζ n ) β var(ζ n-1 ) var(ζ n ) β var(ζ n-1 ) 14.79 2 =22.981.82 2 = 3.3026.28 21.12 2 + 1.05 2 x 22.98 =26.720.56 2 + 0.92 2 x 3.30 = 3.0929.81 31.50 2 + 1.04 2 x 26.72 = 31.400.98 2 + 1.05 2 x 3.09 = 4.3935.79 41.23 2 + 1.02 2 x 31.40 = 34.070.95 2 + 0.85 2 x 4.39 = 4.0838.15 51.39 2 + 1.02 2 x 34.07 = 37.57 0.81 2 + 0.85 2 x 4.08 = 3.5541.12 6? + ? x 37.57 = ?? + ? x 3.55 = ? ? Genetic Simplex Model: Results

30. 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.

31. Time Genetic Variance Environmental VarianceTotal var(ζ n ) β var(ζ n-1 ) var(ζ n ) β var(ζ n-1 ) 14.79 2 =22.981.82 2 = 3.3026.28 21.12 2 + 1.05 2 x 22.98 =26.720.56 2 + 0.92 2 x 3.30 = 3.0929.81 31.50 2 + 1.04 2 x 26.72 = 31.400.98 2 + 1.05 2 x 3.09 = 4.3935.79 41.23 2 + 1.02 2 x 31.40 = 34.070.95 2 + 0.85 2 x 4.39 = 4.0838.15 51.39 2 + 1.02 2 x 34.07 = 37.57 0.81 2 + 0.85 2 x 4.08 = 3.5541.12 61.39 2 + 0.97 2 x 37.57 = 37.401.00 2 + 1.01 2 x 3.55 = 4.6242.02 Genetic Simplex Model: Results