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Power of linkage analysis

Power of linkage analysis. Manuel AR Ferreira. Massachusetts General Hospital. Harvard Medical School. Boston. Egmond, 2006. Outline. 1. Aim. 2. Statistical power. 3. Estimate the power of linkage analysis. Analytically. Empirically. 4. Improve the power of linkage analysis. 1. Aim.

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Power of linkage analysis

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  1. Power of linkage analysis Manuel AR Ferreira Massachusetts General Hospital Harvard Medical School Boston Egmond, 2006

  2. Outline 1. Aim 2. Statistical power 3. Estimate the power of linkage analysis Analytically Empirically 4. Improve the power of linkage analysis

  3. 1. Aim

  4. 1. Know what type-I error and power are 2. Know that you can/should estimate the power of your linkage analysis (analytically or empirically) 3. Be aware that there are MANY factors that increase type-I error and decrease the power of linkage 4. Show how to clean your data to detect presence and minimize the impact of these factors

  5. 2. Statistical power

  6. H0:Person A is not guilty H1:Person A is guilty – send him to jail In reality… H0 is true H1 is true β 1 - α H0 is true Type-2 error We decide… 1 - β α H1 is true Power Type-1 error

  7. H0:There is NO linkage between a marker and a trait H1:There is linkage between a marker and a trait Linkage test statistic has different distributions under H0 and H1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

  8. Where should I set the threshold to determine significance? I decide H0 is true I decide H1 is true (Linkage) x Threshold Power (1 – β) Type-1 error (α) High High To low

  9. Where should I set the threshold to determine significance? I decide H1 is true I decide H0 is true x Threshold Power (1 – β) Type-1 error (α) High High To low Low Low To high

  10. How do I maximise Power while minimising Type-1 error rate? I decide H1 is true I decide H0 is true Power (1 – β) x Type-1 error (α) 1. Set a high threshold for significance (i.e. results in low α [e.g. 0.05-0.00002]) 2. Try to shift the distribution of the linkage test statistic when H1 is true as far as possible from the distribution when H0 is true.

  11. Non-centrality parameter NCP H0 H1 Central Χ2 Non-central Χ2 Mean (μ) df + NCP df Variance (σ2) 2*(df) 2*(df) + 4*NCP These distributions ARE NOT chi-sq with 1df!! Just for illustration.. Switch to R to see what they really look like..

  12. …R

  13. NCP H0 H1 Big overlap between H0 and H1 distributions Small NCP Lower power Small overlap between H0 and H1 distributions Large NCP Greater power

  14. Short practical on GPC Genetic Power Calculator is an online resource for carrying out basic power calculations. http://pngu.mgh.harvard.edu/~purcell/gpc/ For our 1st example we will use the probability function calculator to play with power

  15. Using the Probability Function Calculator of the GPC 1. Go to: ‘http://pngu.mgh.harvard.edu/~purcell/gpc/’ Click the ‘Probability Function Calculator’ tab. 2. We’ll focus on the first 3 input lines. These refer to the chi-sq distribution that we’re interested in right now. NCP Degrees of freedom of your test.E.g. 1df for univariate linkage (ignoring for now that it’s a mixture distribution)

  16. Exercises 1. Let’s start with a simple exercise. Determine the critical value (X) of a chi-square distribution with 1 df and NCP = 0, such that P(X>x) = 0.05. df = 1 NCP = 0 P(X>x) = 0.05 X = ? Determine the P(X>x) for a chi-square distribution with 1 df and NCP = 0 and X = 3.84. df = 1 NCP = 0 P(X>x) = ? X = 3.84

  17. Exercises 2. Find the power when the NCP of the test is 5, degrees of freedom=1, and the critical X is 3.84. NCP = 5 df = 1 NCP = 5 P(X>x) = ? 3.84 X = 3.84 NCP = 10 What if the NCP = 10? df = 1 NCP = 10 P(X>x) = ? 3.84 X = 3.84

  18. Exercises 3. Find the required NCP to obtain a power of 0.8, for degrees of freedom=1 and critical X = 3.84. NCP = ? = 0.8 df = 1 NCP = ? P(X>x) = 0.8 3.84 X = 3.84 What if the X = 13.8? NCP = ? = 0.8 df = 1 NCP = ? P(X>x) = 0.8 13.8 X = 13.8

  19. 2. Estimate power linkage analysis

  20. Why is it important to estimate power? To determine whether the study you’re designing/analysing can in fact localise the QTL you’re looking for. You’ll need to do it for most grant applications. When and how should I estimate power? When? How? Study design stage Theoretically, empirically Analysis stage Empirically

  21. Theoretical power estimation NCP determines the power to detect linkage NCP = μ(H1 is true) - df If we can predict what the NCP of the test will be, we can estimate the power of the test

  22. Theoretical power estimation Variance Components linkage analysis (and some HE extensions) Sham et al. 2000 AJHG 66: 1616 1. The number of sibs in the sibship (s) 2. Residual sib correlation (r) 3. Squared variance due to the additive QTL component (VA) 4. Marker informativeness (i.e. Var(π) and Var(z)) ^ 5. Squared variance due to the dominance QTL component (VD).

  23. Another short practical on GPC The idea is to see how genetic parameters and the study design influence the NCP – and so the power – of linkage analysis

  24. Using the ‘VC QTL linkage for sibships’ of the GPC 1. Go to: ‘http://pngu.mgh.harvard.edu/~purcell/gpc/’ Click the ‘VC QTL linkage for sibships’ tab.

  25. Exercises 1. Let’s estimate the power of linkage for the following parameters: QTL additive variance: 0.2 QTL dominance variance: 0 Residual shared variance: 0.4 Residual nonshared variance: 0.4 Recombination fraction: 0 Sample Size: 200 Sibship Size: 2 User-defined type I error rate: 0.05 User-defined power: determine N : 0.8 Power = 0.36 (alpha = 0.05) Sample size for 80% power = 681 families

  26. Exercises 2. We can now assess the impact of varying the QTL heritability QTL additive variance: 0.4 QTL dominance variance: 0 Residual shared variance: 0.4 Residual nonshared variance: 0.4 Recombination fraction: 0 Sample Size: 200 Sibship Size: 2 User-defined type I error rate: 0.05 User-defined power: determine N : 0.8 Power = 0.73 (alpha = 0.05) Sample size for 80% power = 237 families

  27. Exercises 3. … the residual shared variance QTL additive variance: 0.2 QTL dominance variance: 0 Residual shared variance: 0.2 Residual nonshared variance: 0.6 Recombination fraction: 0 Sample Size: 200 Sibship Size: 2 User-defined type I error rate: 0.05 User-defined power: determine N : 0.8 Power = 0.26 (alpha = 0.05) Sample size for 80% power = 1161 families

  28. Exercises 4. … the sample size QTL additive variance: 0.2 QTL dominance variance: 0 Residual shared variance: 0.4 Residual nonshared variance: 0.2 Recombination fraction: 0 Sample Size:400 Sibship Size: 2 User-defined type I error rate: 0.05 User-defined power: determine N : 0.8 Power = 0.94 (alpha = 0.05) Sample size for 80% power = 294 families

  29. Exercises 4. … the sibship size QTL additive variance: 0.2 QTL dominance variance: 0 Residual shared variance: 0.4 Residual nonshared variance: 0.2 Recombination fraction: 0 Sample Size:200 Sibship Size: 3 User-defined type I error rate: 0.05 User-defined power: determine N : 0.8 Power = 0.99 (alpha = 0.05) Sample size for 80% power = 78 families

  30. Theoretical power estimation Advantages: Fast, GPC Disadvantages: Approximation, may not fit well individual study designs, particularly if one needs to consider more complex pedigrees, missing data, ascertainment strategies, etc…

  31. Empirical power estimation Mx: simulate covariance matrices for 3 groups (IBD 0, 1 and 2 pairs) according to an FQE model (i.e. with VQ > 0) and then fit the wrong model (FE). The resulting test statistic (minus 1df) corresponds to the NCP of the test. See powerFEQ.mx script. Still has many of the disadvantages of the theoretical approach, but is a useful framework for general power estimations. Simulate data: generate a dataset with a simulated phenotype and a marker that explains a proportion of the phenotypic variance. Test the marker for linkage with the phenotype. Repeat this N times. For a given α, Power = proportion of replicates with a P-value < α (e.g. < 0.05).

  32. 3. Improve power of linkage analysis

  33. Factors that influence type-1 error/power linkage 1. Selective sampling 2. Sample size QTL heritability Disease prevalence Residual correlation Sibship size 3. Deviations in trait distribution 4. Outliers 5. Pedigree errors 6. Genotyping errors 7. Marker informativeness 8. Marker density 9. Genetic map

  34. Pedigree errors Definition.When the self-reported familial relationship for a given pair of individuals differs from the real relationship (determined from genotyping data). Similar for gender mix-ups. Impact on linkage analysis.Increase type-1 error rate (can also decrease power) Detection.Can be detected using genome-wide patterns of allele sharing. Some errors are easy to detect. Software: GRR. Boehnke and Cox (1997), AJHG61:423-429; Broman and Weber (1998), AJHG 63:1563-4; McPeek and Sun (2000), AJHG 66:1076-94; Epstein et al. (2000), AJHG 67:1219-31. Correction.If problem cannot be resolved, delete problematic individuals (family)

  35. Pedigree errors *Impact on linkage* • CSGA (1997) A genome-wide search for asthma susceptibility loci in ethnically diverse populations. Nat Genet15:389-92 • ~15 families with wrong relationships • No significant evidence for linkage • Error checking is essential!

  36. Pedigree errors *Detection/Correction* GRR http://www.sph.umich.edu/csg/abecasis

  37. Practical Identify pedigree errors with GRR Aim: 1. Go to: ‘Egmondserver\share\Programs’ Copy entire ‘GRR’ folder into your desktop. 2. Go into the ‘GRR’ folder in your desktop, and run the GRR.exe file. 3. Press the ‘Load’ button, and navigate into the same ‘GRR’ folder on the desktop. Select the file ‘sample.ped’ and press ‘Open’. Note that all sibpairs in ‘sample.ped’ were reported to be fullsibs or half-sibs. I’ll identify one error. Can you identify the other two?

  38. Genotyping errors Definition.When the observed genotype at a given locus does not match the true genotype at that locus. Unavoidable (assay quality, genotyping platform); becoming much lower with most recent genotyping technologies (chip arrays). Impact on linkage analysis.Can substantially decrease power: e.g. 1% genotyping error can result in ~10-50% loss of power for linkage. Can also increase type-1 error rate. Detection.Look at: assay failure rate (e.g. 20%), number of Mendelian errors, number of genotypes that imply unlikely recombination events. Can be hard to detect (SNPs)! Correction.(1) Re-type problematic markers/individuals; (2) Remove the problematic genotypes; (3) leave errors in, but model them appropriately.

  39. Genotyping errors *Impact on linkage* Successive lines for 0, 0.5, 1, 2 and 5% error.

  40. Genotyping errors *Detection/Correction* Detection (1) Assay failure rate (2) Mendelian errors (e.g. SIBPAIR) (3) Genotypes that imply unlikely recombination events Correction (1) Re-type problematic markers/individuals (2) Remove the problematic genotypes (3) Leave errors in, but model them appropriately.

  41. Genotyping errors *Detection/Correction* MERLIN http://www.sph.umich.edu/csg/abecasis

  42. Genotyping errors *Detection/Correction* • Genotype errors can change inferences about gene flow • May introduce additional recombinants • Likelihood sensitivity analysis • How much impact does each genotype have on likelihood of overall data 2 2 2 2 2 1 2 1 2 2 2 2 2 1 2 1 1 2 1 2 2 2 2 2 2 2 1 1 2 1 2 1 1 1 1 1 1 2 1 2 2 1 2 1 1 2 1 2 1 1 1 1

  43. MERLIN demo Detect and correct genotyping errors 1. Use input files: error.dat error.ped error.map You can download these from the MERLIN website 2. I’ll first run the program pedstats to have a look at these files first. pedstats –d error.dat –p error.ped There are 20 markers and 1 affection trait for 200 families with 4 individuals each.

  44. 3. I’ll then test the trait for linkage with each of the 20 markers using MERLIN, using the ‘--npl’ option for the linkage test. Note this is done before detecting/correcting genotyping errors! merlin –d error.dat –p error.ped –m error.map --npl So before correcting any errors, we get a maximum LOD score of 1.69 at position 52.680 cM

  45. 4. But first we should have looked for genotyping errors. Let’s do that using the ‘--error’ option. merlin –d error.dat –p error.ped –m error.map --error MERLIN flagged 7 pairs of unlikely genotypes. There’s a very good chance that these resulted from genotyping errors!

  46. 5. Let’s delete these unlikely genotypes using pedwipe pedwipe –d error.dat –p error.ped pedwipe will read the error.dat and error.ped files, and delete the genotypes that were stored in file merlin.err produced in the previous step. We get 2 new files, wiped.dat and wiped.ped, that do not have those genotyping errors.

  47. 6. We can now check whether we get any improvement in the LOD score after removing those genotyping errors. merlin –d wiped.dat –p wiped.ped –m error.map --npl After deleting those 7 pairs of genotyping errors, the LOD score at position 52.680 cM increases from 1.69 to 2.10, ~24%!

  48. 7. But instead of identifying and removing errors, MERLIN will soon allow the user to leave the genotyping errors in and model them appropriately. This is the ‘--fit’ function. Have a look at the MERLIN documentation for more info.

  49. Summmary 1. Statistical power 2. Estimate the power of linkage analysis 3. Improve the power of linkage analysis

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