Using Twin Data Jacob Hjelmborg - staff.pubhealth.ku.dkstaff.pubhealth.ku.dk/~kkho/undervisning/family2013/biostatgen.pdf · On Biostatistical Genetics Using Twin Data Jacob Hjelmborg

On Biostatistical GeneticsUsing Twin Data

Jacob Hjelmborg

University of Southern Denmark

Fall 2013

Biostatistics (Institute of Public Health) On Biostatistical Genetics Fall 2013 1 / 63

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


The Danish Twin RegistryThe oldest twin registry

Founded in 1953 by doctors Tage Kemp, Mogens Hauge and BentHarvald.Cancer among initial focuses. Rockefeller Foundation and NIH∼100.000 twins born from 1870 till now.Population based with more than 97% completeness.Linked to National Registries.

The Danish Twin Registry

The Danish Twin Registry

Twin Symposium 2013

BRCA gene discoverer

Mary-Claire KingPioneer in research into hereditary breast cancer: New Scientist June2013

’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 ChristianAndersen lecture at the University of Southern Denmark, ThursdayNovember 14th.

http://www.slate.com/articles/health_and_science/new_scientist/2013/06/brca_gene_discovery_mary_claire_king_says_the_supreme_court_is_right_not.html

http://www.slate.com/articles/health_and_science/new_scientist/2013/06/brca_gene_discovery_mary_claire_king_says_the_supreme_court_is_right_not.html

http://www.sdu.dk/en/Om_SDU/Aarets_gang/HCA+Academy/2013+-+Mary-Claire+King

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

Males

Females

3.0

53.1

3.1

53.2

3.2

5M

ean lnB

MI

15 20 25 30 35 40 45 50 55 60 65Mean age

Longitudinal BMI

Body Mass IndexUS 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: alongitudinal study of adult twins.modeling change: BMI(t)i = αi + βi tSee K. Holst ’Day1-estimation’ slide: Longitudinal Biometric Analysis

http://www.ncbi.nlm.nih.gov/pubmed/9285839




Overview

1 Introduction





6 Examples


What is heritability?

Darwin: Origin of the Species (1859)The inherited traits form the basis for natural Selection, Mutation andReplication.

The inherited is a blend

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 randomlyand independently selected, with probability 1

2 , for transmission to theoffspring. The alleles unite randomly to form the offspring’s genotype.Mendel’s Second Law (Independent Assortment): The alleles underlyingtwo or more different traits are transmitted to offspring independently ofeach other. The transmission of each trait separately follows the first lawof segregation.

How can Mendel’s discrete units explain the variation in continuous humancharacteristics?

Statistical geneticsHow 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 twinpairs.

Σ =

(variance of first twin covariance of twinscovariance of twins variance of second twin

)-a measure of twin similarity: ρ, correlation (K. Pearson 1904)

Aims

Difference in correlations between MZ and DZ twins suggests geneticinfluence on trait.What type and magnitude of genetic and environmental influences toexpect?Classical twin analysis using the polygenic model, known as theADCE-model, in which the individual outcome, Yi decomposes into

Yi = Ai + Di + Ii + Ci + Ei ,

whereI A: Additive genetic effects of allelesI D: Dominant genetic effectsI I: Epistasis genetic effectsI C: Shared environmental effectsI E : Unique environmental effects


Biometric analyses - polygenic model (Falconer 1981)

Contributing factors to the variation in outcome:

ΣY =

(σ2

A zσ2A

zσ2A σ2

A

)+

(σ2

D uσ2D

uσ2D σ2

D

)+

(σ2

I kσ2I

kσ2I σ2

I

)+

(σ2

C σ2C

σ2C σ2

C

)+

(σ2

E 00 σ2

E

)where z = u = k = 1 for MZ pairs, z = 1

2 , u = 14 and 0 ≤ k ≤ 1

4 for DZ pairs.

In particular, we obtainHeritability:

H2Y =

σ2A + σ2

D + σ2I

σ2A + σ2

D + σ2I + σ2

C + σ2E

= h2 + d2 + i2

Shared environmental effect:

C2Y =

σ2C

σ2A + σ2

D + σ2I + σ2

C + σ2E


H2 Heuristics {ρMZ = h2 + d2 + i2 + c2

ρDZ = 12 h2 + 1

4 d2 + ki2 + c2

HenceρMZ − ρDZ ≤ H2 ≤ min{ρMZ ,2(ρMZ − ρDZ )}

If no epistasis, {ρMZ = h2 + d2 + c2

ρDZ = 12 h2 + 1

4 d2 + c2

we obtain43

(ρMZ − ρDZ ) ≤ H2 ≤ min{ρMZ ,2(ρMZ − ρDZ )}

Heritability propositionsH2 = 0 always if ρMZ = ρDZ , hence existence of genetic effects can bedetected solely from correlations and is robust towards misspecificationof genetic model, nice!H2 differs by at most ρMZ − ρDZ from true value.

Correlation heuristics

Within pair similarity is measured by correlations.Correlations are further modelled by genetic and environmental variancecomponents via the polygenic ADCE model.For instance, the polygenic ACE model relates to correlations viaρmz = h2 + c2 and ρdz = 1

2 h2 + c2.

Heuristics of MZ and DZ correlationsInterpretation

Relation Genetics Environment Examplesρmz > 4ρdz Epistasis albinismρmz > 2ρdz Genetic dominance Dρmz = 2ρdz Additive effect A (mono- or polygenic) and small D Small C BMI2ρdz > ρmz > ρdz Additive genes A Shared environment C longevityρmz = ρdz > 0 No genetic effect Cρmz = ρdz = 0 No genetic effect No familial aggregation

Biometric analyses - polygenic modelMain 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 assumingbivariate normality of paired observations (as before).

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

● ●●

● ●

●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

● ●

●

●●●

●

●

●

●

● ●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

2.8 3.0 3.2 3.4 3.6

2.8

3.0

3.2

3.4

3.6

ext$logbmi.1[ext$zyg == "MZ"]

ext$

logb

mi.2

[ext

$zyg

==

"MZ"

]

●

●

●

●

●

●

● ●●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

● ●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

● ●

●●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

● ●

●●

● ●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●●

●●

●

●●

●●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

● ●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●●

●

●●

●

●

● ●

●

●●

●

●● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●● ●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●●

●

●

●

●

●

●

●●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●● ●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

2.8 3.0 3.2 3.4 3.6

2.8

3.0

3.2

3.4

3.6

ext$logbmi.1[ext$zyg == "DZ"]

ext$

logb

mi.2

[ext

$zyg

==

"DZ"

]

http://hmg.oxfordjournals.org/cgi/reprint/11/20/2463

Overview

1 Introduction





6 Examples


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?


Example

Height difference between men and womenMen are taller than women.Is this because of genetic differences?


Height difference between men and women

Within pair correlations for same and opposite sex DZ twins

Within pair correlationcountry DZ males DZ females OS twinsAustralia 0.42 0.49 0.46Denmark 0.47 0.55 0.50Finland (old) 0.50 0.49 0.49Finland (young) 0.53 0.53 0.49Italy 0.57 0.49 0.30The Netherlands 0.47 0.49 0.43Norway 0.49 0.49 0.44

source: Silventoinen et al., Twin Res 2000 and 2003

There is no significant difference between correlations within SSDZ andOSDZ twin pairsThere seems to be no sex-specific genetic factor affecting height.



Studying gene×gender interaction.Making use of all DZ pairs.

The within-pair covariance of male and female twins:

Cov(male,female) = gσAmσAf + σCmσCf

Will g = 12 as for DZ same sex pairs?



DZSS Males,

ΣDZSS =

(σ2

Am+ σ2

Cm+ σ2

Em

12σ

2Am

+ σ2Cm

12σ

2Am

+ σ2Cm

σ2Am

+ σ2Cm

+ σ2Em

)DZSS Females,

ΣDZSS =

(σ2

Af+ σ2

Cf+ σ2

Ef

12σ

2Af

+ σ2Cf

12σ

2Af

+ σ2Cf

σ2Af

+ σ2Cf

+ σ2Ef

)DZOS,

ΣDZOS =

(σ2

Am+ σ2

Cm+ σ2

EmgσAmσAf + σCmσCf

gσAmσAf + σCmσCf σ2Af

+ σ2Cf

+ σ2Ef

)



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.


Overview

1 Introduction





6 Examples


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 thevariation?Are the same or different genes influencing the traits?{

Y = Genes + EnvironmentΣY = ΣGenes + ΣEnvironment

However, there are other fruitful use of collections of twin pairs.

SEE SLIDES: "Day02caseCotwin.pdf“.

http://www.biostat.sdu.dk/courses/twincourse/Day02caseCotwin.pdf

Overview

1 Introduction





6 Examples


What is dependence?

AimTwin studies may provide a frame for genetic analysis and etiology.For instance in the etiology of cancer.A measure of dependence becomes utterly important.

CorrelationF. Galton (1890)

’I can only say that there is a vast field of topics that fall underthe laws of correlation, which lies quite open to the research ofany competent person who cares to investigate it’

Pearson 1904The product moment correlation

E{(X − µ)(Y − µ)}Measures linear dependence.

●●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●● ●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●● ●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

● ● ●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

● ●

●

●

●

●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

● ●●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●● ●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

● ●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●●

●

●

●

●

●

●

●

●

●

●●●

●

●●

●

●

● ●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●●

●

●

●

●

●

●●

●

●●●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ● ●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●● ●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

● ●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●● ●

●●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

● ●

●

●

●●

●●

●●

●

●

●

●

●●

●

●●

●

●

●

●●

●

● ●

●

●

●

●

●

●

20 40 60 80 100

2040

6080

100

MZ

time.2

time.

1

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●● ●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●●

● ●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

● ●

● ●

●●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

● ●

●

●

● ●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●●

●

● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

● ●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

● ●●

●

●

●●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

● ●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●● ●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●

● ●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●

●●

●●

●

●

●

●●

●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●●

●

●●

●

●● ●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

● ●●

●

●●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●● ●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

● ●●

●

●

●●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●●

●

●

●●

●

●

● ●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

● ●

●

●●

●

●●

●

●

●

● ●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●●

●

●●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●●

●

●

●

●

● ● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●● ●

●●●

●

●

●

●

● ●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●●

● ●●

●

●●

●

●

●

●

●

●

●

● ●

●

●

●●

●●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

0 20 40 60 80 100

2040

6080

100

DZ

time.2

time.

1

R.A. Fischer 1918Heritability in terms of correlations.Genetic variation causing phenotypic variation

Variance explained by SNPs in metabonomic traits

Addendum


Agouti MZ mice

Equal but not the same

The Epigenetics RevolutionControlled reading of DNADetermines function of cellMay change during lifeTransgenerational inheritance(!) (Lamarckism?).Ovrekalix, Dutch famine, Obesity mice, . . .


Epigeneticsmakes monozygotic twins different from first replicationkey in evolutionary dynamics of cancer

Schewig et. al. , Max Planck Berlin, 2013


In parallel: Evolutionary dynamicsEvolutionary dynamics: Replication, Mutation and Selection.Statistical genetics governs mostly Mutation and Selection.Epigenetics: Replication of genetic Information.

Epigenetic treatment of acute leukemia by5-aza-cytidine


Epigenetics in Twin Studies1 Genetics of Epigenetics (genetic influences on epigenetic effects)2 Epigenetic Epidemiology (case-cotwin discordance)

AimHeritability of complex traits?-for instance taking epigenetic phenomena into account.In general: non-Mendelian inheritance?, genetic variation with time?

’MZ Nokia mobile phones’

What is the conveyed information between X and Y?

Information Theory

The Mutual Information

I(X ,Y ) =

∫∫fXY (x , y) log

fXY (x , y)fX (x)fY (y)

dxdy

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 ) =∫∫

fXY (x , y) log fXY (x,y)fX (x)fY (y)

dxdy is the mutual information,satisfies the seven correlation properties of Renyi.

Renyi propertiesIndependence if and only if ν(X ,Y ) = 0Strict dependence if and only if ν(X ,Y ) = 1Bivariate Gaussian: Linfoot is |ρ(X ,Y )|Invariance under continuous and strictly increasing marginaltransformations

Information

The Informational CorrelationThe Linfoot transform of the Mutual Information is bridging Informationtheory with Statistics.

Generalizing heritability (joint with A. Kryger Jensen)We propose:

The informational heritability coefficient

h2∗ = 2(LMZ(X ,Y | XMZ)− LDZ(X ,Y | XDZ))

L(X ,Y ) =√

1− exp(−2I(X ,Y )) (Linfoot’s correlation)

I(X ,Y ) =

∫∫log

f (x , y)

f (x)f (y)dF (x , y) (Mutual Information)

PropertiesMarginal densities may be constrained to be exchangeable within pairsand equal for MZ and DZ types of pairsProposition: h2

∗ generalizes classic Falconer h2 for bivariate normalsh2∗ is measuring proportion of density explained by genetic variations

transmitted to phenotypic density patterns.topology may be rather complex.

Perspectives

Epigenetics: Key in evolutionary dynamics of cancer. Evidence fortransgenerational inheritance.Calls for more general heritability measure.Linfoot’s correlation is based on the mutual information and fulfills Renyiproperties.

Overview

1 Introduction





6 Examples


Example 1: Bivariate Gaussian simulation

Twinlm Falcon h*

0.0

0.2

0.4

0.6

0.8

1.0

500 simulations with 300 MZ and 300 DZ pairs

Agouti MZ mice

simulation

-3 -2 -1 0 1 2 3

-3-2

-10

12

3

MZ

Twin 1

Twin

2

-4 -2 0 2 4

-4-2

02

4

DZ

Twin 1

Twin

2

Twinlm Falcon Kendall Spearman h*

0.0

0.5

1.0

500 simulations with 300 MZ and 300 DZ pairs

Albumin - h2 = 0 (Bathum et al. 2004)

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

MZ Albumine levels

Twin

Co−

Twin

0.5

0.5

1

1.5

1.5

1.5

2


0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

DZ Albumine levels

Twin

Co−

Twin

0.5

0.5 1

1.5

1.5

2


Proposed heritability for general associationsLMZ = 0.47LDZ = 0.30

h2∗ = 0.34

-estimation needs to be validated!

Time to event twin dataMain objectives:

I Time-varying genetic influenceI Taking censoring and competing risks into account,

Major component in NorTwinCan Study joint with

The Nordic Twin Cancer Study

∼ 400.000 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 uterineadnexa"23 "Other female genital organs"24 "Prostate"25 "Testis"26 "Penisand other male genital organs"27 "Kidney"28 "Bladder and other andunspecified urinary organs"29 "Melanoma of skin"30 "Skin, non-melanoma"31"Eye"32 "Brain, central nervous system"33 "Thyroid"34 "Bone"35 "Softtissues"36 "Non-Hodgkin lymphoma"37 "Hodgkin’s disease"38 "Multiplemyeloma"39 "Acute leukaemia"40 "Other leukaemia"41 "All sites butnon-melanoma skin"42 "All sites - first diagnosis"60 "Rectum"61 "Colorectal"

Multiple outcome at time t .

alive

dead

prostate cancer

α01(t)

α02(t)


Cumulative incidence

40 50 60 70 80 90 100

0.00

0.05

0.10

0.15

Age

Cum

ulat

ive

Inci

denc

e

DenmarkFinlandNorwaySweden

The cumulative incidence function:

Fcancer(t) = Prob(T ≤ t , ε = cancer) =∫ t

0λcancer(s)S(s−)ds,

NB! Same prostate cancer mortality (Bray et al. Eur Jour Cancer 2010).

Concordance

Scheike et al. 2013: Semi-parametric random effects regression model ofthe cumulative incidence function.The concordance function conditional on covariates

P(T1 ≤ t , ε1 = cancer,T2 ≤ t , ε2 = cancer|X ),

Dependence in terms of: Probandwise concordance, relative recurrencerisk, cross-odds ratio and multilocus index.Alternative frailty approach on hazard scale by F. Eriksson (2013).


Prostate cancer - Nordic - Concordance

60 70 80 90 100

0.0

0.1

0.2

0.3

0.4

0.5

Age

conc

orda

nce

60 70 80 90 100

0.0

0.1

0.2

0.3

0.4

0.5

Age

conc

orda

nce

Probandwise concordance

MZ pairsDZ pairsIndependence

Methods: Estimating heritability for cause specific mortality based on twin studies. Scheike,Holst and Hjelmborg, LIDA (2013).

Bivariate extreme value theory

-joint with J. Kaprio, Y. Goegebeur and M. Osmann.M. Osmann receives "Dansk Matematisk Forenings specialepris for 2012"


Extreme value theory for twin dataObesity

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.

●●

●

●

●●

●

●

●

● ●

● ●

●

●●

●

●● ●

● ●● ●

●●

●

●

●●

●●

●●

●

●

● ●

●

●

●

● ●

● ●

●●

●

●●

●

● ●● ●

● ●

●

● ● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

● ● ●

●

●

●

● ●●

●

●

●

●

●

●●

●

●●

●● ●

● ●

●

●●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

●●

●

● ●● ●

●

●

●

●●●

●

●

●

●

●●●

●● ●

●●

●●

●

●

●

●

●

●

●

●

● ● ●

●

●

●

●●

●

● ● ●●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

● ●

● ●

●●

●

●

●

●

● ● ● ●●

●●

●

●

●

●

●

●

●

●

●● ●

●

● ●●

●

●

●

● ●

●

●

●●

●●

●●

●

●

●●

●

●

●

● ●

●●

● ●

●

●

●

●

●

●

●

●

●

● ●

● ●

●

●

●●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●●

●● ●

●

●●

●●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

● ●

● ●

●

●

●

●●

● ●

●

●●

●

●

● ●●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●●

●

●●

●

● ●

●

●

●●● ●

●●

● ●

●

● ●

●●

● ●

●

●●

● ●

●●

●

●

●

●

●

●

●●

● ● ● ●

●

●

●

● ● ● ●

●

●

●

● ●

●

●

●● ● ● ● ● ●

● ●●

●

● ●

●

● ●

● ●

● ●

●

●

● ●

●

●

● ●

●●

●●

●

●●

●

●

●

●

●● ●

●● ●

●

●●

●

●●

●●

● ●

●

●

● ● ● ●●

●

●

●●

●

● ●

●

●

●●

●

●●

●

●

●

●

●

● ●

●

●

●

● ●●● ●

●

●●

●●

●

●

●

●

●

●

● ●

●●

●●

●

●

●●

●

●

● ●

●

● ●

●

● ● ●● ●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●

● ●

● ●

●

●

● ●

●

●●

●

●

●

●

●

●

●

●

●●

●●

● ●

●

●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ● ●

●

● ●●

●

● ●

●

● ●

●

●

● ●

●

● ● ● ●●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●●

● ●

● ●●

●●

● ●

● ●

●●

●

●

● ●

●

●

●

●

●

●

● ●

●

●● ●

● ●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

● ●● ● ●

● ●

●

●● ●

●●

●●

●

●

●

●

●

●

●

●

●

●●

●

● ● ● ●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●●

●●

●

●

●

● ●

●

●●

●

●

●

● ●

●

●

●

●

● ●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

● ●

●

●

●

●●

●●

●●

●

●

●●

● ●

●●

● ● ●

●

● ●●

●

●

●●

●

● ●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

● ●

●

● ●

● ● ●

●●

●

●●

●

●

●

●

●●

●●

●●

●

●●

●

●●

●●

●

●●

●

●

●

●

●

●

●●

● ●

● ●

●●

●

●

●

●

●

●

●

●

●

● ●

●

● ●

●

●

●

●

●

● ●

●

●

●

●

●

●● ●

● ●

●

●

●

●

●● ●

●

●

●●

● ● ●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●●

● ● ●●●

●

●●

●●

● ●

●

● ●

●

●●

●

●● ●

●

●

●

●

●

● ●

●●

●

●●

● ●

●

●

● ●●

● ●

●

●

●

●

●●

●

●

●

●● ●

●●

● ●

●●

●●

●

●

●

●

●

●●

●

● ● ●

● ●

● ●

● ●●

● ●●

●●

●

●

● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●

●●

●●

●

●

● ●

●

●● ● ●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

● ●

● ●

●

●●

●●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●●

●

● ●

●

●

●●

●●

●

● ●

●

●●

●●

●

●●

● ● ● ●

● ●

●

●

●

●

●

● ●

●●

●

● ●

●

●

●

●

● ●●

●

●

●

● ●

●

●●

●

●●

●

●

●

●

● ●

● ● ●●

●●

●

●

●

●●

● ● ●

●

●●

●

● ●

●

●

●

●

●

●

●

● ●

●●

●

● ●

●

● ●

● ●

● ●

●

●

●

●●

●

●

●

●

●●

●

●

●

● ●

●●

●

● ●●

●

●

●

●●

●

●

●

●● ●

●

●

●

●

● ●

●●

●

●●

●

● ●●

● ●● ●

●

●

●

●

● ●

●

●

●●

●

●

● ●

●

●

● ●

●

●

●● ●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

● ●●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

● ●

● ● ●

●

●

●●

●

●

●

●

● ● ●

●

●

●

●●

●

●

●

●●

●●

● ●

●

●

●

●

●

●

●

●●

●●

●

●●

●

●

●

● ●●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●●

●●

●

●

●

●● ●

●

●

●

●

●●

●

●

●● ●

●

●●

●

●

●

●

●

●

●●

●

●

●

● ●●

●● ●

● ●

● ●●

●

●● ● ● ●

●

● ●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●

● ● ● ●

●

●

●

●

●

●

● ●

●

●●

●

●

●●

●

●

●●

●

● ●

●

●

●●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●●

●●

●

●

● ● ●

●

●

● ●

● ●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●●

●

●

●

●

● ●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

● ●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

● ● ●

●

● ● ●● ●

●●

●●

●●

●

●●

●●

●

●

●●

● ● ● ●●

●

● ●

● ●

●

●

● ●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

● ●

●● ●

●

●

● ●

●

●

●●

●

●●

●

●

● ●

● ●

●

●

●

●

● ●

● ●

●

●

●

●●

●

●

●●

●●

● ●

●

●

●●

● ●

● ●

●

●● ●

●

● ●

● ●

●

● ● ●

●

●

● ●

● ● ●

●

●

●

●● ●

●

●

● ●

●

● ● ●●

●

●●

●

●

●

●

● ●

●

●

●

●

●

● ●

●

●

● ●

●

●

●

● ●

●

●

● ●

●

●

●

●

●

● ●

● ●

● ● ● ●●

●

●

● ●

●

● ●

●

●

●●

●

●

●

● ● ●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●●

● ●

● ● ● ●

●

●

●●

●

●●

●

●

●

●

●

●●

●●

●

● ●

●

● ●

●

●

●

●●

●

●●

●

●●

●

●

●

●

● ● ● ●

● ●●

●

●

●

●

●

● ●●

● ● ●

●

●

● ● ●

● ●●

● ●●

● ●

●

●

●

●

●

●

●

● ● ●

●

●

●

●

●

●

●

● ●

●

●● ●

● ●

●

●

● ●

●

●

● ●

●●

●

●

●

●

●

●

●

● ●

●

●

●

● ●

●

●

●

● ●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●● ●●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●●

● ●

●

●

● ●

●

● ●

●

●●

●

●

●

●

●

● ●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

● ●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●● ●

●

●

● ●

● ●

●

● ●●●

●

●

●

●●

●

●

●

●● ●

●

●

●

●●

●

●

●●

●

●●

● ●

●●

●● ●

● ●

●

●

●

●

●

●

●

●

●

●

● ● ● ●

●

●

●●

●

●

●

● ●●

●

● ●

●

●

● ●

●●

● ● ● ● ●

●

●

●●

●●

●

●●

●

●●

●

●

●

●

● ●

●●

●

●

●

●

●

●● ●

● ●

●● ●

●●

●

●

●

●

● ●

●

●●

●

●

●

● ●

●

●

●●

● ●●

●

● ●

●●

●

●

●

●

● ●

●

● ●

●

●

● ●

●●

●●

●

●

●●

●●

●

●

●

●

●

● ●●

●●

●

●

● ●

●●

●

●● ● ●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

● ●

● ●

●

●

●

● ●● ● ●

●

●

●

● ●

●

●

●

● ●

●●

● ●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

● ●

●

● ●

● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

● ●

●●

●

●

●

●

● ● ● ● ●

● ● ●

●

●

●

●

●

●●

● ●●

●

●

● ●●

● ●

● ●

● ●

●

● ●

●

●

●●

●

● ●●

●

●

●

●

●

●

●

● ● ●

●

●●

●

●

●

●

●● ●

●

● ●

●

●

●

●

●

● ● ●

●

●● ● ● ● ●

● ●●

●

●

●●

●

●

● ●

●

●

●●

●

● ● ●

●

● ●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●● ●

●

●

●

● ●●

●●

● ● ●

● ●

● ●●

●●

● ●● ●

● ●

●

●

●

●

●

●●

●

● ●

●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

● ● ●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●

● ●

● ●

●●

●● ●

●

●

●●

● ●

●

●●

●

●

●

●●

●

●

● ●

● ●●

●

●

●●

●

●

●

● ●

●

●

●

● ●

● ● ●

●

●

●

●

●

●

● ●

● ●

●

●

● ●

●●

●●●

●

●

●

●

●

●

●

●

●

● ● ●●

●

●

●●

●●

●

●

●

●●

●

●

● ●

●

●

●● ●

●●

●

●●

● ●

●●

● ● ●●

●

● ●

●

●

●

● ●

●

● ●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●●

● ●

●

● ●

●

●●

●●

●● ●

● ●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●● ●

●

●

● ●●

● ●●

●

●

●

● ● ●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

● ● ●

●

●

●

●

●

●

●

●●

●● ●

●

●●

●

● ●●

●

●

●

●

●

● ●

●

●

●●

● ●

●

●

●

●●

●

●●

●

● ● ● ●●

●

● ● ●

●

●

● ●●

● ●

●

● ●●● ●

●

● ●

●●

●●

●

●●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

● ●

●

● ●●●

●

●●

●

●

●●

● ●● ●

●

●

●

●

●

●

●●

● ●●

●

●●

●

● ●

● ●●

●●

●

●

●

● ●●

●●

●

●●

●

●●

●●

●

●●

●●

●

●

●

●

●

●

●

●

● ● ●

●

●

●

●●

●

●

●●

● ●● ● ●

● ●

●

●

●● ●

●

●

● ●

● ●●

●●

● ●

●

●● ● ●

● ●

● ●

●

●●

●

●

●● ●

●

●

● ●

●

●●

●

●

●

●

● ●

●

●

●

● ●

●

●

●

●

●

●●

●

●●

●●

●

●●

●

●

● ● ● ●●

●

●

●

●

●

●

●

●

●●

●●

● ●●

●

●

●●

●

● ●

●

●● ●

●

●

●

●

●

●

●

●●

●

● ●

●

●

● ●

● ●

●

●

●

● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●● ●

●

●●

●

●

●●

● ●

● ●●

●

● ● ●

●

● ●●

● ● ●

●

● ●

●

●

●●

●

●

●

● ●

●

●

●

●●

●

● ●

●

●

●●

●

●● ●● ● ● ●

● ● ●

●

●●

● ● ●●

●

●

●●

● ●●

● ● ●

●

●

●

●

●

●

●

●

●

●

●●

●● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●● ●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●

●●

●●

●

● ●

●● ●

●

●

●

●

●

● ●

●

●

●

●

●

● ●

●

● ● ● ●

● ●●

●●

●

● ● ●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●

●●

● ●●

●

●

● ● ●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

● ●

●

●

● ●

●●

●●

●

● ●

● ●

●

●

●●

●

●

●

●

●

●●

●●

●

●

● ●●

●

●●

●

●

●● ●

● ● ● ●

●

●

●

●

●●

●

●

●●

● ●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●

● ●

●

●

● ●

●●

●

● ●

●

●● ●

●

●

●

●

●●

●●

●

●

●

● ●●

●●

●

●

●

●

●●

●

● ● ●

●●

● ●●

●

● ●

●

●

●

● ● ●

●

●●

●

●

●

●

●

●

● ● ●

●

●● ●

●

●

●●

●●

● ●●

● ●

●

●●

●

●● ●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

● ●●

●

●● ●

● ● ●

●

●●

●

●●

●

●

●

●

●

●

● ●

●

● ●●

● ● ●

● ● ● ●

●●

●

●

●

●● ●

●

●

●

●● ●

● ●

● ●

●

●

●

●

● ●●

●●

●

●

●

●●

●

● ●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●●

●

●●

●

●

● ●●

● ●

●

●

●

●

●

● ●

● ●

●

●●

● ●

●

●

●

● ●

●

●●

●

●

●

●

●

●

●

● ●

● ●

●●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●

● ● ●

●

● ●

●

●

●●

● ● ●

●●

●●●

●

●●

●●

●

●●

●● ●

●

●

●● ●

●

● ●

●

●●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

● ●

●

●

●

● ●

● ●

●

●

●●

●

● ●●

●● ●

●

●

●

●

●

●

● ●

● ●

●

●

●

●

●

● ●

●

●

● ●●

●

● ●

●

●

● ●●

●

●

●

●●

●

●

●

●

●●

●●

●

●

●● ● ●

●●

●●

●

●●

●

●

●

●

●

●

● ●

●● ●

●

● ●

● ●●

●

●

●

●●

● ●● ●

●●

● ●

●

●

●●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●●

●● ● ●

● ●

●

● ● ●

●● ● ●

● ●

● ●

● ● ●●

●●

● ●

●●

●

●

●●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

● ●●

●

● ●

●

●

●

●

●

●

●● ● ●

●

●

●●

●

●

●

●

●

●

● ●●

●

●

● ●

●●

●

●

●

●

●●

● ●

●●

●●

●

●●

●

●

● ● ● ●

●

●

●

●

●

●

●

●

● ●●

● ●●

●

●

● ●

●

●

●

●

● ●

●●

●

●

● ● ● ●●

●

●

●

●

●

●

● ●

●

●

●● ●

●●

●

● ●

● ●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

● ●

●

● ●

●

●

●●

●● ●

●

● ●●

●●

●

●

●●

●● ●

●

●

● ● ●

● ●

●●

●

● ●●

●

●

●

●

● ● ●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

● ●●

●

●

●

● ●

●●

●

●

● ●

●

●

● ● ●

●

●

●●

●

●

●

●●

●●

● ●

●

●●

●●

●

●

●

●

●

●

●

●

● ●●

●●

●

●

●●

● ●

●●

●● ●

●●

●

●

●

●●

● ●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

● ●

● ●

● ●

●

●

● ●● ● ●

●

●

●

●●

● ●

●

●

●

●

● ●

●●

●

●

●

●

●

●●

● ●

●●

●

●

●

●

●●

●

●

●●

●●

● ●

●

● ●

●

●

●

●

●●

●

●● ●

● ●

●

●

●

●●

●

● ●

● ●

●

●●

●●

●

●

●●

●

● ●

●

● ●

●

● ●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

● ●

●

●

●

● ●

●

● ● ●

●

● ●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●●

●

●●

● ●

●

●

●

●●

●

●

●●

●●

● ●●

●● ● ●

●●

●

●

●●

●●

●●

●

● ●●

●

●●

●

●

●

●●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

● ●

●

● ● ●

●

●●

●

●●

●

●

● ●

● ●

●

●

●●

●

●

●

●

●

●

● ●

●●

●●

●●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

● ● ●●

● ●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●● ● ●

●

●

● ● ● ● ●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●●

●

● ●● ●

●

●

●

●●

● ●

●

●

●●

●

●

●

●

●

● ●

● ● ●●

●

●

●

●

●

●

●

●●

●●

●●

●● ●

● ●●

●

●

●

●

●●

●

●

● ●● ●

●●

●

● ●

● ● ●

●●

● ●

●

●

●●

● ● ●●

● ●●

●

●●

●

● ●●

●

●●

●

●

●

●

● ●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●●

●

●

● ● ●●

●●

●● ●

●● ●

●

●●

● ● ●

●

●

●

●

●

● ●

●●

●

●

●

● ●

●

●

●●

●

● ●

●●

● ●●

●

●

● ●

●

●

●● ●

● ●

●

●

●

●

●

●

●

●

●●

●●

● ●

●

●

●●

● ●●

●●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●●

●●

●●

●

●●

● ●

●

●● ● ●

●

●

●

●

●

● ●●

●

●

●●

● ●

●

●●

●●

●

●

●

●

●●

●

●●

●

● ● ●

● ●●

● ●

●●

●

● ●

●

●

●

●●

● ●

●

●

●●

●

● ●●

●●

●●

● ●

● ● ●

●

●

●●

●

● ● ●

●

●

● ●●

●

●●

●

●

●

●

● ●

● ●●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

● ●●

● ● ●

●

●●

●

●

●●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

● ●

●

●

●

●

●●

● ●●

●

●

●●

●

●

● ●

●

● ●

● ●

●●

●●

●●

●

●

●

●

●●

●

● ● ●

●

●

●

●

● ●

●

●

● ●

●

● ●

●

●

● ●●

●

●

●

●●

●●

● ● ●

●●

●

●

●●

● ●● ●

●

●

●

● ●

●

●

● ●

●●

●

●●

●

●

●

●

●

●

● ●

●●

●

● ●

●●

●● ●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●●

●

● ●

● ●●

●

●●

●

● ●

● ●

●

●●

●

●● ●

●

● ●

●

●

●

●

● ●

●●

●● ●

●

● ●

●

● ●●

● ●

●●

● ●●

●

●

●

●●

●●

●

● ●

● ●●

●●

●

●●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●●

●

●

●

●

● ●

●

●●

●

● ●

●

●

● ●

●

● ●●

●

●

●●

●

● ●

● ●

●

● ● ●

●

●

●

●●

●

●

●

● ● ●

●

●

●

●●

●●

●

●

●

● ●●

● ●

●●

● ●●

●

●

●

●

●

● ●

●

●● ●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

● ● ●

●

●

● ●

●

●

● ●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●●

●●

●

●●

●

●

●●

● ● ●● ●

● ●

●

● ●

●● ●

●● ●

●

●

●

●

●

●

●

● ●

●

●● ●

●

●

●●

●

● ● ●

●

●

●●

●● ●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●●

●

●

● ●

● ●● ● ●

●

●

● ● ●

●

● ● ●

●

●

●

●

●

●● ●

●

●●

●

●

●

● ●

●● ●

● ●●

●

●

●

●

●

● ● ●

●

● ● ● ●●

●

●

● ●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●●

● ●●

●

●●

● ● ●

●

●

● ●

●

●

●

●

●

●

●

● ●

● ●

● ● ●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●● ●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

● ●

●●

●

●●

●●

●

● ● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

● ●

● ●●

●

●

● ●

●

●

●

●

●

●

●

● ●●

● ● ●●

●

●

●

● ●

●

●

●

●

●●

●●

●

●

●●

●

●● ●

●

● ● ●●

●

●

●

● ●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●●

●

●

●● ●

●

●

●

●●

●

●

●

●

●

●●

● ●●

●

●

●●

●● ● ●

●

●

●

● ●

●

●

●

●● ●

●●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

● ● ● ● ●●

●●

●●

●● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●● ●

● ●

●

●

●

●

●

●

●

● ●

●

●

●

● ●

●

●

●

●●

●

●

●

●●

●●

●● ●

●●

● ●

●● ●

● ●●

● ●

●

●●

●

● ● ●●

●

●● ● ● ●

● ●

●●

● ●

●

● ●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●

●●

● ● ●

●

● ●●

●●

●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●

● ●

●

●●● ●

●●

●

●

●●

●●

● ●

●●● ●

●

●●

●●

●● ●

●

●

●

● ●

● ●

●

●

●

● ●●

●

●●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

● ●

●

● ● ●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●●

●

● ●

●

●●● ● ●

●●

●

●

● ●

●

●●

● ●●

●

●●

●

●

●

●

●●

● ●

● ●●

●

●

● ●

●●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●● ●

● ●

● ●

●●

●

●

●

●

●

● ●●

●

●

●

●

● ●

●

●

● ●●

● ●

●

●

● ●●

● ●

● ●

●●

●●

●

● ●

●

●

●

● ●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●● ●

●●

●

● ●

●

●●

●

●

●

●●

●

● ●

●

●

●

●●

●

●

●

●● ● ●

● ●

● ●

●●

● ● ●

●

●●

●● ●

●

●

●●

● ●

●

●●

●●

●

●

●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

● ●

●

●●●

●

●

● ●

● ●

●

●

● ● ●

●

●

●

● ●●

●

●

●●

●

●

● ●

●

●

●●

●

●

● ●●

● ●

●●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●● ●●

●●

●● ●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●●

●

● ●●

●

●●

●

●

● ●

● ●

●●

●●●

●

●

● ●

●

●

●

●

● ●

●●

●●

●

●

● ●

● ●●

● ●

●

●

●

●●

●

●

●●

●

●●

●●

●

● ●

● ●● ●

●

●● ● ● ● ●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

● ● ●

●

●

●●

●

●

●

●●

●

●●

● ●

●

●●

● ●● ●

●

●●

●

●● ●

● ●

●

●●●

● ● ●

●

●●

● ●

●

●

●●

●●

● ●

●

●

●

●●

●●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●●●

●

●

● ●

●●

●

●

●

●

●●

●●

●

●

●

● ●●

●

●

●

●●

●●

●

●●

●

●

●

●

● ●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●● ●

●

●●

●●

● ●

● ●

●●

●

●●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●●

● ●●

● ●●

●

●● ●

● ●

●

●● ● ●

●

● ●

●

● ● ●

●

●●

●

● ●

●

●●

●●

● ●

●

●

●

●●

●●

●●

● ● ●●

●

●

● ● ●● ● ●

●

●● ●

●●

●

●

●

● ● ● ●

●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●●

●

●

●

●

● ●●

●

●●

●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●

●●

●●

●

● ●

●

●

● ● ●

● ●

●

● ●

●

●

●

●●

●

●

●

●

●

●●

●● ●

●● ●

●

● ● ● ●

●

●●

●●

●●

●

●●

●

●

●●

●● ● ● ● ● ●●

●●

● ●

● ●

●

●

● ●

● ●

●

●●

●

● ●● ●

●

●

●●

●

● ●

●

●

●

●●

● ●

●

●

●●

●

● ●

● ●

●●

●

●

●

●●

● ●●

●

●●

● ●●

● ●

●●

●

●

●●

●

●●

●

●●

●

● ●

●

●●

● ●●

●

●

●

● ●

●

●

●

●● ● ●

●

● ●

●

●

● ● ●●

●

●

● ●● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●● ●

● ● ●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●

● ●● ● ●

● ●

● ●●

●

●

●

●

●

●

●

●

●

●●

●

● ●●

●

●

●

●

● ● ●●

●

●

●●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

● ●●

●

●

●●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●●

●●

●

● ● ●

●

●

●

●

●

●

●●

● ● ●

● ●●

● ●

● ●

●

●

● ●● ●

● ●

●● ● ●

●

●

●

●

●

● ●

●● ● ●●

●

●

●

●

● ●

●

●

●●

●

●●

●●

●

●●

● ●

● ●

●●

●

● ●

●

●● ● ●

●

●●

● ● ●

●

●

●●

● ●

●

●

● ●●

●

●●

●

●

●

●●

●●

●● ●

●

●

●

●

●

●

● ●● ●

●

●

●

●

●

●

● ●●

●

●

● ● ●

● ●

●

●

●

●

●

●

●

●

● ● ● ●●

●

●

●●

●● ●

● ●

●

●●

●

●

●

●●

●●

●

●●

●

●

● ●

●

●

●

●

●●

●

●

●

●

●

● ●

●

●●

●

● ●

●

●

●

●●

●

●

●

●

●●

● ●

●

●

● ●

●● ●●

●

●

● ●

●

●

● ●● ●

●

● ●

●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●

● ● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

● ● ●

●

●

● ●

●

● ● ●

● ● ●

● ●

● ●●

● ●

●●

●

●

● ●

●

●●

●●

● ●

●

●

●

● ●

●

●

●

●

●

●

●

●

● ● ●

●

●

●

●

●●

●

●●

● ●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●●

●

●

●

●●

●

● ●

●

●

● ●

● ●

● ●

● ● ●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

● ● ●

●

●

●●

●● ●

●●

●●

●●

●

●

●

● ● ●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●●

● ●● ●

●

●●

●

●

● ● ●

●●

●●

●

●

●●

●● ●

●

● ●

●

●

●

●

●●

●●

●●●

●●

●

●● ●

●

●

● ●

●

●

● ●

●

● ●

● ●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●●

●

●●

●

●

●

●

● ●

●

●

●

●

● ●

●

●

●

● ●

● ●

●

● ●●

● ●

●

●●

● ●

●

● ●

●

●●

●

●●

●

●●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

● ● ● ● ●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●● ●

● ●

●

●

● ● ●

●

●

●●

● ● ●

●●

● ● ● ●

●

●● ●

●

●● ●

●

●

●

●

● ● ●

●

●

●

●

●

●

●

● ● ● ●●

● ●

●●

● ● ●●

●

● ●●

●●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●●

● ●

●

● ●

● ● ● ● ● ● ●

●

●●

●●

●●

● ●

● ●● ●

●●●

● ●●

●

●

●

●● ●

●

●

●

●

●

●●

● ●

●

●

●

● ●

● ●

●

●

●●

● ● ●

●

●●

●

● ●

●

●●

●●

●

●

●

●●

●

●

●

●

●

● ●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●● ● ●

●●

●

●

● ●

●

● ●

● ●

●

●

●● ●

● ●

● ●

● ●●●

● ●

●

●

●●

●

●

●●

●●

●

●

●●

●●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

● ●

●

● ● ● ● ● ●

●

●

●

●

●

● ●●

●

●

● ● ● ●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●

●

●

●●

●

●

●

● ●

●

●●

●●

● ●

●

●

● ●

●

● ●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

● ●

●

●

●

●

● ●●

●

●

●

●

●

●●

●

● ●

●

●

●

●

● ●●

●●

●

●●

●

●

● ●●

●

●

●

●

●

● ●

●●

●

●

●

●

●●

●● ●

●

● ●

●

●

●

●

●

●●

● ●

●

●●

● ●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●● ●

●

●●

●

●

● ●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●●

● ●

● ● ●●

●

●●

● ●● ● ●

● ●●

●● ● ● ● ●

●

●

●

●

●

●●

● ●

●●

●

● ●

●●

● ●● ●

●

● ● ● ● ●

●

● ● ● ● ● ●

●

● ●

●

●● ●

●●

●●

●

●●

●●

●

● ●

● ●●

●

●

●

●

●

●

●

●

●●

● ●●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

● ● ●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●●

●●● ● ● ●

●●

●

●

●●

●

● ●

●●

●●

●

●

●

●● ●

●

●

●

●

●

● ●

●

● ● ●

●

●

●

●

●

●

●

●

●

●

●

● ● ●

●

●

●

●

●

●

●

●

●●

●● ● ●

●●

●

● ●

● ●

●

●

●●

●

●●

●

● ●

●

●

●

●● ●

●

●●

●●

●● ●

●

●

●●

●

●

●

●

●

●

●●

● ●

●●

●

● ●

●

●●

●

●

●

●●

●

●

● ● ● ●

● ●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●● ●

●●

●

●

●● ●

●●

●

●●

● ● ● ●

●

●

● ● ●

● ●

●●

●

●

●● ●

●●

● ●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

● ● ●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

● ●

●

● ●

●●

● ● ●● ●

●●

●

● ● ●

●

●

●●

●

●

●

●

●

●

● ●

●

●●

●

●● ●

●

● ●

● ●

●●

●● ●● ●

● ●

●

●

●

●

●

● ●

●

● ●

●

●●

●

●

● ●

●

● ● ●

● ●

●

●

●

●● ●

●●

●

●

●

● ●

●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

● ●

● ●●

●●

●

● ●

●

●

●

● ●

●

●

● ● ●●

●●

●●

● ●

●

●●

● ●

● ●

●

●●

●● ●

● ●

● ●

●

● ●●

●

● ● ● ●●

●

●

●

●

●

●

● ●● ● ● ●

●

●● ● ● ● ●

●

●●

●●

●

●

●

●

● ● ●

●

●

●

●

●

●●

●●

●

●

● ●

● ●●

●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

● ●● ● ● ●

● ●

●

●

●

●

●

●

●

●

●

● ●

●

● ●●

● ● ●●

●

●● ●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●●

●●

●●

●● ● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●●

● ●

●

●

●

● ●●

●

●

●●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●●●

●● ●

●

●

●

●

●

●

● ●

●●

●● ● ● ●

●

●

● ●

●

●

●● ● ● ●

●●

●

● ●

●● ● ●

●

●

●

●

●●

● ● ●●

●

●

●

●

● ●● ● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ● ●

●

●

●

●

●

●

●

● ●● ●

●

●

●

●

●

●

●

●

● ●

●●

●● ●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●● ●

●●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

● ● ● ●●

●

●●

●

●

● ●

●

●

●

●●

●

●

●●

●●

●

● ●

●

●●

●●

●

●

●

●

●

●

●

●

●

● ● ● ● ●

● ●

●

●

●

●

● ● ●

●

●●

●

●

● ●

●

●

●● ● ●

●

●

●●

●●

●●

●

●

● ●

●

●

●

●●

●●

● ●●

● ●

●

●

●

●●

●

●

●

●

●

●

●

●●

●● ●

●

●

●● ●

● ●

● ● ●

● ●● ●

●●

●●

●

●

●

●●

● ●●

●

●

●

● ●

●

●

●

●

●●

● ●

●

●

●

●

●

● ●

● ●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ● ●

●

●●

●

●

●

●●

● ●

●

●

●

●

●●

●

●

●

●

●

●

● ●

●

●

●●

●● ●

●

●

●●

● ● ●

●

●

●●

●

●●

●

●

●

●

●

● ●●

●

● ●●

●

●

●

● ● ●

●

●●

●

●

●●

●

●

●

● ● ● ●●●

●●

●● ●

●

●

●

● ●

● ● ●

●

●

●

●

●

●●

●

●

●●

●●

● ● ●●

● ● ●

●

● ● ●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●●

●

●

●

●●

●

●● ●

●

●

●

●

●

● ●

●

●● ● ●

● ●●

●

●

●●

●●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●●

●●

●●

●

● ●

● ●

●

●

●

●

●

● ● ●

●

● ●●

●

●

●

●

●

●

●

● ●

●

●

●●

●●

●

● ●

●

●

●

●

● ● ●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ● ●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

● ● ●

●

●●

●●

● ●

●

●●

●

●●

●

●

●

●

● ●●

●

●

●

●

●●

●●

●

●

●

●●

●

●

● ●●

●●

●

●

●

●

●

●

●

●

● ●

● ● ●● ●

●

●

●

●

● ● ●

●

●● ● ●

● ●

●● ●

●

● ● ●●

●

●

● ●

●

●

●●

●

● ●

●

●

●●

●● ●

●●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●

●

●

● ●

●

●

●● ● ●

●●

● ●

●●

●

● ●

● ●●

●

●

●

●

●

●

●

●

● ●

●

●●●

● ●

●●

●

●

●

●

●

●

●

● ● ●

●

●●●

●

●

●

●●

●● ●● ●

●

●●● ●

● ●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

● ● ●

●

●●●

●● ●

● ●

●

●

● ● ●

●

●

●●

● ●●

●

●

●

●● ●

●●

●●●

●

●

●

●

●●

● ●●

●●

●● ● ● ●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●

●●

●

●

●

●

● ●●●

● ● ●●

●

●

●

● ● ●

●

●

●●

●

●●

●

●

●● ●

●●

●

●

●●

●

●

●

●

●

●● ●

●

●

●

●

● ●

●

●

●

●

● ● ●

●●

●● ●

●●

● ●●● ●

● ●

●

●

●

●

● ●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●●● ● ●

●

●●

●

●●

●

●

● ●●●

●

●

●

●

●● ●

●●

● ●

●

●

●● ●●

●

●●

●

●●

●

●

●●

● ● ●

●

● ●

● ●●

●●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●● ●

● ● ●●

● ●●

●

●

● ●

●

●

● ●

●

●

●

●

●

● ●

●

●

●

● ●

●

●

●●

●

●

●

●●

● ●●● ● ●

●

●

●

●

●● ●

●●

●

●●

● ●●●

●

●

●

● ●

●

●

●

●●

●●

●

●

●

●

●

●

●●

● ●

●

● ●●●

●●

●

● ●

●

●

●

●

●● ●

●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●

● ●

●

●●

●●

●

●

●

●

●●

●●

●

●

●

●

●

● ●

●

●

●● ●

● ●

●

●

●

●

●

●

●

●●

●● ●

●

●●

●●

●

● ● ●

● ●

●

●

●●

●

●

●

●●

●

●

●●

●●

●

●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●●

●●●

●

●

●●

●

● ●

●

●

●

●

● ●

●

●

●

● ●

●

●

●●

● ●

●●

●

● ●

●● ●

●

●

● ●

●●

●

●

●

●

●

●

●●● ●

●●

●

●

●

●

● ● ●

●●

●●

●

● ●

●

●

●

●

●

●

●

●●

●● ●

●

●

●●●

●

●

●

●

●

●

●

● ●

●●

●● ●●

●

●

●

●

●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●● ●●●

●

●

●

●

●● ●

● ●

●

●

●

●

●

●●●

●

●

● ●

●

●●

●

●

●

●

●●

●● ● ● ●

●

●

●

●

●●

●

●

● ●

●

● ●

●

● ●

●

●●

●

●

●

●●

● ●

●

●

●

●

● ● ● ●

● ●

●

●●

●

●

●

●

●● ●

● ●

●

●

●

● ● ● ●●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

●

●

●

●

● ●

● ●

● ● ● ● ●

● ●

●

● ●

● ●●

●

●●

●

●●

● ●

●●

●

● ●●●

● ●●

●●

● ●●

● ●

● ●●

●

●

● ● ● ●

●

● ● ● ● ●

●

● ● ●

●

●

●●

● ●

●

●

●

●

●

●

● ●●

●

●

●

●

● ● ●● ●

●●

●

●

● ●

●

● ●

●

●●

● ●

●

●

● ● ● ●

●

● ●● ●

● ●

●

●

●

●

●●

●

●

●

●

● ●

●

●●

●

●

●

●

●

● ● ●

● ●

● ●

●

●

● ●

● ●

●● ●

●●

●

● ● ● ● ● ●

●

●●

● ●●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

●

●

●●

●●

●● ●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

● ● ● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

● ● ●

●●

●●

●

● ●

● ●

● ● ●

●

●● ●

●

●

●

●

●●

●

● ●

●

●●

●

●

●

● ●● ● ●

● ●

●●

●

●

●

●

● ● ●

●

●

●●

●

●

●

●

●●

● ●

● ●

● ●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●●

●●

●

●

●●

●

● ●

●

●

●

●●

● ●●

●

● ●●

●●

●●

● ●

●● ● ● ●

●

●

●●

●

●

●

●

●

●

●

● ●

●

●●

● ● ● ●

●

●

●

● ●

● ●

●●

●●

●

●●

● ●

● ●

●

●

●

●

●

●

● ●

●●

●● ●

● ● ●

●

●●

●

●

●

●

●

● ●

● ●

●●

●

●

●● ●

●

●●

● ●

●

●

●

●

●● ●

● ●

●

● ● ● ●

●

●

●

● ●●

● ● ●

●

●

●

● ●

● ●

● ●

●

●

●

●

●

● ●●

● ●●

● ●

●

●

●

●

●

● ●

● ●

●

●

●

●

●●

●

●●

●

● ●

● ●●

●●

●

●

●

●

●

●● ●

●

● ●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

● ● ●● ● ● ●

●●

●

●

● ●

●

●●

●

● ●

● ●

●●

● ●●

●●

● ●

●

●

●●

●

●●

●

●

●●

●

●

●●

●●

●

●

● ●

●

●

●●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●● ●

● ●

● ●

●

● ● ●

●

●●

● ●

● ●

●

●

●

●

●

●

● ●

●

●

●

●●

●●

●

●

●● ● ● ●

●

●

●

●

●●

●●

●● ● ● ●

● ●

●

●

●

●

●

●

●

●

● ●

●●

●●

● ●

● ●●

● ● ●

● ● ●

●

●

●

●

● ●

●

●

●● ● ● ●

● ●

● ● ●

● ● ●●

●

●

●

● ●

●

●

●

●

●

● ●

●

●

●●

● ●● ●

●

●

●

●

● ●

●●

●●

● ● ● ●

● ● ●● ●

● ●

●●

●●

●●

●

●●

●

●

●● ●

● ● ●

● ●●

●

●●

●●

● ●

●

●

●

●●

●●

●

●

● ●

●

●

●

● ●

●

●●

●

●

●

●

●

●

●

● ●

● ●

●

● ● ●

●

● ●

●● ● ● ●

●

● ●

●

●

●

●

●

●

●●

● ●

●● ●

●●

● ●

●

● ● ●

●

●

●

●

● ● ● ●

● ●

● ●

●

● ● ● ●

●

● ●● ●

● ●

●

●

●

● ● ● ●

●

●

●●

●

●

●

●

●

● ● ●● ●

●

●

● ●

●

●

●

●

●

●

●

● ●

●●

●● ●

● ●

●

●

●● ●

●●●

● ●

●

●

●

●

●●

●

●●

● ●

●●

●

●

●

●●

●

●

●● ● ●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●● ●

●●

● ●

● ●

●

●●

●●

●

●

● ●

●

●●

●● ●

●●

●

●

●●

●

●

●

●●

●

● ● ● ●

● ●

●

● ●

●

●

●

●

● ●

●

●

● ●

●● ● ● ●

●●

●

●

●

●●

●

●● ●

●

●

●

●●

●

●

●

●

●

●

●●

● ● ●●

● ● ●

●

●

●

●

●

●

●

● ●

● ●

●

●

●

●

●

●●

●

●

●

●●

● ●

● ●

●

●●

●

●

● ● ●

● ● ● ●

●

●

●●

●

● ●

●

●

●● ●

●

●

●●

● ●

●

● ●

●

●

●

●

●

●●

●

●

●

● ● ●

● ● ● ●

●

● ●

●●

●

● ● ●

●

●●

●

●

●

●

●●

●

●

●

●

● ●

●

● ●

● ●

●●

●

● ●

●

●

●

●

●

●●

●

●●

●

●●

●●

● ●

●●

●●

●● ●

●

●

● ●

●

●

●●

●

●●

●

● ●●

●

●

●

● ●●

●

●

● ●●

● ●●

●

●

●

●● ●

● ●●

●●

●

●

●●

●

●

●

●

●

●

●

●●

●●

● ● ●

● ●● ● ● ● ●

●

●

●

● ●

●

●

● ●

●

● ●

●

●

●

●●

● ●

● ●

●

●

●

●

● ●●

●●

●● ●

● ●

●

●

●●

●

● ●

●

●●

● ● ●●

●

●

●

●

●

● ●

●

●

●

●

●

● ●

●

● ●

●

●

●

●●

●

●

●

●●

● ●● ●

●●

●

●● ●

●

●●

● ●

●

●

●

●

●

●●

●

●●

● ●●

●

● ●●

● ●

●

●

● ●

●

●●

● ●

● ● ●● ●

● ●

●

●

●

● ● ●●

●

●

●●

●

●

●

● ●

● ● ●

●

●

●

●●

●

●●

●●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●●

●

●●

● ●

●●

●

●

●

● ●●

●

●●

●

●

●

●

●●

● ●

●

●

● ●

●

● ●● ●

●

●

●

●

●

●●●

●●

●

●

● ●

● ●

●● ●

● ●

●●

●●

●● ●

●

●●

● ●● ●●

●● ● ●

● ●

●●

●● ● ● ●

●

●

●● ● ● ●

●●

●

● ●● ●

●●

● ● ●

●

●

●

●

●

● ● ●●

●●

●

● ●● ●

●

●

●

● ● ●

●●●

●

● ●●

● ● ●

●

●

●

●●

● ● ●

●●

●

● ●

●

●

●

●

●

●●

●

●

●

●● ●

●

●

● ●

●

●

●

●

●

●

●

●

● ●●

●

●●

●

● ●

● ●

●

●

●

● ●

●● ●

● ●

● ●

●

●

●

●

●

● ●

●

●●

●

●

●

● ●

●

● ●●

●

●

●

●

●● ●

● ●

●

●

●

● ●

●

●

●●

●

● ●

●

●

●

●

●

● ●

● ●

●●

●

● ● ●●

●

●

● ●●

●

●

●

●

● ●

●

●

●●

●●

●●

●

●●

●

●●

●●

●

●

●

● ●

●

●

● ● ●

● ●

●

● ●

●

● ● ●

●

● ●

●●

●

●

●

●

●

● ● ● ●

●

●

● ●

● ●

●●

●

●

●

●

●

●●

● ●●

●

●

●

●

●

● ●

● ●

● ●●

● ● ●●

●

● ●

●

●

● ●

●

● ●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●

●●

● ●

● ●

●

●

● ● ●

● ●

●●

●● ●

●●●

●

● ● ● ●●

●

●●

● ● ●

●●

●

●● ● ● ●

●

●

● ● ●

● ●

●

●

●● ● ● ●

●

●●

● ●

● ●

●

●●

●●

●●

●

●

●● ●

●

●

● ● ●

● ●●

●

●

● ●

●

●●

●

● ●● ●

●●

●

●

●

●

●

●

●

● ● ●

● ●

●

●

● ● ● ● ●

●

●●

● ● ● ●

●

●

●

●●

● ●●●

●

●

●●

●

●

●●

●●

●

●●

●● ●

● ●●●

●● ●

● ●●● ●

●

●

●●

●●

●

● ●

● ●

●

●

●

●

●

●

●

●●

●

● ●

● ●

●

●

●●

●

●●

● ●●

● ●

● ●

● ●

●● ●

●●

●●

●

● ●

●

●

●

●

●● ●

● ●

●

●

●

●

● ●●

● ●●

● ●

● ●

●

●

●

● ●●

●

●

●

●

●

●

● ●

●●

●

● ●

● ●

●

●

●● ●

● ●

●

●●

● ●

●

●

●

●

●

●●

●●

●

●

● ●

●●

●

●

●

● ●

● ●

●●

●

● ●

● ●

●

●

●● ●

● ●

●

●● ● ● ● ●

●

●

●

●

●

●

●

● ●●

●

●

● ●●

●

●

● ● ● ●

●

●● ●

●●

●●

● ● ●

● ●●

●● ● ●

●

●

● ●

●●

●●

●●

●● ●

● ●

● ●

●

● ●

●

●●

●●

●●

●●

●● ●

●● ● ●

●●

●

● ●

● ●

●

●

●

●●

●●

●●

●

● ●

●

●

●

●

●

●● ●

●

● ● ●

● ●

●

●

● ●

● ● ●

●●

●

●●

● ●

●

●

●

●●

● ●

● ●

●

●

●

● ●●

●

●●

●

●●

●

●

●

● ●●

●

●

●● ● ●

● ●

●●

● ●

●● ●

● ●● ●

● ● ●

●

● ●●

● ●

● ●

● ●

●

●

●

●

●

● ● ●

●● ●

●

●

●

● ● ● ●●

●●

●

● ●● ●

●

●

● ● ●

●

●

●● ●

●

●

●

●

●●

●●

●

●●

●●

●

●

●

●

●

● ●

●●

●●

● ●

●

● ●

●

●● ● ●

●●

●

● ● ● ● ●

●●

● ● ●

●

●

●

●

●

●●

● ●

●

●

●

●● ● ●

●

●

●

●●

●●

●●

●

●●● ●

●

●

●●●

●

●

●● ●

●●

● ●●

●

●

●●●

●

●

●

●● ● ●

●

●●

●●

●

●

●●

●

●●

● ●

●

●

●

●●

●●

●●

●

●●

●

●

●

● ● ●● ● ●

●

●

●

●●

●●

●

●

●●●

● ●●●

●

●

● ●

●● ●

●●

●●●

●

●

●●

●●●

●

●●

●

●

●● ● ●

●●●

●

●

●●

●

●

●●

●●●

●

●●

●

●●

●

●

●

●

●

●●

●

●

● ● ●●

● ●●

●

●

● ●

●● ● ●

●

●● ●●

●

●

●●

●

●●

●

●

●●

●

●●

● ●

●

●

●

●● ●

●●

●●

●

●

●

●●

●

●●

●

●

●

●●

●●

●

●

●

●

●

●●

● ●

●●

●

●

● ● ●●● ●

●● ● ●

●

● ●

●●

●●

●● ●

●●

● ●

●

● ●●●

●●

●● ●

●●

● ●●●

●

●

●

●●

●

●

●●

●●

●

●●●

●

●● ●

●

●●

●●

●●●

●

●

●

●

●●

●

●

●

●●

●●

●●

●

●●

●

●●

●

●

●

● ●

●● ● ●●

●

●●

●

●

●●●

●●

●●

●

●●●

●●

● ●

●●

●

●●

●●● ● ●

●●

●

●●

●

●

●

●● ●●

●●

●

●

●●

●

●

●

●●

●●

●●

●●

●●●

●

●●

●●●

● ●

●●●

●● ● ●●●

●●● ●

●

●●●

●● ●●

●

●●

●●●

●

●●

●

●●

●

●

●●

●

●●

● ●

●

●

●

●●

●●

●

●

●●●

● ●

●

●● ●

●

●●

●

●

● ●● ●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●● ●

●●

● ●

●

●

●

●●● ●●

● ●

●

●

●

●●

●●

●●

●

●●

● ●

●

●

● ●●

●

●●●

●

●

●

●

●

●

●●

●●● ●

●● ●

●●●

●

●●

●●

●

●

●

●

●

●●

● ●

●

● ● ●●

● ●

●

●

●●

● ●

●

●● ● ●●

● ●

●● ●

●

●

●

●

●

●

●●●

●●

●●● ●

●

●●

● ●●●

●

●

●●●

● ●●

●

●●●

●

●●●

●

●●

●

●

●●

●●●

● ●

●

●

● ●●

● ●

●

● ●

●● ● ●

●

●

●

●●●

●

●

●

●

●●

●●

●● ●

●

●

●●

●

●●●

●●

●

●

●

●●

● ●

●

●

●

●●

●

●●●

● ●

●

●

●

●●

● ●●● ●

●●

●●●

●●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●

●●●

● ● ●●●

●

●●

●

●

●

●

●

●

● ●

●●

● ●●

● ●

●

●●

●●

●

●●●

●●●

●●

●

●● ●

●●

● ●

●●

●●

●● ●

●

●

● ●●

●●●

●

●●●

● ●

●●

●●

●

●●

●●● ● ●

●

●

●

●●

●

●

●

●

●

●●

●●

●● ●

●

●

●

●

●

● ● ●●

●

●

●●● ●●

●

●

●

●

●

●●

● ●

●●●

●●

●

●●

●●

●●

●

●

●

●

●●●

● ●

●

●

● ●● ● ●

●

●

●●● ●

●

●●

●●●

● ●●

●

● ●● ●

●

●

●

●●●

● ●● ● ●● ●

●

●

●

●

●●

●●

●●●

●●

● ●

●

●

● ●●● ●

●

●

●

●● ● ●

●

●

● ●●

●

●

●

●

●

●●

●

●

●

● ●

● ●

● ●●

●

● ●

●

● ●

●

●

●

● ●●

●

●

●

●

● ●

●

●

●●

●

● ●

●

●

●

●

● ●●

●

●

● ●

● ●

●●

● ●

●

●● ● ●

●

●

●

● ●

● ●●

●●

●

●

●

●

●

●

● ● ●

●

●

●

●

● ● ●

●

●● ●

●●

●●

●

●

● ● ● ● ●

● ●

●

● ●

●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●● ● ● ●

● ● ●● ● ● ●

●

●

● ●● ●

●

●●

●●

●

●

●

●

●

●

●

●● ● ● ●

●

●●

● ●

●

● ●

●

● ●

● ●

●●

●● ●

●

●●

●

● ●

●

●

● ● ●

● ● ●

● ● ●● ●

●

●●

●

● ●

●

●

●

●●

● ● ●

● ● ●

● ●

●

●

●

● ●

● ●● ●

●

●●

● ● ●

●

● ●

●●

●●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

● ●●

● ● ●●

●●

●

● ●

●

●

●●

● ● ● ●

●

●●

●

●● ● ●

●

●●

●

●

●

●

● ●

●

●

●

●

● ●●

●● ●

● ●

●

●

●●●

●

● ● ●

●

● ●●

● ●●

●

●

●

● ● ●

●

●

●●

●● ●

●

●

●

●

●● ●

●

●

●

● ● ● ● ●

● ●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

● ● ●

●

●

●

●

●

● ●

●

●

●

●

● ●

●

●

●

●

●

● ● ● ●

●●

● ●

●

●

●

●●

●

● ●

●

●

● ●

●

● ●

● ●

●

●

●

●

●

●

●●

●

● ●

● ●

●●

●

● ● ●

●

●●

● ●

●

●

●

●● ● ●

●●

●

●

●● ●

● ●

●

●

●

●

● ●

● ●

● ● ●● ●

●●

●

●

●

●

●

●

●● ● ●

●●

●● ●

●

●●

●

●

●

●

●●

●

●

●●

●● ●

●

●●

●

●

● ●

●

●

●

●

●

● ●

●●

●

●

●

●

●●

●

●

●

●

●● ●

●

●

● ●●

●

●

●

● ●

● ● ●

●

●

●

●

●

● ●

● ●

●

●

●

●

●

● ●

●

●●

● ●

●

●

●

● ●●

●

●

●

●

● ● ●

●

●

● ●

●

●

●

● ●

●●

● ●

●● ● ● ●

●●

●● ●

●●

● ● ●

● ●

●

●● ●

●● ●

● ●

●●

●

●

●

●

●

●

●

●●

●● ●

●●

● ● ● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

● ●

●

● ●

●

●

● ● ●

● ●

●●

●● ●

●●

●●

●

● ●

●

●

●●

● ● ●

●

●

●

●

●

● ●

●

●

●

●

●

● ● ●

●

●

● ●

●

●

● ●● ●

●

●

●●

●

● ● ●●

●

●

●● ●

●

●

●

●● ● ●

● ●

● ● ● ●

●

●●

● ●

●

● ●●

●

●

●●

●

●

●

●

●

●

● ● ●

● ●

●

●

●

● ●

● ●

●

●

●● ●

● ●

●

●

●● ●

●

●

●

●

●

● ●

● ●

●●

● ●●

● ●●

●

●●

●

●

●●

●

●

●

●

● ●

●

●

●

●

● ●●

●

●

●

● ● ●

●

●

●●

● ●

●

●

● ● ● ● ● ● ●

● ●●

●

●

●

●

●●

●● ●

●

●●

●●

● ●

● ●

●

●

●

● ● ● ●

●

●

● ● ●

●

●

●

●

●●

●

●

● ● ●

●

●

●

●● ● ●

●

●

●

●●

● ●

●

●

●

● ●

●

●●

●● ●

●

●

● ●

●● ● ●

●

●

●●

●

●

●

● ●●

●●

●

●

● ● ●●

●

●

●

● ●●

● ●

● ●

●

● ● ●

●

●

● ●

●

●●

● ●

●

●

●

●●

●●

●

●

●

● ●

●

●

● ●

●● ●

●

●

● ●

●

● ● ● ●

●

●

●

●

●

●

●●●

●

● ●●

●● ●

● ●●

● ●

● ●●

●●

●

●

●●

●

●●

● ●●

●

●

● ● ●

●●

●●

●

●

●

●

●●

●● ● ●

●

●

● ● ● ● ●

●

●

●●

● ●

●

●

●● ●

●

●

●

● ● ●

● ●●

●

●

●

●●

● ●

●

●

●●

● ●

●

● ● ●

●

●

● ● ●●

●●

●

●

● ●

●

●●

●

●

● ●

●

●●

● ●

●

●

●

●●

●●

●

●

●

● ●

●●

● ●● ●

●● ●

●

●

●

●

●

●

● ●

● ●●

●● ●

● ●

● ●●

● ●

●

●

●

●

●

●●

● ● ● ● ●

●

●

●●

●

●

●

●

● ●●

● ● ●

●

●●

●

● ● ●

●

●

●

●

●

●● ●

● ● ●

●

●

● ● ●

● ●●

●

●

● ●

● ●●

●●

●●

● ●

●●

● ●

●

●●

●

●

●● ● ● ●●

● ●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●

●●

●●

●

●

● ●● ●

●

●

●

●

● ●● ●

●

●

●

● ●

●

●

●

●

● ●

●●

●

● ●

● ● ● ●

●

● ●

●

●●

●

●

●

●

●

●

● ● ●

●

● ● ●●

● ●

●

●● ●

●●

●●

●

●

●● ●

● ●

●●

●

● ●●

● ●●

●

●

●

●●

●●

●●

● ●

● ●

●

●● ● ● ● ●

●●

●● ●

●

●●

●

●

● ●●

●

●

●

● ● ●

●

● ● ●●

●

●

●

● ● ● ● ● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

● ●

●

● ●

● ●

● ●

●●

●●

●●

●

●

●● ●

●

●●

● ● ● ●●

●●

●

● ● ●

●

●

●

●● ●

●

●●

●

●●

●

● ● ●

●● ●

●

●●

●

● ● ●●

●

●

●

● ● ● ●

●

●

●

●

●●

●

●● ●

●●

●

●

●

●

● ●●

●

●

●

●

●

●●

●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

● ● ●

● ●

●●

● ●

●

●

●

●

● ● ●●

●●

● ●

●

●

●

● ●

●●

●

●

●

●●

●● ● ●

●

●

●

●●

● ●

●● ●

● ●

●

●

●

●

● ●

●

●

● ●●

●●

● ●

●

●

●

●

●● ●

●●

●

● ●

●●

●

●

●

●

●

●

●

● ●●

●

● ● ●●

●●

●●

●

●

● ●

●

●

● ●

●

●

●

● ●

●

●

●

●

●

●●

●

●

● ●

●● ●

●

●

●●

●

●●

●

●●

●

● ●

● ● ●

●

●●

●

●

●●

●

● ●●

●

●

●

●

● ●

●●

●

●

●

●

●

●●

●

● ●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●● ● ● ●

●

●

●

●

●

● ●

● ● ●

● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

● ●

●

●

●●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●

● ●●

●

●

●

●

● ●

●

●

●

● ●

●

●

● ●

●

●●

● ●

●

●

●● ●

●

●●

● ●

●●

●

●● ● ●

● ● ●●

● ● ●

●

● ●

●

● ●

●

●

●

●

●

● ●●

●

●●

● ● ●

●●

●

● ●

● ●

●

● ●● ●

●●

●

●

●

● ● ● ●

●● ● ● ● ● ●

●●

●

●

●●

● ●

●

●

●● ●

●

● ●

●

●

●

●●

●

●

●

●

●

● ●

●●●

●

●●

●

● ●

●●

● ●●

● ●

●●

●

● ●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

● ●●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

● ●

● ●

●● ●

● ●

●

●● ● ●

●●

●

●

● ●

●●

●● ●

●

●

●

●

●

●

●

●

● ●

●●

●●●

● ●

● ●

● ●

●●

● ● ●

● ●●

●

●

● ●

●

●● ● ●

● ●●

● ● ●

●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

● ● ●

●●

● ●

●

●

● ● ● ●●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●● ●

●● ●

●● ● ●

●

●

●

●●

● ●

●

●●

●● ● ●

● ●●

● ●● ●

●

●

● ● ●

●

●●

●● ● ● ● ●

●

●

●

●

●

● ●

●●

● ●

●●

●

● ●

●

●

●

●●

●●

●●

●

●

●

●

●●

● ●●

●

●

● ● ● ●

●

●

●

●●

● ● ●

●

● ●

●

●

●

●

●

●

●

● ●

●●

●●

●

●

●

●●

●●

●●

●

● ●●

●

●

●

●●

●

●●

●

● ● ●●

●

●●

●

●

● ●

●

●

●

● ● ● ●●

●

●

●

● ●

●

●

●

●● ●

●●

●

●

●

●

●

●

●

● ●

● ● ●

●● ●

● ● ● ●●

●● ●

● ●●

●

●

●● ●

●●

●●

●● ●

●●

●

●●

●

●

●

●

●

●

●●

●

● ●●●

●● ●

● ●

●

● ●

● ● ●

●

● ● ●● ●

● ●

●

●

●

● ●

●

●

●●●

●●

● ●

●

●

●●●

●

●

●●

●

● ●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●●

●

●● ●

●●

● ●

●

● ●●●

●●

●

●

●

●●

● ●

●

● ●●●

●●

●●

● ●

●

●

●

● ●

●●

● ●●

●● ● ● ● ●● ●

●

● ● ● ●

●

● ●

●●

●

●

●

● ● ● ● ●●

● ●●

●

● ●

●●

●● ●

● ●

● ● ●

● ●

●

●

●

●

●

● ●

● ●

●

●●

● ●●

●

●●

●

●

●

●

●

● ●

● ●

●

●●

● ●● ●

●●

●● ●

●

●

● ●

●

● ●

●

●

●●

●● ● ● ●

●●

●

● ●

● ● ● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

● ●

●

●

●

●

●●

●

●

●●

● ●● ● ●

● ●●

●

●

●●

●●

●●

● ●

●

●

●

● ●

●

20 30 40 50 60

1520

2530

3540

Age

BM

I

Upper tail copulas

Twin

Co−

Twin

BM

I density

Twin

Co−

Twin

BM

I density

Dependence measures for upper tail

N η χ χ L(X ,Y ) h2∗

MZ Overall 20048 0.80 0.49 0.59 0.23 0.24Men 8212 0.78 0.45 0.56 0.22 0.13

Men (Age 18-29) 2241 0.83 0.44 0.67 0.29 0.57Men (Age 30-39) 3356 0.87 0.41 0.74 0.13 0.21Men (Age 40-60) 2615 0.70 0.39 0.39 0.22 0.14

DZ Overall 37476 0.61 0.26 0.21 0.11 -Men 16354 0.60 0.22 0.20 0.16 -

Men (Age 18-29) 4356 0.60 0.22 0.20 0.00 -Men (Age 30-39) 6657 0.55 0.16 0.10 0.03 -Men (Age 40-60) 5341 0.65 0.14 0.30 0.15 -

Practical: Estimating upper tail dependence measures

This session illustrates how to estimate the upper tail dependence measure

χ = limu→1

Prob(U1 ≥ u | U2 ≥ u)

from the classical bivariate extreme value distribution.

Bivariate Extreme Value DistributionLauch R-code on next slide to obtain an estimate of χEstimate χ for MZ and for DZ pairs.

Reference: Coles S, "An Introduction to Statistical Modeling of Extreme Values", Springer (2001).

Fitting Bivariate Peaks Over a Threshold Using Bivariate Extreme ValueDistributions

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

The R kiosk

’mets’ (K. Holst, T. Scheike)’etm’ (A. Allignol); ’prodlim’ (T. Gerds).

Documents

Using Twin Data Jacob Hjelmborg - staff.pubhealth.ku.dkstaff.pubhealth.ku.dk/~kkho/undervisning/family2013/biostatgen.pdf · On Biostatistical Genetics Using Twin Data Jacob Hjelmborg