On Computation of P-values in Parametric Linkage Analysis

Transcription

1 On Computation of P-values in Parametric Linkage Analysis Azra Kurbašić Centre for Mathematical Sciences Mathematical Statistics Lund University p.1/22

2 Parametric (v. Nonparametric) Analysis The genetic model is assumed to be known. (Although there are parametric models that are mode of inheritance free). p.2/22

3 Parametric (v. Nonparametric) Analysis The genetic model is assumed to be known. (Although there are parametric models that are mode of inheritance free). The parametric analysis is more powerful than nonparametric when the underlying genetic model is known and true, cf. Sham (1998). p.2/22

4 The lod score One marker I Z( ; P(Y, M, ) = log 10 P(Y, M 0.5, ) ) is used for testing H 0 : H 1 : [0, 0.5). = 0.5 against : the genetic model parameters : the recombination fraction (distance) between the marker and disease locus. Y and M denote the collection of disease phenotypes and marker data, respectively p.3/22

5 One marker II The maximal lod score Z max ( ) = sup Z( ; ) = (2 ln 10) 1 2 ln sup P(Y, M, P(Y, M 0.5, ) ) p.4/22

6 One marker II The maximal lod score Z max ( ) = sup Z( ; ) = (2 ln 10) 1 2 ln sup P(Y, M, P(Y, M 0.5, ) ) is asymptotically (2 ln 10) 1 ( (0) (1)) distributed under H 0. p.4/22

7 One marker II The maximal lod score Z max ( ) = sup Z( ; ) = (2 ln 10) 1 2 ln sup P(Y, M, P(Y, M 0.5, ) ) is asymptotically (2 ln 10) 1 ( (0) (1)) distributed under H 0. A mixture of 2 distributions arises because of the structure of the parameter space, cf. Self and Lian (1987) p.4/22

8 One marker III The same asymptotic limit distribution also appears when is misspecified. p.5/22

9 One marker III The same asymptotic limit distribution also appears when is misspecified. The significance level (T ) = P H0 (Z max ( ) T ) is asymptotically independent of the nuisance parameter. T is a given threshold. p.5/22

10 One marker III The same asymptotic limit distribution also appears when is misspecified. The significance level (T ) = P H0 (Z max ( ) T ) is asymptotically independent of the nuisance parameter. T is a given threshold. There is asymptotically a one-to-one correspondence between Z max and the p-value (Z max ). p.5/22

11 Multipoint Analysis I A simultaneous analysis of two/several linked markers on a chromosome is called a multipoint analysis. p.6/22

12 Multipoint Analysis I A simultaneous analysis of two/several linked markers on a chromosome is called a multipoint analysis. The parameter of interest in multipoint analysis is the disease locus x measured in Morgans. p.6/22

13 Multipoint Analysis I A simultaneous analysis of two/several linked markers on a chromosome is called a multipoint analysis. The parameter of interest in multipoint analysis is the disease locus x measured in Morgans. Lod Score is defined as Z(x; P(Y, M x, ) = log 10 P(Y, M, ) ), where x = corresponds to H 0. p.6/22

14 Multipoint Analysis I A simultaneous analysis of two/several linked markers on a chromosome is called a multipoint analysis. The parameter of interest in multipoint analysis is the disease locus x measured in Morgans. Lod Score is defined as Z(x; P(Y, M x, ) = log 10 P(Y, M, ) ), where x = corresponds to H 0. The maximal lod score, Z max ( used as measure of linkage. ) = max x H1 Z(x; ), is p.6/22

15 Multipoint Analysis II The parameter space is no longer connected!... l l l l k } H 1 H0 K : the number of chromosomes l i : the genetic length of chromosome p.7/22

16 Multipoint Analysis III The maximal lod score does no longer converge to a limiting distribution under H 0 as the number of pedigrees grows. p.8/22

17 Multipoint Analysis III The maximal lod score does no longer converge to a limiting distribution under H 0 as the number of pedigrees grows. It has a degenerated limit distribution at as the number of families N. p.8/22

18 Multipoint Analysis III The maximal lod score does no longer converge to a limiting distribution under H 0 as the number of pedigrees grows. It has a degenerated limit distribution at as the number of families N. The significance for a given threshold T depends heavily on, the number of pedigrees, their structure, and the disease and marker phenotypes. p.8/22

19 Multipoint Analysis III The maximal lod score does no longer converge to a limiting distribution under H 0 as the number of pedigrees grows. It has a degenerated limit distribution at as the number of families N. The significance for a given threshold T depends heavily on, the number of pedigrees, their structure, and the disease and marker phenotypes. Let us take a clooser look at the distribution! p.8/22

20 Pointwise Distribution I Multipoint analysis for a family with m meioses: p.9/22

21 Pointwise Distribution I Multipoint analysis for a family with m meioses: Scoring function, cf. Kruglyak et al (1996): S(v) = P(Y v) w V P uniform(w)p(y w) = P(Y v) w V 2 m P(Y w) p.9/22

22 Pointwise Distribution I Multipoint analysis for a family with m meioses: Scoring function, cf. Kruglyak et al (1996): S(v) = P(Y v) w V P uniform(w)p(y w) = P(Y v) w V 2 m P(Y w) Distribution: P x (w) = P(v(x) = w M) p.9/22

23 Pointwise Distribution I Multipoint analysis for a family with m meioses: Scoring function, cf. Kruglyak et al (1996): S(v) = P(Y v) w V P uniform(w)p(y w) = P(Y v) w V 2 m P(Y w) Distribution: P x (w) = P(v(x) = w M) Lod score: Z(x) = log 10 LR(x) = log 10 w S(w)P x (w) p.9/22

24 Pointwise Distribution II Two extreme cases of marker informativity (for one pedigree): perfect marker data: LR(x) = S(v(x)) no marker information at all: P x = P uniform Total multipoint lod score for N families is defined as N Z(x) = Z i (x). i=1 p.10/22

25 Pointwise Distribution II Two extreme cases of marker informativity (for one pedigree): perfect marker data: LR(x) = S(v(x)) no marker information at all: P x = P uniform Total multipoint lod score for N families is defined as N Z(x) = Z i (x). i=1 For two-point analysis we just replace x by Kurbasic & Hössjer (2003)., cf. p.10/22

26 Some Properties of Lod Score From the fact that the H 0 distribution of v(x) is P uniform and LR(x) = E(LR p (x) M) we get (i) E H0 (LR(x)) = E H0 (LR p (x)) = 1 (ii) Var H0 (LR(x)) Var H0 (LR p (x)) when the var. exist (iii) E H0 (Z p (x)) E H0 (Z(x)) 0, p.11/22

27 Some Properties of Lod Score From the fact that the H 0 distribution of v(x) is P uniform and LR(x) = E(LR p (x) M) we get (i) E H0 (LR(x)) = E H0 (LR p (x)) = 1 (ii) Var H0 (LR(x)) Var H0 (LR p (x)) when the var. exist (iii) E H0 (Z p (x)) E H0 (Z(x)) 0, (ii) and (iii) are deduced from Jensens inequality, cf. Royden (1968). p.11/22

28 Some Properties of Lod Score From the fact that the H 0 distribution of v(x) is P uniform and LR(x) = E(LR p (x) M) we get (i) E H0 (LR(x)) = E H0 (LR p (x)) = 1 (ii) Var H0 (LR(x)) Var H0 (LR p (x)) when the var. exist (iii) E H0 (Z p (x)) E H0 (Z(x)) 0, (ii) and (iii) are deduced from Jensens inequality, cf. Royden (1968). We conjecture that in most cases (iv) Var H0 (Z(x)) Var H0 (Z p (x)). p.11/22

29 Model for Lod Score Distribution I Equation (i) is true if the following is fulfilled 1 = E H0 [e ln 10Z ] = M(ln 10) = e c ln 10 (1 a ln 10) b c = b ln(1 a ln 10) ln 10 where M(t) = E H0 (e tz ) = e ct (1 at) b. =: c(a, b), p.12/22

30 Model for Lod Score Distribution I Equation (i) is true if the following is fulfilled 1 = E H0 [e ln 10Z ] = M(ln 10) = e c ln 10 (1 a ln 10) b c = b ln(1 a ln 10) ln 10 where M(t) = E H0 (e tz ) = e ct (1 at) b. =: c(a, b), This gives us a model Z(x) H 0 free parameters: (a, b, c) with two p.12/22

31 Model for Lod Score Distribution I Equation (i) is true if the following is fulfilled 1 = E H0 [e ln 10Z ] = M(ln 10) = e c ln 10 (1 a ln 10) b c = b ln(1 a ln 10) ln 10 where M(t) = E H0 (e tz ) = e ct (1 at) b. =: c(a, b), This gives us a model Z(x) H 0 (a, b, c) with two free parameters: the strength of the genetic model 0 < a < (ln 10) 1 p.12/22

32 Model for Lod Score Distribution I Equation (i) is true if the following is fulfilled 1 = E H0 [e ln 10Z ] = M(ln 10) = e c ln 10 (1 a ln 10) b c = b ln(1 a ln 10) ln 10 where M(t) = E H0 (e tz ) = e ct (1 at) b. =: c(a, b), This gives us a model Z(x) H 0 (a, b, c) with two free parameters: the strength of the genetic model 0 < a < (ln 10) 1 the proportion to the effective number of families in data b > 0 p.12/22

33 Significance Levels for Multipoint Scores Monte Carlo method: very general but can be slow, especially for large pedigrees and low marker informativity. Can be used for both perfect and non perfect data. p.13/22

34 Significance Levels for Multipoint Scores Monte Carlo method: very general but can be slow, especially for large pedigrees and low marker informativity. Can be used for both perfect and non perfect data. Perfect marker data: use analytical methods based on Gaussian extreme value theory i.e. normal approximation and adjusted normal approximation, cf. Lander & Kruglyak (1995), and Ängquist & Hössjer (2003). p.13/22

35 Significance Levels for Multipoint Scores Monte Carlo method: very general but can be slow, especially for large pedigrees and low marker informativity. Can be used for both perfect and non perfect data. Perfect marker data: use analytical methods based on Gaussian extreme value theory i.e. normal approximation and adjusted normal approximation, cf. Lander & Kruglyak (1995), and Ängquist & Hössjer (2003). When using the analytical methods above one should use standardized lod scores, i.e. Z(x) = N Z N i=1 i i (x)/ 2 i=1 i where Z i = (Z i (x) i)/ i for family i. p.13/22

36 Pedigrees used in the study. 1) ) ICEL80002) ) unaffected affected unknown Squares: males Circles: females " " " " " " # # # # # # $ $ $ $ $ $ $ $ & & & & 4) ICEL80004) * * Monozygotic twins are replaced by a single contrived individual p.14/22

37 Results Standardized lod scores Z(x) turned out very useful when calculating p-values in parametric linkage analysis. p.15/22

38 Results Standardized lod scores Z(x) turned out very useful when calculating p-values in parametric linkage analysis. Perfect marker data: stronger genetic model/ larger pedigree/ increased sample size gives a flatter p value curve and lower p value. p.15/22

39 Results Standardized lod scores Z(x) turned out very useful when calculating p-values in parametric linkage analysis. Perfect marker data: stronger genetic model/ larger pedigree/ increased sample size gives a flatter p value curve and lower p value. Imperfect marker data: The effect of marker informativity is stronger for the larger pedigrees. p.15/22

40 Results Standardized lod scores Z(x) turned out very useful when calculating p-values in parametric linkage analysis. Perfect marker data: stronger genetic model/ larger pedigree/ increased sample size gives a flatter p value curve and lower p value. Imperfect marker data: The effect of marker informativity is stronger for the larger pedigrees. Comparisons show that the perfect marker data approximations for lod score p-values can be either conservative or anticonservative, depending on the genetic model. p.15/22

41 Summary As a practical implication of our work, we suggest, if traditional lod scores are to be used in multipoint linkage analysis, that they are reported together with p-values. Alternatively, the standardized version Z could be used. p.16/22

42 Summary As a practical implication of our work, we suggest, if traditional lod scores are to be used in multipoint linkage analysis, that they are reported together with p-values. Alternatively, the standardized version Z could be used. For imperfect data we suggest to use nonparametric scores defined in Lander & Kruglyak (1995) or the approach in Kong & Cox (1997). p.16/22

43 Summary As a practical implication of our work, we suggest, if traditional lod scores are to be used in multipoint linkage analysis, that they are reported together with p-values. Alternatively, the standardized version Z could be used. For imperfect data we suggest to use nonparametric scores defined in Lander & Kruglyak (1995) or the approach in Kong & Cox (1997). How Z compares with other scores S in terms of power and robustness to model misspecification will depend on both the true and assumed genetic model, pedigree structure(s), sample size, etc. This is certainly a topic for future research. p.16/22

44 Acknowledgements Ola Hössjer, Pär-Ola Bendahl, Adalgeir Arason, Rosa Barkardottir, Lars Ängquist, Henrik Bengtsson, James Hakim and Joakim Lübeck. p.17/22

45 Acknowledgements Ola Hössjer, Pär-Ola Bendahl, Adalgeir Arason, Rosa Barkardottir, Lars Ängquist, Henrik Bengtsson, James Hakim and Joakim Lübeck. Thank you for your attention! p.17/22

46 References Kong A and Cox NJ: Allele-Sharing Models: LOD Scores and Accurate Linkage Tests. Am J Hum Genet 1997;61: Kruglyak L, Daly MJ, Reeve-Daly MP and Lander ES: Parametric and non-parametric Linkage Analysis: A Unified Multipoint Approach. Am J Hum Genet 1996;58: Kurbašić A and Hössjer O: On Computation of P-values in Parametric Linkage Analysis. To appear in Human Heredity. Lander E and Kruglyak L: Genetic dissection of Complex Traits: Guidelines for Interpreting and Reporting Linkage Results. Nature Genetics 1995;11: p.18/22

47 Self SG and Lian KY: Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests Under Nonstandard Conditions. Journal of the American Statistical Association 1987;82: Sham P: Statistics in Human Genetics. Arnold, a Member of the Holder Headline group Royden HL: Real analysis, 2d ed. Macmillan Publishing, New York Ängquist L and Hössjer O: Improving the Calculation of Statistical Significance in Genome-Wide Scans. Preprints in Mathematical Sciences, Centre for Mathematical Sciences, Lund University p.19/22

48 p.20/22 Appendix

49 Model for Lod Score Distribution II The addition of the lod score stays in the model, i.e. Z = Z 1 + Z 2 H 0 (a, b 1 + b 2, c(a, b 1 + b 2 ) The expectation under the H 0 0 = E H0 (Z) = c + ab = b(a + ln(1 a ln 10) ln 10 ) < 0 Further: 0 = a b 1 = ELOD = E H1 (Z) = zdp H1 (z) = zlr(z)dp H0 (z) = ze ln 10z dp H0 (z) = E H0 (Ze ln 10Z ) = M (ln 10) = M(ln 10)(c + ab ln(1 a ln 10) a 1 a ln 10 ) = b( ln a ln 10 ) p.21/22

50 Model for Lod Score Distribution III Imperfect Data: 1 Let a (0, ln 10 ) be a genetic model A weakened genetic model is a, 0 < 1 Equations (ii) (iv) must always be fullfilled. It follows from the fact that Z H 0 ( a, b, c( a, b)) Var H0 (LR) = E H0 (LR 2 ) E H0 (LR) 2 = E H0 (e 2 ln 10Z ) 1 2 = M Z (2 ln 10) 1 = e c( a,b)2 ln 10 (1 2 a ln 10) b 1 = (1 a ln 10) 2b (1 2 a ln 10) b 1 < (1 a ln 10) 2b (1 2a ln 10) b 1 = V H0 (LR p ) This assumes that both variances exist, that is 0 < a < (2 ln 10) 1. In the inequality we used V H0 (LR) = (1 x) 2b /(1 2x) b 1, where x = ln 10, which is an increasing function of x. p.22/22