# Power in QTL linkage analysis

## Power in QTL linkage analysis

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1. Power in QTL linkage analysis Shaun Purcell & Pak Sham SGDP, IoP, London, UK F:\pshaun\power.ppt

2. YES NO Test statistic Power primer • Statistics (e.g. chi-squared, z-score) are continuous measures of support for a certain hypothesis YES OR NO decision-making : significance testing Inevitably leads to two types of mistake : false positive (YES instead of NO) (Type I) false negative (NO instead of YES) (Type II)

3. Hypothesis testing • Null hypothesis : no effect • A ‘significant’ result means that we can reject the null hypothesis • A ‘nonsignificant’ result means that we cannot reject the null hypothesis

4. Statistical significance • The ‘p-value’ • The probability of a false positive error if the null were in fact true • Typically, we are willing to incorrectly reject the null 5% or 1% of the time (Type I error)

5. Misunderstandings • p - VALUES • that the p value is the probability of the null hypothesis being true • that high p values mean large and important effects • NULL HYPOTHESIS • that nonrejection of the null implies its truth

6. Limitations • IF A RESULT IS SIGNIFICANT • leads to the conclusion that the null is false • BUT, this may be trivial • IF A RESULT IS NONSIGNIFICANT • leads only to the conclusion that it cannot be concluded that the null is false

7. Alternate hypothesis • Neyman & Pearson (1928) • ALTERNATE HYPOTHESIS • specifies a precise, non-null state of affairs with associated risk of error

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

9. 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- )

10. Power • The probability of rejection of a false null-hypothesis • depends on • - the significance crtierion () • - the sample size (N) • - the effect size (NCP) “The probability of detecting a given effect size in a population from a sample of size N, using significance criterion ”

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

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

13. Applications • POWER SURVEYS / META-ANALYSES • - low power undermines the confidence that can be placed in statistically significant results • EXPERIMENTAL DESIGN • - avoiding false positives vs. dealing with false negatives • MAGNITUDE VS. SIGNIFICANCE • - highly significant  very important • INTERPRETING NONSIGIFICANT RESULTS • - nonsignficant results only meaningful if power is high

14. Practical Exercise 1 • Calculation of power for simple case-control association study. • DATA : allele frequency of “A” allele for cases and controls • TEST : 2-by-2 contingency table : chi-squared • (1 degree of freedom)

15. Step 1 : determine expected chi-squared • Hypothetical allele frequencies • Cases P(A) = 0.68 • Controls P(A) = 0.54 • Sample 150 cases, 150 controls • Excel spreadsheet : faculty drive:\pshaun\chisq.xls Chi-squared statistic = 12.36

16. Step 2. Determine the critical value for a given type I error rate,  - inverse central chi-squared distribution P(T) Critical value T

17. http://workshop.colorado.edu/~pshaun/gpc/pdf.html • df = 1 , NCP = 0 •  X • 0.05 • 0.01 • 0.001 3.84146 6.63489 10.82754

18. Step 3. Determine the power for a given critical value and non-centrality parameter - non-central chi-squared distribution P(T) Critical value T

19. Determining power • df = 1 , NCP = 12.36 •  X Power • 0.05 3.84146 • 0.01 6.6349 • 0.001 10.827 0.94 0.83 0.59

20. Exercises • Using the spreadsheet and the chi-squared calculator, what is power (for the 3 levels of alpha) • 1. … if the sample size were 300 for each group? • 2. … if allele frequencies were 0.24 and 0.18 for 750 cases and 750 controls?

21. Answers • 1. NCP = 24.72  Power • 0.05 1.00 • 0.01 0.99 • 0.001 0.95 • 2. NCP = 16.27 Power • 0.05 0.98 • 0.01 0.93 • 0.001 0.77 • nb. Stata : di 1-nchi(df,NCP,invchi(df,))

22. POWER Allele frequencies Genetic values Type I errors Type II errors Sample N Effect Size Variance explained QTL linkage

23. Power of tests • For chi-squared tests on large samples, power is determined by non-centrality parameter () and degrees of freedom (df) •  = E(2lnL1 - 2lnL0) • = E(2lnL1 ) - E(2lnL0) • where expectations are taken at asymptotic values of maximum likelihood estimates (MLE) under an assumed true model

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

25. Expected log likelihood under H0 Expectation of the quadratic product is simply s, the sibship size (note: standarised trait)

26. Expected log likelihood under HA

27. For sib-pairs under complete marker information Determinant of 2-by-2 standardised covariance matrix = 1 - r2 Linkage test Expected NCP

28. Approximation of NCP NCP per sib pair 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)

29. Allele frequencies Genetic values Recombination fraction QTL linkage POWER Type I errors Type II errors Sample N Effect Size Variance explained Marker vs functional variant

30. Incomplete linkage • The previous calculations assumed analysis was performed at the QTL. • - imagine that the test locus is not the QTL • but is linked to it. • Calculate sib-pair IBD distribution at the QTL, conditional on IBD at test locus, • - a function of recombination fraction

31.  at QTL 0 1/2 1  at M 0 1/2 1

32. P(M=0 | QTL) r VS VA / 2 + VS VA + VD + VS + VS VA / 2 + VS + VA + VD + VS • Use conditional probabilities to calculate the sib correlation conditional on IBD sharing at the test marker. For example : for IBD 0 at marker :  at QTL 0 1/2 1 C0 =

33. The noncentrality parameter per sib pair is then given by

34. If the QTL is additive, then • attenuation of the NCP is by a factor of (1-2)4 • = square of the correlation • between the proportions of alleles IBD • at two loci with recombination fraction 

37. Comparison to H-E • Amos & Elston (1989) H-E regression • - 90% power (at significant level 0.05) • - QTL variance 0.5 • - marker and major gene are completely linked •  320 sib pairs •  778 sib pairs if  = 0.1

38. 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 - 5 ) • Type I error rate • Type II error rate

39. 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

40. From GPC • Modelling additive effects only • Sibships Individuals • Pairs 265 (320) 530 • Pairs ( = 0.1) 666 (778) 1332 Trios ( = 0.1) 220 660 Quads ( = 0.1) 110 440 Quints ( = 0.1) 67 335

41. Practical Exercise 2 • What is the effect on power to detect linkage of : • 1. QTL variance? • 2. residual sibling correlation? • 3. marker-QTL recombination fraction?

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

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

44. 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

45. Individuals required

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

47. Selective genotyping • The power calculations so far assume an unselected population. • - calculate expected NCP per sibship • If we have a sample with trait scores • - calculate expected NCP for each sibship conditional on trait values • - this quantity can be used to rank order the sample for genotying

48. 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

49. Sibship NCP Sibship NCP 2 2 1.5 1.5 1 1 0.5 0.5 0 4 0 Sibship NCP 3 2 4 1 2 3 -4 0 -3 2 Sib 2 trait -1 -2 1.5 1 -1 -2 0 -4 0 1 1 -3 -3 2 Sib 1 trait Sib 2 trait -2 -1 3 -4 -1 0.5 4 -2 0 1 -3 2 0 Sib 1 trait 3 -4 4 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 dominance rare recessive unequal allele frequencies

50. Selective genotyping SEL T ASP PS ED EDAC MaxD MDis SEL B p d/a .5 15.82 0 .1 0 17.10 .25 15.45 0 .1 16.88 1 .25 15.76 1 .5 1 18.89 .75 27.64 1 43.16 .9 1