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Power in QTL linkage: single and multilocus analysis

Power in QTL linkage: single and multilocus analysis. Shaun Purcell 1,2 & Pak Sham 1 1 SGDP, IoP, London, UK 2 Whitehead Institute, MIT, Cambridge, MA, USA. Overview. 1) Brief power primer 2) Calculating power for QTL linkage analysis

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Power in QTL linkage: single and multilocus analysis

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  1. Power in QTL linkage: single and multilocus analysis Shaun Purcell1,2 & Pak Sham1 1SGDP, IoP, London, UK 2Whitehead Institute, MIT, Cambridge, MA, USA

  2. Overview • 1) Brief power primer • 2) Calculating power for QTL linkage analysis • Practical 1 : Using GPC for linkage power calculations • 3) The adequacy of additive single locus analysis • Practical 2 : Using Mx for linkage power calculations

  3. YES NO Test statistic 1) Power primer

  4. Sampling distribution if H0 were true Sampling distribution if HA were true Critical value   P(T) T

  5. STATISTICS Nonrejection of H0 Rejection of H0 Type I error at rate  Nonsignificant result H0 true R E A L I T Y Type II error at rate  Significant result HA true POWER =(1- )

  6. Critical value Impact of  effect size, N P(T) T  

  7. Critical value Impact of  alpha P(T) T  

  8. 2) Power for QTL linkage • For chi-squared tests on large samples, power is determined by non-centrality parameter () and degrees of freedom (df) •  = E(2lnLA - 2lnL0) • = E(2lnLA ) - E(2lnL0) • where expectations are taken at asymptotic values of maximum likelihood estimates (MLE) under an assumed true model

  9. Linkage test • HA • H0 for i=j for ij for i=j for ij

  10. For sib-pairs under complete marker information Determinant of 2-by-2 standardised covariance matrix = 1 - r2 Linkage test Expected NCP Note: standardised trait See Sham et al (2000) AJHG, 66. for further details

  11. Concrete example • 200 sibling pairs; sibling correlation 0.5. • To calculate NCP if QTL explained 10% variance: • 200 × 0.002791 = 0.5581

  12. Approximation of NCP NCP per sibship is proportional to - the # of pairs in the sibship (large sibships are powerful) - the square of the additive QTL variance (decreases rapidly for QTL of v. small effect) - the sibling correlation (structure of residual variance is important)

  13. P(IBD at M | IBD at QTL) IBD at QTL 0 1 2 IBD at M 0 1 2

  14. Using GPC • Comparison to Haseman-Elston regression linkage • Amos & Elston (1989) H-E regression • - 90% power (at significant level 0.05) • - QTL variance 0.5 • - marker & major gene completely linked (θ= 0) •  320 sib pairs • - if θ= 0.1 •  778 sib pairs

  15. GPC input parameters • Proportions of variance • additive QTL variance • dominance QTL variance • residual variance (shared / nonshared) • Recombination fraction ( 0 - 0.5 ) • Sample size & Sibship size ( 2 - 8 ) • Type I error rate • Type II error rate

  16. GPC output parameters • Expected sibling correlations • - by IBD status at the QTL • - by IBD status at the marker • Expected NCP per sibship • Power • - at different levels of alpha given sample size • Sample size • - for specified power at different levels of alpha given power

  17. GPC http://ibgwww.colorado.edu/~pshaun/gpc/

  18. From GPC • Modelling additive effects only • Sibships Individuals • Pairs 216 (320) 432 • Pairs ( = 0.1) 543 (778) 1086 Trios ( = 0.1) 179 537 Quads ( = 0.1) 90 360 Quints ( = 0.1) 55 275

  19. Practical 1 • Using GPC, what is the effect on power to detect linkage of : • 1. QTL variance? • 2. residual sibling correlation? • 3. marker-QTL recombination fraction?

  20. Pairs required (=0, p=0.05, power=0.8)

  21. Pairs required (=0, p=0.05, power=0.8)

  22. Effect of residual correlation • QTL additive effects account for 10% trait variance • Sample size required for 80% power (=0.05) • No dominance •  = 0.1 • A residual correlation 0.35 • B residual correlation 0.50 • C residual correlation 0.65

  23. Individuals required

  24. Effect of incomplete linkage

  25. Effect of incomplete linkage

  26. Some factors influencing power • 1. QTL variance • 2. Sib correlation • 3. Sibship size • 4. Marker informativeness & density • 5. Phenotypic selection

  27. Marker informativeness: • Markers should be highly polymorphic • - alleles inherited from different sources are likely to be distinguishable • Heterozygosity (H) • Polymorphism Information Content (PIC) • - measure number and frequency of alleles at a locus

  28. Polymorphism Information Content • IF a parent is heterozygous, • their gametes will usually be informative. • BUT if both parents & child are heterozygous for the same genotype, • origins of child’s alleles are ambiguous • IF C = the probability of this occurring,

  29. Singlepoint θ1 Marker 1 Trait locus Multipoint T1 T2 T3 T4 T5 T11 T12 T13 T14 T15 T6 T7 T8 T9 T10 T16 T17 T18 T19 T20 Marker 1 Trait locus Marker 2

  30. Multipoint PIC: 10 cM map

  31. Multipoint PIC: 5 cM map

  32. The Singlepoint Information Content of the markers: • Locus 1 PIC = 0.375Locus 2 PIC = 0.375Locus 3 PIC = 0.375 • The Multipoint Information Content of the markers: • Pos MPIC • -10 22.9946 • -9 24.9097 • -8 26.9843 • -7 29.2319 • -6 31.6665 • -5 34.304 • -4 37.1609 • -3 40.256 • -2 43.6087 • -1 47.2408 • 0 51.1754 • 1 49.6898 • … • meaninf 50.2027

  33. Selective genotyping Unselected Proband Selection EDAC Maximally Dissimilar ASP Extreme Discordant EDAC Mahanalobis Distance

  34. Sibship NCP 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 4 3 2 1 -4 0 -3 Sib 2 trait -1 -2 -1 -2 0 1 -3 2 Sib 1 trait 3 -4 4 Sibship informativeness : sib pairs

  35. Impact of selection

  36. E(-2LL) Sib 1 Sib 2 Sib 3 • 0.00121621   1.00    1.00    • 0.14137692 -2.00   2.00   0.00957190  2.00    1.80   2.20 0.00005954  -0.50 0.50   

  37. 3) Single additive locus model • locus A shows an association with the trait • locus B appears unrelated Locus B Locus A

  38. Joint analysis • locus B modifies the effects of locus A: epistasis

  39. Partitioning of effects • Locus A • Locus B M P M P

  40. 4 main effects M Additive effects P M P

  41. 6 twoway interactions M P  Dominance M P 

  42. 6 twoway interactions M M  P P Additive-additive epistasis  M P  M P 

  43. 4 threeway interactions M P M   Additive-dominance epistasis P P M   M P M   M P P  

  44. 1 fourway interaction Dominance-dominance epistasis M P M P   

  45. One locus • Genotypic • means • AA m + a • Aa m + d • aa m - a 0 -a +a d

  46. m m m – aA + aA – aa + aB + aB + da + dA + aa + aB m m m + aA – aA + dB + ad + dA + dB + dB – ad m m m – aA – aB – aa – da – aB – aB + aA + dA + aa Two loci • AA Aa aa • BB • Bb • bb + dd

  47. IBD locus 12 Expected Sib Correlation 00 2S 012A/2 + 2S 022A + 2D + 2S 102A/2 + 2S 112A/2 + 2A/2 + 2AA/4 + 2S 122A/2 + 2A + 2D + 2AA/2 + 2AD/2 + 2S 202A + 2D + 2S 212A + 2D + 2A/2 + 2AA/2 + 2DA/2 + 2S 222A + 2D + 2A + 2D+ 2AA + 2AD + 2DA + 2DD + 2S

  48. Estimating power for QTL models • Using Mx to calculate power • i. Calculate expected covariance matrices under the full model • ii. Fit model to data with value of interest fixed to null value • i.True model ii. Submodel • Q 0 • S S • N N • -2LL 0.000 =NCP

  49. Model misspecification • Using the domqtl.mx script • i.True ii. Full iii. Null • QA QA 0 • QD 0 0 • S S S • N N N • -2LL 0.000 T1 T2 • Test: dominance only T1 • additive & dominance T2 • additive onlyT2-T1

  50. Results • Using the domqtl.mx script • i.True ii. Full iii. Null • QA 0.1 0.217 0 • QD 0.10 0 • S 0.4 0.367 0.475 • N 0.4 0.417 0.525 • -2LL 0.000 1.269 12.549 • Test: dominance only (1df) 1.269 • additive & dominance (2df) 12.549 • additive only (1df)12.549 - 1.269 = 11.28

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