1. Social interaction March 7 th, 2002 Boulder, Colorado John Hewitt

2. ACE P1P1 ace ACE P2P2 ace 1.01.0 or 0.5 P 1 = aA 1 + cC 1 + eE 1 P 2 = aA 2 + cC 2 + eE 2

3. In matrix form we can write: A 1 P 1 a c e 0 0 0 C 1 =E 1 P 2 0 0 0 a c e A 2 C 2 E 2

4. or as a matrix expression y = Gx

5. ACE P1P1 ace ACE P2P2 ace 1.01.0 or 0.5 P 1 = sP 2 +aA 1 +cC 1 +eE 1 P 2 =sP 1 +aA 2 +cC 2 +eE 2 s s

6. In matrix form we can write: A 1 P 1 0 s P 1 a c e 0 0 0 C 1 = + E 1 P 2 s 0 P 2 0 0 0 a c e A 2 C 2 E 2

7. or as a matrix expression y = By + Gx y-By = Gx (I-B)y = Gx (I-B )-1 (I-B)y = (I-B )-1 Gx Iy = (I-B )-1 Gx y = (I-B )-1 Gx

8. X1X1 P1P1 X2X2 P2P2 x P 1 = sP 2 +xX 1 P 2 =sP 1 +xX 2 s s x

9. In matrix form we can write: P 1 0 s P 1 x 0 X 1 = + P 2 s 0 P 2 0 x X 2

10. or as a matrix expression y = By + Gx y-By = Gx (I-B)y = Gx (I-B )-1 (I-B)y = (I-B )-1 Gx Iy = (I-B )-1 Gx y = (I-B )-1 Gx

11. In this case the matrix (I – B) is 1 -s -s 1 Which has determinant 1-s 2. So (I-B )-1 is 1 1 s 1-s 2 s 1

12. So, {yy’} = {(I - B) -1 Gx} {(I - B) -1 Gx}’ = (I - B) -1 G{xx’}G’(I - B) -1 ’

13. The effects of sibling interaction on variance and covariance components between pairs of relatives. SourceVarianceCovariance Additive geneticw(1+2sr a +s 2 )a 2 w(r a +2s+r a s 2 )a 2 Dominancew(1+2sr d +s 2 )d 2 w(r d +2s+r d s 2 )d 2 Shared envw(1+2s+s 2 )c 2 Non-shared envw(1+s 2 )e 2 w2se 2 w represents the scalar 1/(1-s 2 ) 2

14. Effects of strong sibling interaction on the variance and covariance between MZ and DZ and unrelated individuals reared together. The interaction s takes the values 0, 0.5, and –0.5 for no interaction, co-operation, and competition, respectively MZ twinsDZ twinsUnrelated Inter- action varcovrvarcovrvarcovr None 1.000.50 1.000.25 1.000.00 Coop- eration 3.112.890.932.672.330.882.221.780.80 Com- petition 1..330.440.331.78-.67-.382.22-1.78-.80

15. Now let’s look at the Mx script for fitting this model to data. The basic program is in your handout and in F:\jkh\siblings\sibint.mx

16. Model fitting to externalizing scores without social interaction Fit statistics Parameter estimates MODEL df Chi- square AICace AE432.5724.6.78--.33 CE429.8021.8--.78.43 ACE34.95.50.64.34

17. Model fitting to externalizing scores with social interaction Fit statisticsParameter estimates MODEL df Chi- square AICaces E,s429.821.8-- ** AE,s31.8-4.2.611--.419.230 CE,s329.821.8--.882.282-.101 ACE,s 21.8-2.2.611.000.419.230