ACDE model and estimability

1
ACDE model and estimability

• Why cant we estimate (co)variances due to A, C,
  D and E simultaneously in a standard twin design?

2
Covariances MZ

• cov(yi1,yi2MZ) cov(MZ)
• sA2 sD2 sC2

3
Covariance DZ

• cov(yi1,yi2DZ) cov(DZ)
• ½sA2 ¼sD2 sC2

4
Functions of covariances

• 2cov(DZ) cov(MZ) sC2 - ½sD2
• 2(cov(MZ) cov(DZ)) sA2 3/2sD2

5
Linear model

• yij m bi wij
• sy2 sb2 sw2
• y, b and w are random variables
• t sb2/sy2
• intra-class correlation fraction of total
  variance that is attributable to differences
  among pairs

6
Data Sufficient statistics( Sums of Squares
/ Mean Squares)

• MZ
• variation between pairs ( covariance)
• variation within pairs ( residual)
• DZ
• variation between pairs (covariance)
• variation within pairs (residual)
• 4 summary statistics, so why cant we estimate
  all four underlying components?

7
Causal components

• Between pairs Within pairs
• MZ sA2 sD2 sC2 sE2
• DZ ½sA2 ¼sD2 sC2 ½sA2 3/4sD2 sE2
• Difference ½sA2 3/4sD2 ½sA2 3/4sD2
• Different combinations of values of sA2 and sD2
  will give the same observed difference in between
  and within MZ and DZ (co)variance confounding
  (dependency), can only estimate 3 components

8
In terms of (co)variances

• Observed Expected
• MZ var sA2 sD2 sC2 sE2
• MZ cov sA2 sD2 sC2
• DZ var sA2 sD2 sC2 sE2
• DZ cov ½sA2 ¼sD2 sC2
• MZ DZ variance have the same expectation. Left
  with two equations and three unknowns

9
Assumption sD2 0 the ACE model

• Between pairs Within pairs
• MZ sA2 sC2 sE2
• DZ ½sA2 sC2 ½sA2 sE2
• 4 Mean Squares, 3 unknowns
• Maximum likelihood estimation (e.g., Mx)