1. June 1999NCSU QTL Workshop © Brian S. Yandell 1 Bayesian Inference for QTLs in Inbred Lines II Brian S Yandell University of Wisconsin-Madison www.stat.wisc.edu/~yandell NCSU Statistical Genetics June 1999

2. NCSU QTL Workshop © Brian S. Yandell 2 Many Thanks Michael Newton Daniel Sorensen Daniel Gianola Jaya Satagopan Patrick Gaffney Fei Zou Tom Osborn David Butruille Marcio Ferrera Josh Udahl Pablo Quijada USDA Hatch Grants

3. June 1999NCSU QTL Workshop © Brian S. Yandell 3 Overview II quick review of trait model –single & multiple QTL –details of Gibbs sampler full conditionals –vector notation reversible jump MCMC –multiple regression –number of QTLs deconstructing Bayesian LODs

4. June 1999NCSU QTL Workshop © Brian S. Yandell 4 Quick Review of trait Model single QTL details of Gibbs sampler –normal priors & likelihoods mean, additive effects –inverse gamma prior for variance or inverse chi-square –vague priors lead to usual estimates as posterior means multiple QTL trait model –model with vector notation

5. June 1999NCSU QTL Workshop © Brian S. Yandell 5 Single QTL trait Model trait = mean + additive + error trait = effect_of_geno + error prob( trait | geno, effects )

6. June 1999NCSU QTL Workshop © Brian S. Yandell 6 Gibbs Sampler updates variance mean traits additive genos

7. June 1999NCSU QTL Workshop © Brian S. Yandell 7 Full Conditional for mean normal prior with large variance leads to normal posterior posterior mean posterior variance

8. June 1999NCSU QTL Workshop © Brian S. Yandell 8 Full Conditional for additive Effect normal prior with large variance leads to normal posterior posterior mean posterior variance

9. June 1999NCSU QTL Workshop © Brian S. Yandell 9 Full Conditional for variance inverse gamma prior with large v/a posterior distribution posterior mean

10. June 1999NCSU QTL Workshop © Brian S. Yandell 10 MCMC run for variance

11. June 1999NCSU QTL Workshop © Brian S. Yandell 11 Alternative for Variance: use Inverse Chi-square inverse chi-square prior with large d,v posterior distribution

12. June 1999NCSU QTL Workshop © Brian S. Yandell 12 Markov chain updates effects locus traits genos

13. June 1999NCSU QTL Workshop © Brian S. Yandell 13 Prior for locus no prior information on locus –uniform prior over genome –use framework map choose interval proportional to length then pick uniform position within interval prior information from other studies concentrate on credible regions use posterior of previous study as new prior

14. June 1999NCSU QTL Workshop © Brian S. Yandell 14 Metropolis-Hastings Step pick new locus based upon current locus –propose new locus from distribution q( ) pick value near current one? pick uniformly across genome? –accept new locus with probability a() Gibbs sampler is special case of M-H –always accept new proposal acceptance insures right stable distribution

15. June 1999NCSU QTL Workshop © Brian S. Yandell 15 Full Conditional for genos full conditional for geno type depends on –effects via trait model –locus via recombination model can explicitly decompose by individual j –binomial (or trinomial) probability

16. June 1999NCSU QTL Workshop © Brian S. Yandell 16 Missing marker Data sample missing marker data a la QT geno types full conditional for missing markers depends on –flanking markers –possible flanking QTL can explicitly decompose by individual j –binomial (or trinomial) probability

17. June 1999NCSU QTL Workshop © Brian S. Yandell 17 Multiple QTL model trait = mean + add1 + add2 + error trait = effect_of_genos + error prob( trait | genos, effects )

18. June 1999NCSU QTL Workshop © Brian S. Yandell 18 Vector Notation for QTLs inner product for sum condense notation

19. June 1999NCSU QTL Workshop © Brian S. Yandell 19 Multiple loci vector of loci across linkage map careful bookkeeping during update –identifiability & bump hunting –possibility of two loci in one marker interval ordered loci are sufficient

20. June 1999NCSU QTL Workshop © Brian S. Yandell 20 Posterior: Multiple QTLs posterior = likelihood * prior / constant posterior( paramaters | data ) prob( genos, effects, loci | traits, map ) is proportional to

21. June 1999NCSU QTL Workshop © Brian S. Yandell 21 MCMC for Multiple QTLs construct Markov chain around posterior update one (or several) components at a time –update effects given geno types & traits –update loci given geno types & traits –update geno types give loci & effects update all terms for each locus at one time? –open questions of efficient mixing

22. June 1999NCSU QTL Workshop © Brian S. Yandell 22 MCMC Updates effects loci traits genos

23. June 1999NCSU QTL Workshop © Brian S. Yandell 23 MCMC Conditions construct Markov chain with stable distribution ergodic Markov chain –reversibile (detailed balance) –irreducible (can get from any value to any other) –aperiodic (no fixed pattern) –positive recurrent (chance to visit all possible values)

24. June 1999NCSU QTL Workshop © Brian S. Yandell 24 Reversible Jump MCMC basic idea of Green(1995) model selection in regression how many QTLs? –number of QTL is random –estimate the number m RJ-MCMC vs. Bayes factors other similar ideas

25. June 1999NCSU QTL Workshop © Brian S. Yandell 25 Jumping the Number of QTL model changes with number of QTL –almost analogous to stepwise regression –use reversible jump MCMC to change number book keeping helps in comparing models change of variables between models prior on number of QTL –uniform over some range –Poisson with prior mean

26. June 1999NCSU QTL Workshop © Brian S. Yandell 26 Posterior: Number of QTL posterior = likelihood * prior / constant posterior( paramaters | data ) prob( genos, effects, loci, m | traits, map ) is proportional to

27. June 1999NCSU QTL Workshop © Brian S. Yandell 27 Reversible Jump Choices action step: draw one of three choices update step with probability 1-b(m+1)-d(m) –update current model –loci, effects, geno types as before add a locus with probability b(m+1) –propose a new locus –innovate effect and geno types at new locus –decide whether to accept the “birth” of new locus drop a locus with probability d(m) –pick one of existing loci to drop –decide whether to accept the “death” of locus

28. June 1999NCSU QTL Workshop © Brian S. Yandell 28 Markov chain for number m add a new locus drop a locus update current model 01mm-1m+1... m

29. June 1999NCSU QTL Workshop © Brian S. Yandell 29 Jumping QTL number & loci

30. June 1999NCSU QTL Workshop © Brian S. Yandell 30 RJ-MCMC Updates effects loci traits genos add locus drop locus b(m+1) d(m)d(m) 1-b(m+1)-d(m)

31. June 1999NCSU QTL Workshop © Brian S. Yandell 31 Propose to Add a locus propose a new locus –similar proposal to ordinary update uniform chance over genome easier to avoid interval with another QTL –need geno types at locus & model effect innovate effect & geno types at new locus –draw geno types based on recombination (prior) no dependence on trait model yet –draw effect as in Green’s reversible jump adjust for collinearity modify other parameters accordingly check acceptance...

32. June 1999NCSU QTL Workshop © Brian S. Yandell 32 Propose to Drop a locus choose an existing locus –equal weight for all loci ? –more weight to loci with small effects ? “drop” effect & geno types at old locus –adjust effects at other loci for collinearity –this is reverse jump of Green (1995) check acceptance … –do not drop locus, effects & geno types –until move is accepted

33. June 1999NCSU QTL Workshop © Brian S. Yandell 33 Acceptance of Reversible Jump accept birth of new locus with probability min(1,A) accept death of old locus with probability min(1,1/A)

34. June 1999NCSU QTL Workshop © Brian S. Yandell 34 Acceptance of Reversible Jump move probabilities birth & death proposals Jacobian between models –fudge factor –see stepwise regression example mm+1

35. June 1999NCSU QTL Workshop © Brian S. Yandell 35 RJ-MCMC: Number of QTL

36. June 1999NCSU QTL Workshop © Brian S. Yandell 36 Posterior # QTL for 8-week Data 98% credible region for m : (1,3) based on 1 million steps with prior mean of 3

37. June 1999NCSU QTL Workshop © Brian S. Yandell 37 How Good is RJ-MCMC? simulations with 0, 1 or 2 QTL –strong effects (additive = 2, variance = 1) –linked loci 36cM apart differences with number of QTL –clear differences by actual number –works well with 100,000, better with 1M effect of Poisson prior mean –larger prior mean shifts posterior up –but prior does not take over

38. June 1999NCSU QTL Workshop © Brian S. Yandell 38 Posterior for Simulated Data 0,1 or 2 large QTL prior Poisson mean of 2 100,000 RJ-MCMC runs

39. June 1999NCSU QTL Workshop © Brian S. Yandell 39 Effect of Prior Mean

40. June 1999NCSU QTL Workshop © Brian S. Yandell 40 # QTL in Brassica Data 4-week & 8-week vernalization –log( days to flower) –105 lines, 10 markers –modest effects –evidence of 1 or 2 QTL using Bayes factors histograms of posterior number of QTL –depends somewhat on prior –mode is 1 or 2 QTL 90% credible sets –all include 2 QTL –include 1 QTL if prior not huge

41. June 1999NCSU QTL Workshop © Brian S. Yandell 41 #QTL for Brassica 8-week

42. June 1999NCSU QTL Workshop © Brian S. Yandell 42 Brassica #QTL 90% Credible Sets 8-week 4-week

43. June 1999NCSU QTL Workshop © Brian S. Yandell 43 Brassica #QTL Comparison

44. June 1999NCSU QTL Workshop © Brian S. Yandell 44 Reversible Jump II reversible jump MCMC details –can update model with m QTL –have basic idea of jumping models –now: careful bookkeeping between models RJ-MCMC & Bayes factors –Bayes factors from RJ-MCMC chain –components of Bayes factors

45. June 1999NCSU QTL Workshop © Brian S. Yandell 45 RJ-MCMC Updates effects loci traits genos add locus drop locus b(m+1) d(m)d(m) 1-b(m+1)-d(m)

46. June 1999NCSU QTL Workshop © Brian S. Yandell 46 Reversible Jump Idea expand idea of MCMC to compare models adjust for parameters in different models –augment smaller model with innovations –constraints on larger model calculus “change of variables” is key –add or drop parameter(s) –carefully compute the Jacobian consider stepwise regression –Mallick (1995) & Green (1995) –efficient calculation with Hausholder decomposition

47. June 1999NCSU QTL Workshop © Brian S. Yandell 47 Model Selection in Regression known regressors (e.g. markers ) –models with 1 or 2 regressors jump between models –centering regressors simplifies calculations

48. June 1999NCSU QTL Workshop © Brian S. Yandell 48 Slope Estimate for 1 Regressor recall least squares estimate of slope note relation of slope to correlation

49. June 1999NCSU QTL Workshop © Brian S. Yandell 49 2 Correlated Regressors slopes adjusted for other regressors

50. June 1999NCSU QTL Workshop © Brian S. Yandell 50 Gibbs Sampler for Model 1 mean slope variance

51. June 1999NCSU QTL Workshop © Brian S. Yandell 51 Gibbs Sampler for Model 2 mean slopes variance

52. June 1999NCSU QTL Workshop © Brian S. Yandell 52 Updates from 2->1 drop 2nd regressor adjust other regressor

53. June 1999NCSU QTL Workshop © Brian S. Yandell 53 Updates from 1->2 add 2nd slope, adjusting for collinearity adjust other slope & variance

54. June 1999NCSU QTL Workshop © Brian S. Yandell 54 Model Selection in Regression known regressors (e.g. markers ) –models with 1 or 2 regressors jump between models –augment with new innovation z

55. June 1999NCSU QTL Workshop © Brian S. Yandell 55 Change of Variables change variables from model 1 to model 2 calculus issues for integration –need to formally account for change of variables –infinitessimal steps in integration (db) –involves partial derivatives (next page)

56. June 1999NCSU QTL Workshop © Brian S. Yandell 56 Jacobian & the Calculus Jacobian sorts out change of variables –careful: easy to mess up here!

57. June 1999NCSU QTL Workshop © Brian S. Yandell 57 Geometry of Reversible Jump

58. June 1999NCSU QTL Workshop © Brian S. Yandell 58 QT additive Reversible Jump

59. June 1999NCSU QTL Workshop © Brian S. Yandell 59 Credible Set for additive 90% & 95% sets based on normal regression line corresponds to slope of updates

60. June 1999NCSU QTL Workshop © Brian S. Yandell 60 Efficient Updating of additive more computations when m > 2 want to avoid matrix inverses –decompose matrix instead –solve linear system of equations use linear algebra –Hausholder (QR) decomposition –LAPACK User’s Guide (1995, 2nd ed) Anderson et al., SIAM.

61. June 1999NCSU QTL Workshop © Brian S. Yandell 61 Hausholder (QR) Decomposition decomposition –G is upper triangular –F is orthogonal orthogonality design matrix

62. June 1999NCSU QTL Workshop © Brian S. Yandell 62 QR & Regression model error piece model piece estimators

63. June 1999NCSU QTL Workshop © Brian S. Yandell 63 Absorbing Old Model old model –m regressors –QR decomposition new model –m+1 regressor –use QR to absorb old model

64. June 1999NCSU QTL Workshop © Brian S. Yandell 64 Adjusted Slope Estimators old slopes –note m=1 case added slope –note sum of squares variance –note Jacobian new slopes

65. June 1999NCSU QTL Workshop © Brian S. Yandell 65 How To Infer loci ? if m is known, use fixed MCMC –histogram of loci –issue of bump hunting combining loci estimates in RJ-MCMC –some steps are from wrong model too few loci (bias) too many loci (variance/identifiability) –condition on number of loci subsets of Markov chain

66. June 1999NCSU QTL Workshop © Brian S. Yandell 66 Brassica 8-week Data locus MCMC with m=2

67. June 1999NCSU QTL Workshop © Brian S. Yandell 67 Jumping QTL number & loci

68. June 1999NCSU QTL Workshop © Brian S. Yandell 68 RJ-MCMC loci chain

69. June 1999NCSU QTL Workshop © Brian S. Yandell 69 Raw Histogram of loci

70. June 1999NCSU QTL Workshop © Brian S. Yandell 70 Conditional Histograms

71. June 1999NCSU QTL Workshop © Brian S. Yandell 71 Bayes Factors ratio of posterior odds to prior odds –RJ-MCMC gives posterior on number of QTL –prior is Poisson

72. June 1999NCSU QTL Workshop © Brian S. Yandell 72 #QTL for Brassica 8-week

73. June 1999NCSU QTL Workshop © Brian S. Yandell 73 RJ-Bayes Factors (8-week Brassica data)

74. June 1999NCSU QTL Workshop © Brian S. Yandell 74 Simulation Study of Prior Effect how dramatic is the effect of prior? simulations of 0, 1 or 2 QTL –QTL have large effect additive = 2, variance = 1 –2 QTL spaced 36cM apart –sample sized of 105 RJ-MCMC runs of 100,000

75. June 1999NCSU QTL Workshop © Brian S. Yandell 75 Effect of Prior Mean

76. June 1999NCSU QTL Workshop © Brian S. Yandell 76 Bayes Factor prior of 2 prior of 4

77. June 1999NCSU QTL Workshop © Brian S. Yandell 77 Computing Bayes Factors arithmetic mean –using samples from prior –mean across Monte Carlo or MCMC runs –can be inefficient if prior differs from posterior harmonic mean –using samples from posterior –more efficient but less stable –careful choice of weight h() close to posterior

78. June 1999NCSU QTL Workshop © Brian S. Yandell 78 Stable Bayes Factors Satagopan, Raftery & Newton (1999) –weighted harmonic mean –absorb variance (normal to t dist) replace by

79. June 1999NCSU QTL Workshop © Brian S. Yandell 79 Bayes Factors & LODs others have tried arithmetic & harmonic mean why not geometric mean? terms that are averaged are log likelihoods...

80. June 1999NCSU QTL Workshop © Brian S. Yandell 80 Bayesian LOD Bayesian “LOD” computed at each step –based on LR given sampled geno types and effects –can be larger or smaller than profile LOD –informal diagnostic of fit –combine to for geometric estimates of Bayes factors

81. June 1999NCSU QTL Workshop © Brian S. Yandell 81 Compare LODs scatter plot of loci and Bayesian LODs –same BLOD for all loci of step –overlay LOD from interval mapping (red) –overlay LOD from CIM (green) –vertical lines at true or inferred loci (blue) steps with higher BLOD –may have more likely genotypes –basis for MCMC step always up to higher likelihood sometimes down to lower likelihood

82. June 1999NCSU QTL Workshop © Brian S. Yandell 82 BLODs with no QTL LOD profile roughly zero most BLOD values should be negative –no pattern across linkage group distribution similar to rescaled chi- square with 1 df –-2log(LR) approximately chi-square –assignment of genotypes “irrelevant” RJ-MCMC skewed with inferred QTL –numbers indicate #QTL

83. June 1999NCSU QTL Workshop © Brian S. Yandell 83 Simulated Data with No QTL

84. June 1999NCSU QTL Workshop © Brian S. Yandell 84 BLOD: no QTL

85. June 1999NCSU QTL Workshop © Brian S. Yandell 85 RJ-MCMC BLOD: no QTL

86. June 1999NCSU QTL Workshop © Brian S. Yandell 86 BLODs with 1 QTL LOD peaks at correct locus most BLOD values near locus –some considerably larger than LOD –inferred genotype vs EM average non-central distribution of BLOD –rescaled non-central chi-square? RJ-MCMC dispersed from peak –locus proposal has local dispersion

87. June 1999NCSU QTL Workshop © Brian S. Yandell 87 LOD for 1 QTL

88. June 1999NCSU QTL Workshop © Brian S. Yandell 88 RJ-MCMC BLOD: 1 QTL

89. June 1999NCSU QTL Workshop © Brian S. Yandell 89 BLODs with 2 QTL incorrect fit of 1 QTL model –LOD peaks at ghost locus –BLOD values near LOD peaks correct 2-QTL model –IM misses, CIM gets loci –BLOD at loci –BLOD approx. 2x LOD simultaneous fit of both loci RJ-MCMC dispersed from peak –need to look conditional on m=2

90. June 1999NCSU QTL Workshop © Brian S. Yandell 90 LOD for 2 QTL with 1 Fit

91. June 1999NCSU QTL Workshop © Brian S. Yandell 91 LOD for 2 QTL with 2 Fit

92. June 1999NCSU QTL Workshop © Brian S. Yandell 92 RJ-MCMC BLOD: 2 QTL

93. June 1999NCSU QTL Workshop © Brian S. Yandell 93 Brassica BLODs 4-week clearer than 8-week –ghost QTL or smear for 8-week? MCMC with m=2 fairly clear RJ-MCMC dispersed –conditioning on m=2 similar to MCMC not shown –mixing models together –local proposal moves hamper mixing

94. June 1999NCSU QTL Workshop © Brian S. Yandell 94 Brassica 4-week BLOD Map

95. June 1999NCSU QTL Workshop © Brian S. Yandell 95 Brassica 8-week BLOD Map

96. June 1999NCSU QTL Workshop © Brian S. Yandell 96 4-week RJ-MCMC BLOD

97. June 1999NCSU QTL Workshop © Brian S. Yandell 97 8-week RJ-MCMC BLOD

98. June 1999NCSU QTL Workshop © Brian S. Yandell 98 The Art of MCMC convergence issues –burn-in period & when to stop proper mixing of the chain –smart proposals & smart updates frequentist approach –simulated annealing: reaching the peak –simulated tempering: heating & cooling the chain Bayesian approach –influence of priors on posterior –Rao-Blackwell smoothing bump-hunting for mixtures (e.g. QTL)

99. June 1999NCSU QTL Workshop © Brian S. Yandell 99 RJ-MCMC Software General MCMC software –U Bristol links http://www.stats.bris.ac.uk/MCMC/pages/links.html –BUGS (Bayesian inference Using Gibbs Sampling) http://www.mrc-bsu.cam.ac.uk/bugs/ Our MCMC software for QTLs –C code using LAPACK ftp://ftp.stat.wisc.edu/pub/yandell/revjump.tar.gz –coming soon: perl preprocessing (to/from QtlCart format) Splus post processing Bayes factor computation

100. June 1999NCSU QTL Workshop © Brian S. Yandell 100 RJ-MCMC Software Details input files –marker.dist –marker.mark –trait.y output file –result.write –result.error distances between loci (cM) marker genotypes (-1,1) –one line per marker trait phenotypes results errors if any

101. June 1999NCSU QTL Workshop © Brian S. Yandell 101 trait.y file missing value = -99.000000 8.097789 7.928476 12.538893 8.370929 8.347937 11.252023 10.687911 6.961958 12.318708 11.985510 11.841586 8.339165 9.347868 11.933833 7.593694 12.248663 13.183897 9.928043 8.326296...

102. June 1999NCSU QTL Workshop © Brian S. Yandell 102 marker.dist file 8.8 11.8 6.8 8.7 10.7 10.5 5.1 14.7

103. June 1999NCSU QTL Workshop © Brian S. Yandell 103 marker.mark file one row per marker, one column per line -1 -1 -1 1 -1 1 1... -1 -1 1 1 -1 1 1... -1 -1 1 -1 -1 1 1... -1 -1 1 -1 -1 1 -1... -1 1 1 1 -1 1 -1...

104. June 1999NCSU QTL Workshop © Brian S. Yandell 104 nval.dat file 1# 1=revjump,0=no 105 10# n=individuals markers 10000 10# N=MCMC_runs skips 2# m 0 0.5# mu sigmasq 0 0# initial b’s 3.5 7.5# initial loci 0 10 1 1# prior(mu)~N(0,10) prior(sigmasq)~IG(1,1) 0 10# prior(b)~N(0,1) 4# prior(m)~Poisson(4)

105. June 1999NCSU QTL Workshop © Brian S. Yandell 105 result.write file m mu b(1)..b(m) sigmasq lambda(1)..lambda(m) move LOD 2 3.0702 -0.0820 0.0975 0.2085 10.5301 83.7381 1 -0.3943 3 3.0512 0.0343 -0.1166 -0.0600 0.0853 18.4030 51.2329 76... 2 3.0400 -0.0415 -0.0946 0.0619 28.6062 74.0904 3 6.1284... 1 3.1045 -0.1412 0.0851 76.9420 3 6.1310 2 3.0669 -0.0434 -0.1576 0.1091 17.4566 80.7973 3 4.3184 propose accept birth 384474 100625 death 227999 100625 locus 387527 259705 m=0 m=1 m=2 m=3 m=4 m=5... 1936 428306 425105 126871 16703 1047 32 0 0 0

106. June 1999NCSU QTL Workshop © Brian S. Yandell 106 Bayes Factor References MA Newton & AE Raftery (1994) “Approximate Bayesian inference with the weighted likelihood bootstrap”, J Royal Statist Soc B 56: 3-48. RE Kass & AE Raftery (1995) “Bayes factors”, J Amer Statist Assoc 90: 773-795. JM Satagopan, MA Newton & AE Rafter (1999) “On the harmonic mean estimator of marginal probability”, ms in prep, mailto:satago@biosta.mskcc.org.

107. June 1999NCSU QTL Workshop © Brian S. Yandell 107 Reversible Jump MCMC References PJ Green (1995) “Reversible jump Markov chain Monte Carlo computation and Bayesian model determination”, Biometrika 82: 711-732. S Richardson & PJ Green (1997) “On Bayesian analysis of mixture with an unknown of components”, J Royal Statist Soc B 59: 731-792. BK Mallick (1995) “Bayesian curve estimation by polynomials of random order”, TR 95-19, Math Dept, Imperial College London. L Kuo & B Mallick (1996) “Bayesian variable selection for regression models”, ASA Proc Section on Bayesian Statistical Science, 170-175.

108. June 1999NCSU QTL Workshop © Brian S. Yandell 108 QTL Reversible Jump MCMC: Inbred Lines JM Satagopan & BS Yandell (1996) “Estimating the number of quantitative trait loci via Bayesian model determination”, Proc JSM Biometrics Section. DA Stephens & RD Fisch (1998) “Bayesian analysis of quantitative trait locus data using reversible jump Markov chain Monte Carlo”, Biometrics 54: 1334-1347. MJ Sillanpaa & E Arjas (1998) “Bayesian mapping of multiple quantitative trait loci from incomplete inbred line cross data”, Genetics 148: 1373-1388. R Waagepetersen & D Sorensen (1999) “Understanding reversible jump MCMC”, mailto:sorensen@inet.uni2.dk.

109. June 1999NCSU QTL Workshop © Brian S. Yandell 109 QTL Reversible Jump MCMC: Pedigrees S Heath (1997) “Markov chain Monte Carlo segregation and linkage analysis for oligenic models”, Am J Hum Genet 61: 748-760. I Hoeschele, P Uimari, FE Grignola, Q Zhang & KM Gage (1997) “Advances in statistical methods to map quantitative trait loci in outbred populations”, Genetics 147:1445-1457. P Uimari and I Hoeschele (1997) “Mapping linked quantitative trait loci using Bayesian analysis and Markov chain Monte Carlo algorithms”, Genetics 146: 735-743. MJ Sillanpaa & E Arjas (1999) “Bayesian mapping of multiple quantitative trait loci from incomplete outbred offspring data”, Genetics 151, 1605-1619.

110. June 1999NCSU QTL Workshop © Brian S. Yandell 110 QTLs and Polygenes phenotype = design + QTLs + polygenes + error QTLs:quantitative trait loci polygenes:many genes of small effect spread throughout genome distinction is arbitrary, depending on sample size magnitude of effects design/cross/marker polymorphism analogy to multiple regression

111. June 1999NCSU QTL Workshop © Brian S. Yandell 111 Polygenes and Inbred Lines same (raw) genetic correlation across cohort: –1/2 for DH –2/3 for F2 –4/7 for F3 –1/2 for RI modified by specific information: –major & minor QTLs –marker surrogates for polygenes (CIM)

112. June 1999NCSU QTL Workshop © Brian S. Yandell 112 Composite QTL model trait = mean + add + dom + other + error trait = effect_of_geno + other + error prob( trait | genos, effects, other ) other ( ): other linked and unlinked QTLs, and polygenes