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Developmental Models in Genetic Research

This study explores the use of developmental models in genetic research, specifically in analyzing repeated measures data and investigating genetic and environmental influences on traits over time. The study utilizes data from genetically informative individuals, such as twins, and examines changes in the magnitude of genetic and environmental effects, as well as the consistency of these effects over time.

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Developmental Models in Genetic Research

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  1. Developmental Models in Genetic Research David M. EvansSarah E. Medland Wellcome Trust Centre for Human Genetics Oxford United Kingdom Queensland Institute of Medical Research Brisbane Australia Twin Workshop Boulder 2004

  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 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 Y3 Y4 Y1 • “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

  6. ζ3 ζ1 ζ2 ζ4 β2 β3 β4 η2 η1 η3 η4 λ3 λ4 λ2 λ1 Y2 ε2 ε3 ε4 Y3 ε1 Y4 Y1 Y - “indicator variable” ζ - “innovations” η - “latent variable” λ - “factor loadings” ε - “measurement error” β - “transmission coefficients” Phenotypic Simplex Model

  7. ζ1 ζ3 ζ2 ζ4 β2 β3 β4 η2 η1 η3 η4 λ3 λ4 λ2 λ1 Y2 ε2 ε3 ε4 Y3 ε1 Y4 Y1 Measurement Model: Yi = λi ηi + εi Latent Variable Model: ηi = βi ηi-1 + ζi

  8. 1 1 1 1 ζ1 ζ4 ζ3 ζ2 β2 β3 β4 η2 η1 η3 η4 λ3 λ4 λ2 λ1 Y2 Y3 Y4 Y1 ε2 ε3 ε4 ε1 ζ - Innovations are standardized to unit variance λ - Factor loadings are estimated

  9. ? ? ? ? ζ1 ζ4 ζ3 ζ2 β2 β3 β4 η2 η1 η3 η4 1 1 1 1 Y2 Y3 Y4 Y1 ε2 ε3 ε4 ε1 ζ -Variance of the innovations are estimated λ - Factor loadings are constrained to unity

  10. ? ? ? ζ1 ζ4 ζ3 ζ2 ? β2 β3 β4 η2 η1 η3 η4 1 1 1 1 Y2 Y3 Y4 Y1 ε2 ε3 ε4 ε1 CONSTRAINTS (1) var (ε1) = var (ε4) (2) Need at the VERY MINIMUM three measurement occasions

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

  12. 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

  13. The Rules of Path Analysis Adapted from Neale & Cardon (1992) 1 η1 λ1 λ2 Y2 Y1 cov (Y1, Y2) = λ1 λ2 (2) The contribution of each chain traced between two variables is the product of its path coefficients

  14. The Rules of Path Analysis 1 1 η1 η2 Adapted from Neale & Cardon (1992) λ1 λ2 λ3 λ4 Y2 Y1 cov (Y1, Y2) = λ1λ2 + λ3λ4 (3) The expected covariance between two variables is the sum of all legitimate routes between the two variables

  15. The Rules of Path Analysis Adapted from Neale & Cardon (1992) ζ1 β2 η2 η1 1 1 Y2 Y1 cov (Y1, Y2) = β2var(ζ1) (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

  16. ζ1 ζ3 ζ2 ζ4 β2 β3 β4 η2 η1 η3 η4 1 1 1 1 Y2 ε2 ε3 ε4 Y3 ε1 Y4 Y1 cov(y1, y2) = ??? var(y1) = ??? var(y2) = ???

  17. ζ1 ζ4 ζ3 ζ2 β2 β3 β4 η2 η1 η3 η4 1 1 1 1 Y2 ε2 ε3 ε4 Y3 ε1 Y4 Y1 cov(y1, y2) = β2 var (ζ1) (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. ζ1 ζ4 ζ3 ζ2 β2 β3 β4 η2 η1 η3 η4 1 1 1 1 Y2 ε2 ε3 ε4 Y3 ε1 Y4 Y1 cov(y1, y2) = var(y1) = var(y2) = β2 var (ζ1) var (ζ1) + var (ε1) β22 var (ζ1) + var (ζ2) + var (ε2)

  19. Y1 Y2 Y3 Y4 Y1 var (ζ1) + var (ε1 ) β2 var (ζ1) β22 var (ζ1) + var (ζ2) + var (ε2 ) β2 β3 var (ζ1) β3 var (ζ2) β32(β22 var (ζ1) + var (ζ2)) + var(ζ3) + var (ε3 ) β2 β3 β4var (ζ1) β3 β4var (ζ2) β4var (ζ3) β42(β32(β22 var (ζ1) + var (ζ2)) +var(ζ3)) + var(ζ4) + var (ε4 ) Y2 Y3 Y4 Expected Phenotypic Covariance Matrix

  20. 0 0 0 0 β2 0 0 0 0 β3 0 0 0 0 β4 0 B = var(ζ1) 0 0 0 0 var(ζ2) 0 0 0 0 var(ζ3) 0 0 0 0 var(ζ4) Ψ = 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(ε1) 0 0 0 0 var(ε2) 0 0 0 0 var(ε3) 0 0 0 0 var(ε4) Θε=

  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. Phenotypic Simplex Model: Results 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

  24. βa4 A4 λa4 ε4 y4 λc4 βc4 C4 λe4 ζc4 βe4 E4 ζe4 ζa1 ζa2 ζa3 ζa4 βa2 βa3 A2 A1 A3 λa1 λa2 λa3 y2 ε2 ε3 y3 ε1 y1 λc3 λc2 λc1 βc3 βc2 C1 C3 C2 ζc1 λe1 ζc2 λe3 ζc3 λe2 βe3 βe2 E1 E2 E3 ζe2 ζe1 ζe3

  25. 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

  26. βa4 A4 1 ε4 y4 1 βc4 C4 1 ζc4 βe4 E4 ζe4 ζa1 ζa2 ζa3 ζa4 βa2 βa3 A2 A1 A3 1 1 1 y2 ε2 ε3 y3 ε1 y1 1 1 1 βc3 βc2 C1 C3 C2 ζc1 1 ζc2 1 ζc3 1 βe3 βe2 E1 E2 E3 ζe2 ζe1 ζe3

  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. Genetic Simplex Model: Results 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 = ? ?

  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. Genetic Simplex Model: Results 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

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