Twin 1

1. Can heritability be modeled as a function of DNA marker sharing at a particular chromosomal location using pihat?

2. rMZ = rDZ = 1 rMZ = 1, rDZ = 0.5 E E rMZ = 1, rDZ = 0, 0.5 or 1 C C e e A A c c a a Q Q q q Twin 1 Twin 2 4 group linkage analysis (3 IBD DZ groups and 1 MZ group)

3. Pihat • Merlin estimates the 3 IBD probabilities for each sib pair at each location • Remember: pibd2 + pibd1 + pibd0 = 1 • Pihat = pibd2*1 + pibd1*0.5 + pibd0*0 • Estimate of the probability of sharing alleles identical by descent • Proxy additive genetic correlation at the locus

4. rMZ = rDZ = 1 rMZ = 1, rDZ = 0.5 E E ^ rMZ = 1, rDZ =  C C e e A A c c a a Q Q q q Twin 1 Twin 2

5. Pihat Mx Script for 1 marker • !QTL analysis via Pihat method • !Using marker1 • !Using MZ and DZ twins • !Analysis of LDL • !Dutch Adults • #define nvar 1 !different for multivariate • #define nsib 2 !number of siblings • #NGroups 3 • G1: Parameter Estimates • G2: Monozygotic twins • G3: Dizygotic twins

6. Group 1: specify parameters • Begin Matrices; • X Lower nvar nvar Free !additive genetic parameter • Y Lower nvar nvar Free !shared environmental parameter • Z Lower nvar nvar Free !unique environmental parameter • L Full nvar 1 Free !QTL effect • Begin Algebra; • A= X*X'; !additive genetic variance • C= Y*Y'; !shared environmental variance • E= Z*Z'; !unique environmental variance • Q= L*L'; !variance due to QTL • V= A+C+Q+E; !total variance • T= A|C|Q|E; !parameters in one matrix for standardizing • S= T@V~; !standardized variance component estimates

7. Group 2 : MZ twins • See Group 1 • M Full 1 nvar Free !means • U Unit 1 nsib • #include lipidmz.dat • Select ldl1 ldl2; • Begin Matrices = Group 1; • Means U@M; • Covariance A+C+Q+E | A+C+Q _ • A+C+Q | A+C+Q+E; • End

8. Group 3: DZ twins • #include lipiddz.dat • Select ibd0m1 ibd1m1 ibd2m1 ldl1 ldl2; • Definition ibd0m1 ibd1m1 ibd2m1; • Begin Matrices = Group 1; • K Full 3 1 !IBD probabilities (from Merlin) • J Full 1 3 !coefficients 0,0.5,1 for pihat • End Matrices; • Specify K ibd0m1 ibd1m1 ibd2m1 • Matrix J 0. 0.5 1.0 • Begin Algebra; • P= J*K; !estimate of pihat • End Algebra;

9. Group 3 End + Multiple Fit • Means U@M; • Covariance A+C+Q+E | H@A+C+P@Q _ • H@A+C+P@Q | A+C+Q+E; • Start 1 All • Start 2.8 M 1 1 1 • Option NDecimals=3 • Option Multiple Issat • End • ! Test significance of QTL effect • Drop L 1 1 1 • End

10. Exercise • Fit ACEQ model to MZ/DZ data for one location at a time • Fit ACE model to (drop Q) • Is the QTL effect significant? • Plot the results

11. Basic script and data (LDL) • ACEQ model in MZ/DZ twins: lipidpihatmzdz1.mx • f:\hmaes\a16\martin\linkage • IBD probabilities every 2 cM > 54 locations • Rectangular files: DZ: lipiddz.rec MZ:lipidmz.rec Dat files: DZ: lipiddz.dat MZ: lipidmz.dat • FEQ model for DZ twins only: lipidpihat1.mx

12. Fit to marker number according to table position

13. Chi-square for all locations

14. Pihat Mx Script for N locations • #repeat 54 !total number of markers N • Begin Script • Group 3 • #include lipiddz.dat • #include seldef$repeat_number • Select ibd0m? Ibd1m? Ibd2m? ldl1 ldl2; • Definition ibd0m? ibd1m? Ibd2m?; • #include specify$repeat_number • Specify K ibd0m? ibd1m? ibd2m? • Rest Script • #end repeat

15. Extended script and data (LDL) • ACEQ model in MZ/DZ twins: lipidpihatmzdz54.mx • IBD probabilities every 2 cM > 54 locations • Rectangular files: DZ: lipiddz.rec MZ:lipidmz.rec Dat files: DZ: lipiddz.dat MZ: lipidmz.dat • Help files: seldef1 … seldef54 specify1 … specify54 • FEQ model for DZ twins only: lipidpihat54.mx

16. Get summary of results • Unix commands in Cygwin on PC • grep ‘ Difference Chi-squared’ lipidpihat54.mxo > chi54 • awk {‘print $4’} chi54 > chisq54 • pico chisq54