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Understanding Extended Twin Family Designs Using Path Tracing

Deriving variances and covariances in Extended Twin Family Designs (ETFDs) can be complex, but path tracing offers a systematic approach. This lecture explains how to use path diagrams to model assortative mating, vertical transmission, and passive gene-environment covariance, providing insights into expectations of variances and covariances. By following specific rules for tracing paths, researchers can navigate the intricate relationships in ETFDs and address otherwise challenging mathematical problems arising from assortative mating and vertical transmission.

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Understanding Extended Twin Family Designs Using Path Tracing

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  1. Extended Twin Family Designs: Path Tracing Matthew Keller International Statistical Genetics Workshop 2022

  2. The point of this lecture Extending Twin Family Designs (ETFDs) can model assortative mating, vertical transmission, and passive G-E covariance, along with other parameters This is extremely difficult to do without the magic of path tracing This lecture shows how to derive expectations of variances and covariances in ETFDs using path tracing rules   

  3. Path Diagrams Path diagrams pictorially represent causal models. They aid in deriving the variances and covariances implied by the model. • Observed Variables Latent Variables Causal Paths (Co)variance Paths

  4. Deriving variances & covariances 1. Identify all legitimate chains (a series of paths) that connect one variable to another (covariances) or connect a variable back to itself (variances) 2. The expected value of a chain is the product of all coefficients associated with each path making up that chain 3. The final expected variance or covariance equals the sum of the values of all legitimate chains

  5. Path Tracing Rules. Legitimate chains: All chains begin by travelling backwards against the direction of a (single or double-headed) arrow, head to tail. 1.

  6. Path Tracing Rules. Legitimate chains: All chains begin by travelling backwards against the direction of a (single or double-headed) arrow, head to tail. Once a double headed arrow has been traversed, the direction reverses such that the chain travels forward 1. 2.

  7. Path Tracing Rules. Legitimate chains: All chains begin by travelling backwards against the direction of a (single or double-headed) arrow, head to tail. Once a double headed arrow has been traversed, the direction reverses such that the chain travels forward All chains must include exactly one double-headed arrow. This implies a chain must change directions exactly once. 1. 2. 3.

  8. Path Tracing Rules. Legitimate chains: All chains begin by travelling backwards against the direction of a (single or double-headed) arrow, head to tail. Once a double headed arrow has been traversed, the direction reverses such that the chain travels forward All chains must include exactly one double-headed arrow. This implies a chain must change directions exactly once. All chains must be counted exactly once and each must be unique. However, order matters: abc is a distinct chain from cba. 1. 2. 3. 4.

  9. Path Tracing Rules. Legitimate chains: All chains begin by travelling backwards against the direction of a (single or double-headed) arrow, head to tail. Once a double headed arrow has been traversed, the direction reverses such that the chain travels forward All chains must include exactly one double-headed arrow. This implies a chain must change directions exactly once. All chains must be counted exactly once and each must be unique. However, order matters: abc is a distinct chain from cba. Copaths are special. They may only be traversed once per chain, but once crossed, rule 3 resets. A chain therefore must contain a double-headed arrow before traversing a copath and one after traversing a copath. 1. 2. 3. 4. 5.

  10. Path Tracing Example

  11. Special considerations in ETFDs Typically, we assume that AM and VT have reached equilibrium - i.e., the levels of AM and VT have been stable across multiple generations This allows us to set the variances of A & F and the covariances between A & F in the parental generation to be equal to those in the offspring generation This results in non-linear constraints: parameters are functions of those same parameters Solving these requires iterative procedures, e.g., ML. Closed-form solutions of these estimates are typically not possible.    

  12. Nuclear Twin Family Design (NTFD) w w q q A A F F x x S S E E f f a a s s e e D D d d PFa PMa µ m m m m zs A A F F zd S S E E f a a f s s e e D D d d PT2 PT1

  13. Nuclear Twin Family Design (NTFD) w w q q A A F F x x S S E E f f a a s s e e D D d d PFa PMa µ m m m m zs A A F F zd S S E E f a a f s s e e D D d d PT2 PT1

  14. Nuclear Twin Family Design (NTFD) w w q q A A F F x x S S E E f f a a s s e e D D d d PFa PMa µ m m m m zs A A F F zd S S E E f a a f s s e e D D d d PT2 PT1

  15. Conclusions A few simple rules from path tracing makes it possible to derive expected variances and covariances when there is VT and AM. This would lead to otherwise intractable math arising from recursive relationships caused by AM and VT  

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