On Biostatistical Genetics

Transcription

1 On Biostatistical Genetics Using Twin Data Jacob Hjelmborg University of Southern Denmark Fall 2013 Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

2 Overview 1 Introduction 2 Heritability Heuristics and History 3 Opposite versus Same Sexed Pairs 4 The Matched Case-Cotwin Design and Analysis 5 On Epigenetics and Information Theory in Twin Studies 6 Examples Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

3 The Danish Twin Registry The oldest twin registry Founded in 1953 by doctors Tage Kemp, Mogens Hauge and Bent Harvald. Cancer among initial focuses. Rockefeller Foundation and NIH twins born from 1870 till now. Population based with more than 97% completeness. Linked to National Registries.

4 The Danish Twin Registry

5 The Danish Twin Registry

6 Twin Symposium 2013

7 BRCA gene discoverer Mary-Claire King Pioneer in research into hereditary breast cancer: New Scientist June 2013 Supreme court is right - the BRCA1 gene cannot be patented. Portrayed in forthcoming film Decoding Annie Parker. Tales of a Minstrel Geneticist - Title of this year s Hans Christian Andersen lecture at the University of Southern Denmark, Thursday November 14th.

8 Longitudinal BMI Mean lnbmi Males Females Mean age Body Mass Index US Twin Study 1997: Genetic influences on changes in body mass index: a longitudinal analysis of women twins. (Mary-Claire King) Finnish Twin Study 2007: Genetic influences on growth traits of BMI: a longitudinal study of adult twins. modeling change: BMI(t) i = α i + β i t See K. Holst Day1-estimation slide: Longitudinal Biometric Analysis

9 Overview 1 Introduction 2 Heritability Heuristics and History 3 Opposite versus Same Sexed Pairs 4 The Matched Case-Cotwin Design and Analysis 5 On Epigenetics and Information Theory in Twin Studies 6 Examples Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

10 What is heritability? Darwin: Origin of the Species (1859) The inherited traits form the basis for natural Selection, Mutation and Replication. The inherited is a blend

11 Mendel: Experiments in Plant Hybridization (1865) The inherited is a fifty-fifty mix of discrete units. Mendel s First Law (Segregation): One allele of each parent is randomly and independently selected, with probability 1 2, for transmission to the offspring. The alleles unite randomly to form the offspring s genotype. Mendel s Second Law (Independent Assortment): The alleles underlying two or more different traits are transmitted to offspring independently of each other. The transmission of each trait separately follows the first law of segregation. How can Mendel s discrete units explain the variation in continuous human characteristics?

12 Statistical genetics How is variation at phenotypic level governed by variation at genetic level? Two structures for modelling: mean and variance-covariance. R.A. Fisher (1918): The variance-covariance matrix varies by type of twin pairs. ( ) variance of first twin covariance of twins Σ = covariance of twins variance of second twin -a measure of twin similarity: ρ, correlation (K. Pearson 1904)

13 Aims Difference in correlations between MZ and DZ twins suggests genetic influence on trait. What type and magnitude of genetic and environmental influences to expect? Classical twin analysis using the polygenic model, known as the ADCE-model, in which the individual outcome, Y i decomposes into Y i = A i + D i + I i + C i + E i, where A: Additive genetic effects of alleles D: Dominant genetic effects I: Epistasis genetic effects C: Shared environmental effects E: Unique environmental effects Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

14 Biometric analyses - polygenic model (Falconer 1981) Contributing factors to the variation in outcome: ( ) ) σ 2 Σ Y = A zσa 2 σ 2 +( D uσ 2 zσa 2 σa 2 D σ 2 +( I kσ 2 uσd 2 σd 2 I kσi 2 σi 2 ) σ 2 +( C σc 2 σc 2 σc 2 ) +( σ 2 E 0 where z = u = k = 1 for MZ pairs, z = 1 2, u = 1 4 and 0 k 1 4 for DZ pairs. In particular, we obtain Heritability: HY 2 σa 2 = + σ2 D + σ2 I σa 2 + σ2 D + σ2 I + σc 2 + σ2 E = h 2 + d 2 + i 2 Shared environmental effect: σ 2 C CY 2 = σa 2 + σ2 D + σ2 I + σc 2 + σ2 E Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

15 H 2 Heuristics { ρmz = h 2 + d 2 + i 2 + c 2 ρ DZ = 1 2 h d 2 + ki 2 + c 2 Hence ρ MZ ρ DZ H 2 min{ρ MZ, 2(ρ MZ ρ DZ )} If no epistasis, { ρmz = h 2 + d 2 + c 2 ρ DZ = 1 2 h d 2 + c 2 we obtain 4 3 (ρ MZ ρ DZ ) H 2 min{ρ MZ, 2(ρ MZ ρ DZ )} Heritability propositions H 2 = 0 always if ρ MZ = ρ DZ, hence existence of genetic effects can be detected solely from correlations and is robust towards misspecification of genetic model, nice! H 2 differs by at most ρ MZ ρ DZ from true value.

16 Correlation heuristics Within pair similarity is measured by correlations. Correlations are further modelled by genetic and environmental variance components via the polygenic ADCE model. For instance, the polygenic ACE model relates to correlations via ρ mz = h 2 + c 2 and ρ dz = 1 2 h2 + c 2. Heuristics of MZ and DZ correlations Interpretation Relation Genetics Environment Examples ρmz > 4ρ dz Epistasis albinism ρmz > 2ρ dz Genetic dominance D ρmz = 2ρ dz Additive effect A (mono- or polygenic) and small D Small C BMI 2ρ dz > ρmz > ρ dz Additive genes A Shared environment C longevity ρmz = ρ dz > 0 No genetic effect C ρmz = ρ dz = 0 No genetic effect No familial aggregation

17 Biometric analyses - polygenic model Main assumptions Equal environments assumption for MZ and DZ twins. No gene-environment interaction and correlation. No gene-gene interaction (link: epistasis). Equal mean and variance of twin 1 and twin 2, MZ and DZ. Estimation and inference by maximum likelihood principle assuming bivariate normality of paired observations (as before) ext$logbmi.2[ext$zyg == "MZ"] ext$logbmi.2[ext$zyg == "DZ"]

18 Overview 1 Introduction 2 Heritability Heuristics and History 3 Opposite versus Same Sexed Pairs 4 The Matched Case-Cotwin Design and Analysis 5 On Epigenetics and Information Theory in Twin Studies 6 Examples Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

19 Sex-limitation model to include OS DZ s Should we include opposite sexed DZ s? Are the same genes in males and females affecting the trait of interest? (genetic pleiotropy) -howto? Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

20 Example Height difference between men and women Men are taller than women. Is this because of genetic differences? Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

21 Height difference between men and women Within pair correlations for same and opposite sex DZ twins Within pair correlation country DZ males DZ females OS twins Australia Denmark Finland (old) Finland (young) Italy The Netherlands Norway source: Silventoinen et al., Twin Res 2000 and 2003 There is no significant difference between correlations within SSDZ and OSDZ twin pairs There seems to be no sex-specific genetic factor affecting height. Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

22 Sex-limitation model to include OS DZ s Studying gene gender interaction. Making use of all DZ pairs. The within-pair covariance of male and female twins: Cov(male,female) = gσ Am σ Af + σ Cm σ Cf Will g = 1 2 as for DZ same sex pairs? Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

23 Sex-limitation model to include OS DZ s DZSS Males, Σ DZSS = ( σ 2 Am + σc 2 m + σe 2 m 1 2 σ2 A m + σc 2 m 1 2 σ2 A m + σc 2 m σa 2 m + σc 2 m + σe 2 m ) DZSS Females, Σ DZSS = ( σ 2 Af + σc 2 f + σe 2 f 1 2 σ2 A f + σc 2 f 1 2 σ2 A f + σc 2 f σa 2 f + σc 2 f + σe 2 f ) DZOS, Σ DZOS = ( σ 2 Am + σ 2 C m + σ 2 E m gσ Am σ Af + σ Cm σ Cf gσ Am σ Af + σ Cm σ Cf σ 2 A f + σ 2 C f + σ 2 E f ) Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

24 Sex-limitation model to include OS DZ s We will encounter more examples as we go along. A value of g = 1 2 for OS twins suggest no gender specific genetic effect. -a lower value will indicate such effects. R package mets estimates g when OS pairs are available. See K. Holst Day1-estimation slide: Gene-Environment Interactions. Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

25 Overview 1 Introduction 2 Heritability Heuristics and History 3 Opposite versus Same Sexed Pairs 4 The Matched Case-Cotwin Design and Analysis 5 On Epigenetics and Information Theory in Twin Studies 6 Examples Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

26 Review of classic twin analysis aims and methodology Inferring genetic influence without observing any gene. What is the contribution of genetic and environmental factors to the variation? Are the same or different genes influencing the traits? { Y = Genes + Environment Σ Y = Σ Genes + Σ Environment However, there are other fruitful use of collections of twin pairs. SEE SLIDES: "Day02caseCotwin.pdf.

27 Overview 1 Introduction 2 Heritability Heuristics and History 3 Opposite versus Same Sexed Pairs 4 The Matched Case-Cotwin Design and Analysis 5 On Epigenetics and Information Theory in Twin Studies 6 Examples Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

28 What is dependence? Aim Twin studies may provide a frame for genetic analysis and etiology. For instance in the etiology of cancer. A measure of dependence becomes utterly important. Correlation F. Galton (1890) I can only say that there is a vast field of topics that fall under the laws of correlation, which lies quite open to the research of any competent person who cares to investigate it

29 Pearson 1904 The product moment correlation E{(X µ)(y µ)} Measures linear dependence MZ time.2 time DZ time.2 time.1 R.A. Fischer 1918 Heritability in terms of correlations. Genetic variation causing phenotypic variation

30 Variance explained by SNPs in metabonomic traits

31 Addendum Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

32 Agouti MZ mice

33 Equal but not the same The Epigenetics Revolution Controlled reading of DNA Determines function of cell May change during life Transgenerational inheritance(!) (Lamarckism?). Ovrekalix, Dutch famine, Obesity mice,...

34 Equal but not the same Epigenetics makes monozygotic twins different from first replication key in evolutionary dynamics of cancer Schewig et. al., Max Planck Berlin, 2013

35 Equal but not the same In parallel: Evolutionary dynamics Evolutionary dynamics: Replication, Mutation and Selection. Statistical genetics governs mostly Mutation and Selection. Epigenetics: Replication of genetic Information.

36 Epigenetic treatment of acute leukemia by 5-aza-cytidine

37 Equal but not the same Epigenetics in Twin Studies 1 Genetics of Epigenetics (genetic influences on epigenetic effects) 2 Epigenetic Epidemiology (case-cotwin discordance) Aim Heritability of complex traits? -for instance taking epigenetic phenomena into account. In general: non-mendelian inheritance?, genetic variation with time?

38 MZ Nokia mobile phones What is the conveyed information between X and Y?

39 Information Theory The Mutual Information I(X, Y ) = f XY (x, y) log f XY (x, y) f X (x)f Y (y) dxdy

40 E. H. Linfoot 1957 Theorem: E.H. Linfoot (1957) For (X, Y ) F, ν(x, Y ) = (1 exp( 2I(X, Y ))) where I(X, Y ) = f XY (x, y) log f XY (x,y) f X (x)f Y (y) dxdy is the mutual information, satisfies the seven correlation properties of Renyi. Renyi properties Independence if and only if ν(x, Y ) = 0 Strict dependence if and only if ν(x, Y ) = 1 Bivariate Gaussian: Linfoot is ρ(x, Y ) Invariance under continuous and strictly increasing marginal transformations

41 Information The Informational Correlation The Linfoot transform of the Mutual Information is bridging Information theory with Statistics.

42 Generalizing heritability (joint with A. Kryger Jensen) We propose: The informational heritability coefficient Properties h 2 = 2(L MZ (X, Y X MZ ) L DZ (X, Y X DZ )) L(X, Y ) = 1 exp( 2I(X, Y )) (Linfoot s correlation) f (x, y) I(X, Y ) = log df(x, y) (Mutual Information) f (x)f (y) Marginal densities may be constrained to be exchangeable within pairs and equal for MZ and DZ types of pairs Proposition: h 2 generalizes classic Falconer h 2 for bivariate normals h 2 is measuring proportion of density explained by genetic variations transmitted to phenotypic density patterns. topology may be rather complex.

43 Perspectives Epigenetics: Key in evolutionary dynamics of cancer. Evidence for transgenerational inheritance. Calls for more general heritability measure. Linfoot s correlation is based on the mutual information and fulfills Renyi properties.

44 Overview 1 Introduction 2 Heritability Heuristics and History 3 Opposite versus Same Sexed Pairs 4 The Matched Case-Cotwin Design and Analysis 5 On Epigenetics and Information Theory in Twin Studies 6 Examples Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

45 Example 1: Bivariate Gaussian simulation 500 simulations with 300 MZ and 300 DZ pairs Twinlm Falcon h*

46 Agouti MZ mice

47 simulation MZ DZ 500 simulations with 300 MZ and 300 DZ pairs Twin Twin Twin Twin 1 Twinlm Falcon Kendall Spearman h*

48 1.5 Albumin - h 2 = 0 (Bathum et al. 2004) MZ Albumine levels Co Twin Twin

49 Albumin - h 2 = 0 (Bathum et al. 2004) DZ Albumine levels Co Twin Twin

50 Albumin - h 2 = 0 (Bathum et al. 2004) Proposed heritability for general associations L MZ = 0.47 L DZ = 0.30 h 2 = estimation needs to be validated!

51 Time to event twin data Main objectives: Time-varying genetic influence Taking censoring and competing risks into account, Major component in NorTwinCan Study joint with

52 The Nordic Twin Cancer Study Nordic twins linked with national cancer registries. 42 common cancer sites: 1 "Lip"2 "Tongue"3 "Salivary glands"4 "Mouth"5 "Pharynx"6 "Oesophagus"7 "Stomach"8 "Small intestine"9 "Colon"10 "Rectum and anus"11 "Liver"12 "Gallbladder and extrahepatic bile ducts"13 "Pancreas"14 "Nose, sinuses"15 "Larynx"16 "Lung (incl. trachea and bronchus)"17 "Pleura"18 "Breast"19 "Cervix uteri"20 "Corpus uteri"21 "Uterus, other"22 "Ovary and uterine adnexa"23 "Other female genital organs"24 "Prostate"25 "Testis"26 "Penis and other male genital organs"27 "Kidney"28 "Bladder and other and unspecified urinary organs"29 "Melanoma of skin"30 "Skin, non-melanoma"31 "Eye"32 "Brain, central nervous system"33 "Thyroid"34 "Bone"35 "Soft tissues"36 "Non-Hodgkin lymphoma"37 "Hodgkin s disease"38 "Multiple myeloma"39 "Acute leukaemia"40 "Other leukaemia"41 "All sites but non-melanoma skin"42 "All sites - first diagnosis"60 "Rectum"61 "Colorectal"

53 Multiple outcome at time t. α 01 (t) dead alive α 02 (t) prostate cancer Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

54 Cumulative incidence Cumulative Incidence Denmark Finland Norway Sweden Age The cumulative incidence function: t F cancer(t) = Prob(T t, ɛ = cancer) = λ cancer(s)s(s )ds, 0 NB! Same prostate cancer mortality (Bray et al. Eur Jour Cancer 2010).

55 Concordance Scheike et al. 2013: Semi-parametric random effects regression model of the cumulative incidence function. The concordance function conditional on covariates P(T 1 t, ɛ 1 = cancer, T 2 t, ɛ 2 = cancer X), Dependence in terms of: Probandwise concordance, relative recurrence risk, cross-odds ratio and multilocus index. Alternative frailty approach on hazard scale by F. Eriksson (2013). Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

56 Prostate cancer - Nordic - Concordance Probandwise concordance concordance MZ pairs DZ pairs Independence Age Methods: Estimating heritability for cause specific mortality based on twin studies. Scheike, Holst and Hjelmborg, LIDA (2013).

57 Bivariate extreme value theory -joint with J. Kaprio, Y. Goegebeur and M. Osmann. M. Osmann receives "Dansk Matematisk Forenings specialepris for 2012" Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall / 63

58 Extreme value theory for twin data Obesity Evidence for epigenetic effects governing extreme BMI (Obesity). Twin studies can provide a frame for genetic analysis and etiology. Modelling the dependence of extremes becomes utterly important Age BMI

59 BMI density BMI density Upper tail copulas Co Twin Co Twin Twin Twin

60 Dependence measures for upper tail N η χ χ L(X, Y ) h 2 MZ Overall Men Men (Age 18-29) Men (Age 30-39) Men (Age 40-60) DZ Overall Men Men (Age 18-29) Men (Age 30-39) Men (Age 40-60)

61 Practical: Estimating upper tail dependence measures This session illustrates how to estimate the upper tail dependence measure χ = lim u 1 Prob(U 1 u U 2 u) from the classical bivariate extreme value distribution. Bivariate Extreme Value Distribution Lauch R-code on next slide to obtain an estimate of χ Estimate χ for MZ and for DZ pairs. Reference: Coles S, "An Introduction to Statistical Modeling of Extreme Values", Springer (2001).

62 Fitting Bivariate Peaks Over a Threshold Using Bivariate Extreme Value Distributions library(pot) library(mets) data(twinbmi) # a number, 1 or 2, is assigned to each twin in a pair. twinbmi$nr <- with(twinbmi, ave(bmi,tvparnr, FUN = seq_along)) twinbmi$logbmi <- log(twinbmi$bmi) bmiwide <- reshape(twinbmi,idvar="tvparnr",timevar="nr", v.names=c("bmi","logbmi","age"), direction="wide") th = quantile(twinbmi$bmi,probs=0.80, na.rm=true) extlog <- fitbvgpd(cbind(bmiwide$bmi.1,bmiwide$bmi.2),c(th,th), cshape=true, cscale=true,model = "log") extlog$chi

63 The R kiosk mets (K. Holst, T. Scheike) etm (A. Allignol); prodlim (T. Gerds).