1. QTL Mapping Using Mx Michael C Neale Virginia Institute for Psychiatric and Behavioral Genetics Virginia Commonwealth University

2. Overview  Alternative approach  Linkage as Mixture  Univariate/Multivariate  One/more loci  Practical considerations  Power - Pihat vs covs - Larger Sibships

3. Schematic of Genome Marker 1Marker 2Marker 3Marker 4 QTL d1 d2 d3 d4

4. Genetic Heterogeneity Sib pairs IBD at a locus, parents AB and CD ACADBCBD AC2110 AD1201 BC1021 BD0112

5. Pi hat approach  1 Pick a putative QTL location  2 Compute p(IBD0) p(IBD1) p(IBD2) given  marker data [Mapmaker/sibs]  3 Compute = p(IBD2) +.5p(IBD1)  4 Fit model  Repeat 1-4 as necessary for different locations Elston & Stewart B ^

6. Major QTL effects DZ twins A1C1D1E1 P1 Q1Q2E2D2C2 P2 A2 B ^.51.25

7. Normal Theory Likelihood Function For raw data in Mx j=1 ln L i = f i ln [ 3 w j g(x i,: ij, G ij )] m x i - vector of observed scores on n subjects : ij - vector of predicted means G ij - matrix of predicted covariances - functions of parameters

8. General Likelihood Function ) Model for Means can differ ) Model for Covariances can differ ) Weights can differ ) Frequencies can differ Things that may differ over subjects i = 1....n subjects (families) j=1 ln L i = f i ln [ 3 w ij g(x i,: ij, G ij )] m

9. Normal distribution N(: ij, G ij ) Likelihood is height of the curve 01234-2-3-4 0 0.1 0.2 0.3 0.4 0.5 : G xixi N likelihood

10. Weighted mixture of models Finite mixture distribution j=1 m j = 1....m models w ij Weight for subject i model j e.g., Segregation analysis ln L i = f i ln [ 3 w ij g(x i,: ij, G ij )]

11. Mixture of Normal Distributions Two normals, propotions w1 & w2, different means But Likelihood Ratio not Chi-Squared - what is it? 0123-2-3-4 0 0.1 0.2 0.3 0.4 0.5 :1:1 xixi g :2:2 w 1 x l 1 w 2 x l 2

12. Weighted Likelihood Method  1 Pick a putative QTL location  2 Compute p(IBD0) p(IBD1) p(IBD2) given marker data  these are "WEIGHTS"  3 Compute likelihood of phenotype data under each of 3 IBD conditions  4 Maximize weighted likelihood of 3  Repeat 1-4 as necessary for different locations

13. Mixture method Add them up A1C1D1E1 P1 Q1Q2E2D2C2 P2 A2.51.25.5 A1C1D1E1 P1 Q1Q2E2D2C2 P2 A2.51.25 A1C1D1E1 P1 Q1Q2E2D2C2 P2 A2.51.25 10 p(IBD1) x p(IBD2) xp(IBD0) x

14. Dataset structure Rectangular format Id sex age P1 P2 IBD0 IBD1 IBD2 IBD0 IBD1 IBD2 Locus 1 Locus 2 1231 1 24 103.5 115.6.81.13.06.28.51.21 1781 0 29 127.4 145.6.23.65.11.08.57.35 1952 1 39 98.5..81.13.06.28.51.21 2056 1 19 93.5 100.3....20.40.40 Missing data: Phenotypes ML Markers Listwise

15. Mx Script Mixture method !QTL analysis via Mixture Distribution method !Using marker1 !Using DZ twins only !Analysis of LDL !Dutch Adults #define nvar 1 !different for multivariate #define nsib 2 !number of siblings #NGroups=2

16. Mx Script Mixture part 2 G1: Parameter Estimates Calculation Begin Matrices; X Lower nvar nvar Free !familial background Z Lower nvar nvar Free !unique environment L Full 1 1 Free !QTL effect M Full 1 nvar Free !means H Full 1 1 End Matrices; Matrix H.5 Begin Algebra; F= X*X'; !familial variance E= Z*Z'; !unique environmental variance Q= L*L'; !variance due to QTL V= F+Q+E; !total variance T= F|Q|E; !parameters in one matrix for standardizing S= T@V~; !standardized variance component estimates End Algebra; Labels Row S standest Labels Col S f^2 q^2 e^2 Labels Row T unstandest Labels Col T f^2 q^2 e^2 End

17. Mx Script G2: Dizygotic twins #include lipiddzmix.dat Select ibd0m1 ibd1m1 ibd2m1 ldl1 ldl2; Definition ibd0m1 ibd1m1 ibd2m1; Begin Matrices = Group 1; K Full 3 1 !IBD probabilities (from Merlin) U Unit 3 2 End Matrices; Specify K ibd0m1 ibd1m1 ibd2m1 Means U@M; Covariance F+Q+E | F _ F | F+Q+E _ ! IBD 0 Covariance matrix F+Q+E | F+ h@Q_ F+h@Q | F+Q+E _ ! IBD 1 Covariance matrix F+Q+E | F+Q _ F+Q | F+Q+E; ! IBD 2 Covariance matrix Weights K; ! IBD probabilities Start 1 All Start 2.8 M 1 1 1 Option NDecimals=3 Option Multiple Issat End

18. Mx Script Mixture part 4 ! Test significance of QTL effect Drop L 1 1 1 End

19. Output Pihat Method Summary of VL file data for group 1 Code -3.000 -2.000 -1.000 1.000 2.000 Number 190.000 190.000 190.000 190.000 190.000 Mean 0.234 0.510 0.256 4.927 4.928 Variance 0.104 0.096 0.096 1.092 1.325 MATRIX F This is a LOWER TRIANGULAR matrix of order 1 by 1 1 1 0.898 MATRIX Q This is a FULL matrix of order 1 by 1 1 1 0.540

20. Output Your model has 4 estimated parameters and 950 Observed statistics -2 times log-likelihood of data >>> 1057.064 Degrees of freedom >>>>>>>>>>>>>>>> 946 Your model has 3 estimated parameters and 950 Observed statistics -2 times log-likelihood of data >>> 1059.025 Degrees of freedom >>>>>>>>>>>>>>>> 947 QTL Effect Present QTL Effect Absent Difference chi-squared = 1.961 (1 df)

21. Output Pihat Method Your model has 4 estimated parameters and 950 Observed statistics -2 times log-likelihood of data >>> 1057.500 Degrees of freedom >>>>>>>>>>>>>>>> 946 Your model has 3 estimated parameters and 950 Observed statistics -2 times log-likelihood of data >>> 1059.025 Degrees of freedom >>>>>>>>>>>>>>>> 947 QTL Effect Present QTL Effect Absent Difference chi-squared = 1.525 (1 df)

22. Summary  SEM - QTL direct relationship  Mx graphical/script approaches  Mixture vs Pihat  Multivariate treatment  Multilocus  Missing Data  Ascertainment

23. How much more power?  Large sibships much more powerful  Dolan et al 1999  Pihat simple with large sibships - Solar, Genehunter etc · Pihat shows substantial bias with missing data

24. Expected IBD Frequencies TypeConfigurationFrequency 124/16 218/16 304/16 Sibships of size 2

25. Expected IBD Frequencies TypeConfigurationFrequency 12224/64 22118/64 32004/64 41218/64 51128/64 61108/64 71018/64 80204/64 90118/64 100024/64 Sibships of size 3

26. More power in large sibships Dolan, Neale & Boomsma (2000) +Size 2 o Size 3 * Size 4

27. Number of IBD Combinations As a function of number of sibs in family Sibship SizeNumber of combinations 23 310 436 5136 6528 72080 87196

28. Mixture Approach for Pedigrees  Iterate configurations within families  Only use non-zero IBD probabilities  Set threshold?  Improves with genotype data  Allows moderated genotypes Some ideas

29. Strategy 2  Families within combinations  Limited # of IBD configurations  Depends on max sibship size  Usually Faster - Can do missing data - Cannot do moderator variables

30. Multivariate QTL Vectors of variables, Matrices of paths Three component mixture B ^.51.25 Q1Q2A2C2D2E2E1D1C1A1 P1P2

31. Two locus model R1C1A1E1 P1 Q1Q2E2A2C2 P2 R2 1.25 B1B1 ^ B2B2 ^

32. Two locus model mixture p(ibd0 R) p(ibd1 R) p(ibd2 R) R1C1A1E1 P1 Q1Q2E2A2C2 P2 R2 1.25 R1C1A1E1 P1 Q1Q2E2A2C2 P2 R2 1.25 R1C1A1E1 P1 Q1Q2E2A2C2 P2 R2 1.25 1 R1C1A1E1 P1 Q1Q2E2A2C2 P2 R2 1.25 R1C1A1E1 P1 Q1Q2E2A2C2 P2 R2 1.25 R1C1A1E1 P1 Q1Q2E2A2C2 P2 R2 1.25 R1C1A1E1 P1 Q1Q2E2A2C2 P2 R2 1.25 0 1 R1C1A1E1 P1 Q1Q2E2A2C2 P2 R2 1.25 0.5 R1C1A1E1 P1 Q1Q2E2A2C2 P2 R2 1.25 0 0 p(ibd0 Q) p(ibd1 Q) p(ibd2 Q) 1.50 11 1 0

33. Multivariate multilocus multipoint )Eaves Neale & Maes 1996 )10 minutes for 5 phenotypes )Restart at previous solution )Only fit null model (q=0) once

34. Not dead yet )Latent variable qtls )Multiple rater )Comorbidity )Repeated measures