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Intro to Quantitative Genetics HGEN502, 2011 Hermine H. Maes

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Intro to Quantitative Genetics 1/18: Course introduction; Introduction to Quantitative Genetics & Genetic Model Building 1/20: Study Design and Genetic Model Fitting 1/25: Basic Twin Methodology 1/27: Advanced Twin Methodology and Scope of Genetic Epidemiology 2/1: Quantitative Genetics Problem Session

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Aims of this talk Historical Background Genetical Principles Genetic Parameters: additive, dominance Biometrical Model Statistical Principles Basic concepts: mean, variance, covariance Path Analysis Likelihood

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Quantitative Genetics Principles Analysis of patterns and mechanisms underlying variation in continuous traits to resolve and identify their genetic and environmental causes Continuous traits have continuous phenotypic range; often polygenic & influenced by environmental effects Ordinal traits are expressed in whole numbers; can be treated as approx discontinuous or as threshold traits Some qualitative traits; can be treated as having underlying quantitative basis, expressed as a threshold trait (or multiple thresholds)

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Types of Genetic Influence Mendelian Disorders Single gene, highly penetrant, severe, small % affected (e.g., Huntington’s Disease) Chromosomal Disorders Insertions, deletions of chromosomal sections, severe, small % affected (e.g., Down’s Syndrome) Complex Traits Multiple genes (of small effect), environment, large % population, susceptibility – not destiny (e.g., depression, alcohol dependence, etc)

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Genetic Disorders

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Great 19th Century Biologists Gregor Mendel ( ): Mathematical rules of particulate inheritance (“Mendel’s Laws”) Charles Darwin ( ): Evolution depends on differential reproduction of inherited variants Francis Galton ( ): Systematic measurement of family resemblance Karl Pearson ( ): “Pearson Correlation”; graduate student of Galton

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Family Measurements

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Standardize Measurement Pearson and Lee’s diagram for measurement of “span” (finger-tip to finger-tip distance)

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From Pearson and Lee (1903) p.378 Parent Offspring Correlations

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From Pearson and Lee (1903) p.387 Sibling Correlations

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© Lindon Eaves, 2009 Nuclear Family Correlations

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Quantitative Genetic Strategies Family Studies Does the trait aggregate in families? The (Really!) Big Problem: Families are a mixture of genetic and environmental factors Twin Studies Galton’s solution: Twins One (Ideal) solution: Twins separated at birth But unfortunately MZA’s are rare Easier solution: MZ & DZ twins reared together

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Twin Studies Reared Apart Minnesota Study of Twins Reared Apart (T. Bouchard et al, 1979 >100 sets of reared-apart twins from across the US & UK All pairs spent formative years apart (but vary tremendously in amount of contact prior to study) 56 MZAs participated

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Types of Twins Monozygotic (MZ; “identical”): result from fertilization of a single egg by a single sperm; share 100% of genetic material Dizygotic (DZ, “fraternal” or “non- identical”): result from independent fertilization of two eggs by two sperm; share on average 50% of their genes

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Logic of Classical Twin Study MZs share 100% genes, DZs (on avg) 50% Both twin types share 100% environment If rMZ > rDZ, then genetic factors are important If rDZ > ½ rMZ, then growing up in the same home is important If rMZ < 1, then non-shared environmental factors are important

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Causes of Twinning For MZs, appears to be random For DZs, Increases with mother’s age (follicle stimulating hormone, FSH, levels increase with age) Hereditary factors (FSH) Fertility treatment Rates of twins/multiple births are increasing, currently ~3% of all births

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Zygosity of Twins

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Chorionicity of Twins 100% of DZ twins are dichorionic ~1/3 of MZ twins are dichorionic and ~2/3 are monochorionic

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Virginia Twin Study of Adolescent Behavioral Development Twin Correlations MZ Stature DZ Stature © Lindon Eaves, 2009

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Ronald Fisher ( ) 1918: On the Correlation Between Relatives on the Supposition of Mendelian Inheritance 1921: Introduced concept of “likelihood” 1930: The Genetical Theory of Natural Selection 1935: The Design of Experiments Fisher developed mathematical theory that reconciled Mendel’s work with Galton and Pearson’s correlations

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Fisher (1918): Basic Ideas Continuous variation caused by lots of genes (polygenic inheritance) Each gene followed Mendel’s laws Environment smoothed out genetic differences Genes may show different degrees of dominance Genes may have many forms (multiple alleles) Mating may not be random (assortative mating) Showed that correlations obtained by Pearson & Lee were explained well by polygenic inheritance [“Mendelian” Crosses with Quantitative Traits]

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Biometrical Genetics Lots of credit to: Manuel Ferreira, Shaun Purcell Pak Sham, Lindon Eaves

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Revisit common genetic parameters - such as allele frequencies, genetic effects, dominance, variance components, etc Use these parameters to construct a biometrical genetic model Model that expresses the: (1)Mean (2)Variance (3)Covariance between individuals for a quantitative phenotype as a function of genetic parameters. Building a Genetic Model

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Population level Transmission level Phenotype level G G G G G G G G G G G G G G G G G G G G G G GG PP Allele and genotype frequencies Mendelian segregation Genetic relatedness Biometrical model Additive and dominance components Genetic Concepts

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Population level 1. Allele frequencies A single locus, with two alleles - Biallelic / diallelic - Single nucleotide polymorphism, SNP Alleles A and a - Frequency of A is p - Frequency of a is q = 1 – p A a A a Every individual inherits two alleles - A genotype is the combination of the two alleles - e.g. AA, aa (the homozygotes) or Aa (the heterozygote)

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2. Genotype frequencies (Random mating) A (p)a (q) A (p) a (q) Allele 1 Allele 2 AA (p 2 ) aA (qp) Aa (pq) aa (q 2 ) Hardy-Weinberg Equilibrium frequencies P (AA) = p 2 P (Aa) = 2pq P (aa) = q 2 p 2 + 2pq + q 2 = 1 Population level

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Transmission level Pure Lines AAaa F1 Aa AAAa aa 3:1 Segregation Ratio Intercross Mendel’s experiments

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Aa aa Aaaa F1Pure line Back cross 1:1 Segregation ratio Transmission level

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Pure Lines AAaa F1 Aa AAAa aa 3:1 Segregation Ratio Intercross

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Aa aa Aaaa F1Pure line Back cross 1:1 Segregation ratio Transmission level

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Segregation, Meiosis Mendel’s law of segregation A3 (½)A3 (½)A4 (½)A4 (½) A1 (½)A1 (½) A2 (½)A2 (½) Mother (A 3 A 4 ) A1A3 (¼)A1A3 (¼) A2A3 (¼)A2A3 (¼) A1A4 (¼)A1A4 (¼) A2A4 (¼)A2A4 (¼) Gametes Father (A 1 A 2 ) Transmission level

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1. Classical Mendelian traits Dominant trait (D - presence, R - absence) - AA, Aa D - aa R Recessive trait (D - absence, R - presence) - AA, Aa D - aa R Codominant trait (X, Y, Z) - AA X - Aa Y - aa Z Phenotype level

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2. Dominant Mendelian inheritance D (½)D (½)d (½)d (½) D (½) d (½) Mother (Dd) DD (¼)DD (¼) dD (¼)dD (¼) Dd (¼)Dd (¼) dd (¼)dd (¼) Father (Dd) Phenotype level

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3. Dominant Mendelian inheritance with incomplete penetrance and phenocopies D (½)D (½)d (½)d (½) D (½) d (½) Mother (Dd) DD (¼)DD (¼) dD (¼)dD (¼) Dd (¼)Dd (¼) dd (¼)dd (¼) Father (Dd) Phenocopies Incomplete penetrance Phenotype level

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4. Recessive Mendelian inheritance D (½)D (½)d (½)d (½) D (½) d (½) Mother (Dd) DD (¼)DD (¼) dD (¼)dD (¼) Dd (¼)Dd (¼) dd (¼)dd (¼) Father (Dd) Phenotype level

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Two kinds of differences Continuous Graded, no distinct boundaries e.g. height, weight, blood-pressure, IQ, extraversion Categorical Yes/No Normal/Affected (Dichotomous) None/Mild/Severe (Multicategory) Often called “threshold traits” because people “affected” if they fall above some level of a measured or hypothesized continuous trait Phenotype level

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Polygenic Traits 1 Gene 3 Genotypes 3 Phenotypes 2 Genes 9 Genotypes 5 Phenotypes 3 Genes 27 Genotypes 7 Phenotypes 4 Genes 81 Genotypes 9 Phenotypes Mendel’s Experiments in Plant Hybridization, showed how discrete particles (particulate theory of inheritance) behaved mathematically: all or nothing states (round/wrinkled, green/yellow), “Mendelian” disease How do these particles produce a continuous trait like stature or liability to a complex disorder? Phenotype level

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Quantitative traits AA Aa aa Phenotype level

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m d+a P(X)P(X) X AA Aa aa -a AA Aaaa Genotypic means Biometric Model Genotypic effect Phenotype level m -am +dm +a

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Very Basic Statistical Concepts 1. Mean (X) 2. Variance (X) 3. Covariance (X,Y) 4. Correlation (X,Y)

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Mean, variance, covariance 1. Mean (X)

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Mean, variance, covariance 2. Variance (X)

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Mean, variance, covariance 3. Covariance (X,Y)

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Mean, variance, covariance (& correlation) 4. Correlation (X,Y)

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Biometrical model for single biallelic QTL Biallelic locus - Genotypes: AA, Aa, aa - Genotype frequencies: p 2, 2pq, q 2 Alleles at this locus are transmitted from P-O according to Mendel’s law of segregation Genotypes for this locus influence the expression of a quantitative trait X (i.e. locus is a QTL) Biometrical genetic model that estimates the contribution of this QTL towards the (1) Mean, (2) Variance and (3) Covariance between individuals for this quantitative trait X

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Biometrical model for single biallelic QTL Biallelic locus - Genotypes: AA, Aa, aa - Genotype frequencies: p 2, 2pq, q 2 Alleles at this locus are transmitted from P-O according to Mendel’s law of segregation Genotypes for this locus influence the expression of a quantitative trait X (i.e. locus is a QTL) Biometrical genetic model that estimates the contribution of this QTL towards the (1) Mean, (2) Variance and (3) Covariance between individuals for this quantitative trait X

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1. Contribution of the QTL to the Mean (X) aa Aa AA Genotypes Frequencies, f(x) Effect, x p2p2 2pqq2q2 a d-a = a(p 2 ) + d(2pq) – a(q 2 )Mean (X)= a(p-q) + 2pqd Biometrical model for single biallelic QTL

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2. Contribution of the QTL to the Variance (X) aa Aa AA Genotypes Frequencies, f(x) Effect, x p2p2 2pqq2q2 a d-a = (a-m) 2 p 2 + (d-m) 2 2pq + (-a-m) 2 q 2 Var (X) = V QTL Broad-sense heritability of X at this locus = V QTL / V Total Broad-sense total heritability of X = ΣV QTL / V Total Biometrical model for single biallelic QTL

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= (a-m) 2 p 2 + (d-m) 2 2pq + (-a-m) 2 q 2 Var (X) = 2pq[a+(q-p)d] 2 + (2pqd) 2 = V A QTL + V D QTL m d+a–a AA aa Aa Additive effects: the main effects of individual alleles Dominance effects: represent the interaction between alleles d = 0 Biometrical model for single biallelic QTL

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= (a-m) 2 p 2 + (d-m) 2 2pq + (-a-m) 2 q 2 Var (X) = 2pq[a+(q-p)d] 2 + (2pqd) 2 = V A QTL + V D QTL AA aaAa Additive effects: the main effects of individual alleles Dominance effects: represent the interaction between alleles m –a+ad d > 0 Biometrical model for single biallelic QTL

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= (a-m) 2 p 2 + (d-m) 2 2pq + (-a-m) 2 q 2 Var (X) = 2pq[a+(q-p)d] 2 + (2pqd) 2 = V A QTL + V D QTL AA aaAa Additive effects: the main effects of individual alleles Dominance effects: represent the interaction between alleles m –a+ad d < 0 Biometrical model for single biallelic QTL

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aa Aa AA Var (X) = Regression Variance + Residual Variance = Additive Variance + Dominance Variance m –a +a d Biometrical model for single biallelic QTL

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Var (X) = 2pq[a+(q-p)d] 2 + (2pqd) 2 V A QTL + V D QTL Demonstrate 2A. Average allelic effect 2B. Additive genetic variance NOTE: Additive genetic variance depends on allele frequency p & additive genetic value a as well as dominance deviation d Additive genetic variance typically greater than dominance variance Biometrical model for single biallelic QTL

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2A. Average allelic effect (α) The deviation of the allelic mean from the population mean a(p-q) + 2pqd A a αaαa αAαA ? ? Mean (X) Allele a Allele APopulation AAAaaa ad-a Apq ap+dq q(a+d(q-p)) apq dp-aq -p(a+d(q-p)) Allelic meanAverage allelic effect (α) 1/3 Biometrical model for single biallelic QTL

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Denote the average allelic effects - α A = q(a+d(q-p)) - α a = -p(a+d(q-p)) If only two alleles exist, we can define the average effect of allele substitution - α = α A - α a - α = (q-(-p))(a+d(q-p)) = (a+d(q-p)) Therefore: - α A = qα - α a = -pα 2/3 Biometrical model for single biallelic QTL

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2B. Additive genetic variance The variance of the average allelic effects 2αA2αA Additive effect 2A. Average allelic effect (α) Freq. AA Aa aa p2p2 2pq q2q2 α A + α a 2αa2αa = 2qα= 2qα = (q-p)α = -2pα V A QTL = (2qα) 2 p 2 + ((q-p)α) 2 2pq + (-2pα) 2 q 2 = 2pqα 2 = 2pq[a+d(q-p)] 2 d = 0, V A QTL = 2pqa 2 p = q, V A QTL = ½a 2 3/3 α A = qα α a = -pα Biometrical model for single biallelic QTL

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2B. Additive genetic variance 2A. Average allelic effect (α) 3. Contribution of the QTL to the Covariance (X,Y) 2. Contribution of the QTL to the Variance (X) 1. Contribution of the QTL to the Mean (X) Biometrical model for single biallelic QTL

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AA Aa aa AA Aaaa (a-m)(a-m) (d-m)(d-m) (-a-m) (a-m)(a-m) (d-m)(d-m) (a-m)2(a-m)2 (a-m)(a-m) (d-m)(d-m) (a-m)(a-m) (d-m)2(d-m)2 (d-m)(d-m) (-a-m) 2 3. Contribution of the QTL to the Cov (X,Y) Biometrical model for single biallelic QTL

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AA Aa aa AA Aaaa (a-m)(a-m) (d-m)(d-m) (-a-m) (a-m)(a-m) (d-m)(d-m) (a-m)2(a-m)2 (a-m)(a-m) (d-m)(d-m) (a-m)(a-m) (d-m)2(d-m)2 (d-m)(d-m) (-a-m) 2 p2p pq 0 q2q2 3A. Contribution of the QTL to the Cov (X,Y) – MZ twins = (a-m) 2 p 2 + (d-m) 2 2pq + (-a-m) 2 q 2 Covar (X i,X j ) = V A QTL + V D QTL = 2pq[a+(q-p)d] 2 + (2pqd) 2 Biometrical model for single biallelic QTL

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AA Aa aa AA Aaaa (a-m)(a-m) (d-m)(d-m) (-a-m) (a-m)(a-m) (d-m)(d-m) (a-m)2(a-m)2 (a-m)(a-m) (d-m)(d-m) (a-m)(a-m) (d-m)2(d-m)2 (d-m)(d-m) (-a-m) 2 p3p3 p2qp2q 0 pq pq 2 q3q3 3B. Contribution of the QTL to the Cov (X,Y) – Parent-Offspring Biometrical model for single biallelic QTL

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e.g. given an AA father, an AA offspring can come from either AA x AA or AA x Aa parental mating types AA x AA will occur p 2 × p 2 = p 4 and have AA offspring Prob()=1 AA x Aa will occur p 2 × 2pq = 2p 3 q and have AA offspring Prob()=0.5 and have Aa offspring Prob()=0.5 therefore, P(AA father & AA offspring) = p 4 + p 3 q = p 3 (p+q) = p 3 Biometrical model for single biallelic QTL

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AA Aa aa AA Aaaa (a-m)(a-m) (d-m)(d-m) (-a-m) (a-m)(a-m) (d-m)(d-m) (a-m)2(a-m)2 (a-m)(a-m) (d-m)(d-m) (a-m)(a-m) (d-m)2(d-m)2 (d-m)(d-m) (-a-m) 2 p3p3 p2qp2q 0 pq pq 2 q3q3 = (a-m) 2 p 3 + … + (-a-m) 2 q 3 Cov (X i,X j ) = ½V A QTL = pq[a+(q-p)d] 2 3B. Contribution of the QTL to the Cov (X,Y) – Parent-Offspring Biometrical model for single biallelic QTL

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AA Aa aa AA Aaaa (a-m)(a-m) (d-m)(d-m) (-a-m) (a-m)(a-m) (d-m)(d-m) (a-m)2(a-m)2 (a-m)(a-m) (d-m)(d-m) (a-m)(a-m) (d-m)2(d-m)2 (d-m)(d-m) (-a-m) 2 p4p4 2p 3 q p2q2p2q2 4p 2 q 2 2pq 3 q4q4 = (a-m) 2 p 4 + … + (-a-m) 2 q 4 Cov (X i,X j ) = 0 3C. Contribution of the QTL to the Cov (X,Y) – Unrelated individuals Biometrical model for single biallelic QTL

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Cov (X i,X j ) 3D. Contribution of the QTL to the Cov (X,Y) – DZ twins and full sibs ¼ genome ¼ (2 alleles) + ½ (1 allele) + ¼ (0 alleles) MZ twins P-O Unrelateds ¼ genome # identical alleles inherited from parents 0 1 (mother) 1 (father) 2 = ¼ Cov(MZ) + ½ Cov(P-O) + ¼ Cov(Unrel) = ¼(V A QTL +V D QTL ) + ½ (½ V A QTL ) + ¼ (0) = ½ V A QTL + ¼V D QTL Biometrical model for single biallelic QTL

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Biometrical model predicts contribution of a QTL to the mean, variance and covariances of a trait Var (X) = V A QTL + V D QTL 1 QTL Cov (MZ) = V A QTL + V D QTL Cov (DZ) = ½V A QTL + ¼V D QTL Var (X) = Σ(V A QTL ) + Σ(V D QTL ) = V A + V D Multiple QTL Cov (MZ) Cov (DZ) = Σ(V A QTL ) + Σ(V D QTL ) = V A + V D = Σ(½V A QTL ) + Σ(¼V D QTL ) = ½V A + ¼V D Summary

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Biometrical model underlies the variance components estimation performed in Mx Var (X) = V A + V D + V E Cov (MZ) Cov (DZ) = V A + V D = ½V A + ¼V D Summary

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Path Analysis HGEN502, 2011 Hermine H. Maes

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Model Building Write equations for means, variances and covariances of different type of relative or Draw path diagrams for easy derivation of expected means, variances and covariances and translation to mathematical formulation

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Method of Path Analysis Allows us to represent linear models for the relationship between variables in diagrammatic form, e.g. a genetic model; a factor model; a regression model Makes it easy to derive expectations for the variances and covariances of variables in terms of the parameters of the proposed linear model Permits easy translation into matrix formulation as used by statistical programs

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Path Diagram Variables Squares or rectangles denote observed variables Circles or ellipses denote latent (unmeasured) variables Upper-case letters are used to denote variables Lower-case letters (or numeric values) are used to denote covariances or path coefficients

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Variables latent variables observed variables

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Path Diagram Arrows Single-headed arrows or paths (–>) are used to represent causal relationships between variables under a particular model - where the variable at the tail is hypothesized to have a direct influence on the variable at the head Double-headed arrows ( ) represent a covariance between two variables, which may arise through common causes not represented in the model. They may also be used to represent the variance of a variable

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Arrows double-headed arrows single-headed arrows

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Path Analysis Tracing Rules Trace backwards, change direction at a 2- headed arrow, then trace forwards (implies that we can never trace through two-headed arrows in the same chain). The expected covariance between two variables, or the expected variance of a variable, is computed by multiplying together all the coefficients in a chain, and then summing over all possible chains.

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Non-genetic Example

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Cov AB Cov AB = kl + mqn + mpl

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Expectations Cov AB = Cov BC = Cov AC = Var A = Var B = Var C =

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Expectations Cov AB = kl + mqn + mpl Cov BC = no Cov AC = mqo Var A = k 2 + m kpm Var B = l 2 + n 2 Var C = o 2

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Genetic Examples MZ Twins Reared Together DZ Twins Reared Together MZ Twins Reared Apart DZ Twins Reared Apart Parents & Offspring

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MZ Twins Reared Together

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Expected Covariance Twin 1Twin 2 Twin 1a 2+ c 2+ e 2 variance a 2+ c 2 Twin 2a 2+ c 2 covariance a 2+ c 2+ e 2 MZ Twins RT

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DZ Twins Reared Together

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Expected Covariance Twin 1 Twin 2 Twin 1a 2+ c 2+ e 2.5a 2+ c 2 Twin 2.5a 2+ c 2 a 2+ c 2+ e 2 DZ Twins RT

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MZ Twins Reared Apart

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DZ Twins Reared Apart

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Twins and Parents

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Role of model mediating between theory and data

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