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Basic Statistics for Linkage and Association Studies of Quantitative Traits Boulder Colorado Workshop March 5 2007 Overview • • • • A brief history of SEM Regression Maximum likelihood estimation Models – Twin data – Sib pair linkage analysis – Association analysis • Mixture distributions • Some extensions Origins of SEM • Regression analysis – ‘Reversion’ Galton 1877: Biological phenomenon – Yule 1897 Pearson 1903: General Statistical Context – Initially Gaussian X and Y; Fisher 1922 Y|X • Path Analysis – Sewall Wright 1918; 1921 – Path Diagrams of regression and covariance relationships Structural Equation Model basics • Two kinds of relationships – Linear regression X -> Y single-headed – Unspecified covariance X<->Y double-headed • Four kinds of variable – Squares – observed variables – Circles – latent, not observed variables – Triangles – constant (zero variance) for specifying means – Diamonds -- observed variables used as moderators (on paths) Linear Regression Model Var(X) Res(Y) b X Y Models covariances only Of historical interest Linear Regression Model Var(X) Res(Y) b Y X Mu(x) 1 Mu(y*) Models Means and Covariances Linear Regression Model Res(Y) X 1 i D b Yi 1 Mu(y*) Models Mean and Covariance of Y only Must have raw (individual level) data X is a definition variable Mean of Y different for every observation Single Factor Model 1.00 F lm l1 l2 l3 S1 S2 S3 Sm e1 e2 e3 e4 Factor Model with Means MF 1.00 1.00 mF2 mF1 F1 F2 lm l1 S1 e1 l2 l3 S2 e2 mS2 mS1 S3 mS3 B8 e3 Sm mSm e4 Factor model essentials • The factor itself is typically assumed to be normally distributed: SEM • May have more than one latent factor • The error variance is typically assumed to be normal as well • May be applied to binary or ordinal data – Threshold model Multifactorial Threshold Model Normal distribution of liability. ‘Affected’ when liability x > t t 0.5 0.4 0.3 0.2 0.1 0 -4 -3 -2 -1 0 x 1 2 3 4 Measuring Variation • Distribution – Population – Sample – Observed measures • Probability density function ‘pdf’ – Smoothed out histogram – f(x) >= 0 for all x One Coin toss 2 outcomes 0.6 Probability 0.5 0.4 0.3 0.2 0.1 0 Heads Tails Outcome Two Coin toss 3 outcomes 0.6 Probability 0.5 0.4 0.3 0.2 0.1 0 HH HT/TH Outcome TT Four Coin toss 5 outcomes 0.4 Probability 0.3 0.2 0.1 0 HHHH HHHT HHTT Outcome HTTT TTTT Ten Coin toss 9 outcomes 0.3 Probability 0.25 0.2 0.15 0.1 0.05 0 Outcome Fort Knox Toss Infinite outcomes 0.5 0.4 0.3 0.2 0.1 0 -4 -3 -2 -1 0 1 Heads-Tails De Moivre 1733 Gauss 1827 2 3 4 Variance Measure of Spread Easily calculated Individual differences Average squared deviation Normal distribution xi di -3 -2 -1 0 1 2 Variance = di /N 2 3 Measuring Variation Weighs & Means • Absolute differences? • Squared differences? • Absolute cubed? • Squared squared? Measuring Variation Ways & Means • Squared differences Fisher (1922) Squared has minimum variance under normal distribution Concept of “Efficiency” emerges Deviations in two dimensions x + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + y Deviations in two dimensions x dx + dy y Covariance • Measure of association between two variables • Closely related to variance • Useful to partition variance • Analysis of variance coined by Fisher Summary Formulae for sample statistics; i=1…N observations x = (xi)/N x = 2 xy = rxy = (xi - x ) / (N) (xi-x )(yi-y ) / (N) 2 xy 2 2 x x Variance covariance matrix Univariate Twin/Sib Data Var(Twin1) Cov(Twin1,Twin2) Cov(Twin2,Twin1) Var(Twin2) Only suitable for complete data Good conceptual perspective Summary • Means and covariances • Basic input statistics for “Traditional SEM” • Notion of probability density function Maximum Likelihood Estimates: Nice Properties 1. Asymptotically unbiased • Large sample estimate of p -> population value 2. Minimum variance “Efficient” • Smallest variance of all estimates with property 1 3. Functionally invariant • If g(a) is one-to-one function of parameter a • and MLE (a) = a* • then MLE g(a) = g(a*) • See http://wikipedia.org Likelihood computation Calculate height of curve -1 Height of normal curve: x = 0 Probability density function x (xi) -3 -2 -1 0 1 xi 2 3 (xi) is the likelihood of data point xi for particular mean & variance estimates Height of normal curve at xi: x = .5 Function of mean x (xi) -3 -2 -1 0 1 xi 2 3 Likelihood of data point xi increases as x approaches xi Height of normal curve at x1 Function of variance x (xi var = 1) (xi var = 2) (xi var = 3) -3 -2 -1 0 x1 1 2 3 Likelihood of data point xi changes as variance of distribution changes Height of normal curve at x1 and x2 x (x1 var = 1) (x1 var = 2) -3 -2 -1 0 x1 1 2 3 x2 (x2 var = 2) (x2 var = 1) x1 has higher likelihood with var=1 whereas x2 has higher likelihood with var=2 Likelihood of xi as a function of Likelihood function L(xi) -3 MLE -2 -1 0 1 xi 2 ^ x 3 L(xi) is the likelihood of data point xi for particular mean & variance estimates Likelihood as a measure of “outlierness” • Unlikely observation may be an outlier – Genuine – Data entry error – Model-specific • Can use Mx feature to obtain case-wise likelihoods – Raw data – Option mx%p= uni_pi.out – Output for each case: the contribution to the -2ll as well as z-score statistic and Mahalanobis distance, weight and weighted likelihood – Generates R syntax to read in file, and sort by z-score – Beeby Medland & Martin (2006) ViewPoint and ViewDist: utilities for rapid graphing of linkage distributions and identification of outliers. Behav Genet. 2006 Jan;36(1):7-11 Height of bivariate normal density function An unlikely pair of (x,y) values 0.5 0.4 0.3 -3 -2 -1 0 x 1 2 x1 3 -3 -2 -1 y1 1 y 0 2 3 Height of bivariate normal density function A more likely pair of (x,y) values 0.5 0.4 0.3 -3 -2 -1 0 x2 1 2 3 -3 -2 -1 0 y2 1 2 3 Likelihood of Independent Observations • Chance of getting two heads • L(x1…xn) = Product (L(x1), L(x2) , … L(xn)) • L(xi) typically < 1 • Avoid vanishing L(x1…xn) • Computationally convenient log-likelihood • ln (a * b) = ln(a) + ln(b) • Minimization more manageable than maximization – Minimize -2 ln(L) Likelihood Ratio Tests • • • • Comparison of likelihoods Consider ratio L(data,model 1) / L(data, model 2) -3 -2 Ln(a/b) = ln(a) - ln(b) Log-likelihood lnL(data, model 1) - ln L(data, model 2) l -1 0 1 2 3 • Useful asymptotic feature when model 2 is a submodel of model 1 -2 (lnL(data, model 1) - lnL(data, model 2)) ~ df = # parameters of model 1 - # parameters of model 2 • BEWARE of gotchas! – Estimates of a2 q2 etc. have implicit bound of zero – Distributed as 50:50 mixture of 0 and Exercises: Compute Normal PDF • Get used to Mx script language • Use matrix algebra • Taste of likelihood theory Mx script part 1: Declare groups and matrices #NGroups 1 Title figure out likelihood by hand Calculation Begin Matrices; E symm 2 2 ! Expected Covariance Matrix H full 1 1 ! One half T full 1 1 ! Two M full 2 1 ! Mean vector P full 1 1 ! Pi X full 2 1 ! Observed Data End Matrices; Mx script part 2: Put values in matrices Matrix E 1 .0 1 Matrix H .5 Matrix M 0 0 Matrix P 3.141592 Matrix T 2 Matrix X 1 2 Mx script part 3: Matrix Algebra Begin Algebra; O=T*P*\sqrt\det(E); ! Fractional part, 2pi*sqrt(det(e)) Q=(X-M)'&(E~); ! Mahalanobis Distance R=\exp(-H*Q); ! e to the power -.5*Mahalanobis distance S=-T*\ln(R%O);! minus twice log-likelihood Z=-T*\ln(\pdfnor(X'_M'_E)); ! A simpler way End Algebra; End Group; Exercises 1 • Bivariate normal distribution – Means [110.28 112.00] – Covariance matrix [299.40 174.20 281.18] • Compute likelihood of observed vector x = [87 89] Exercises 2 • Bivariate normal distribution – Means [1 1] – Covariance matrix [1 .3 .3 1 ] • Compute likelihood of observed vector x = [1 2] • Compute likelihood with correlation of .0 instead • Optional compute likelihood of observed vector x = [-2 -2] with correlations .5, .0, and 0 • Which is the most likely combination of model and data? Exercises 3 • Univariate normal distribution – Mean [1] – Variance [1 ] • Compute likelihood of observed vector x = [1] • Compute likelihood of observed vector x = [2] • Compute their product • Which bivariate case does this equal? Two Group Model: ACE 1 1 A .5 1 1 1 1 C a 1 c E 1 E e 1 C e T1 c T2 m1 m2 1 MZ twins 7 parameters 1 A a 1 A 1 1 C a c E 1 E e 1 C e T1 c T2 m3 m4 1 DZ twins A a DZ by IBD status 1 1 Q .5 1 1 1 1 F q 1 f E E e T1 1 f Q 1 Q q 1 1 F q T2 m2 1 1 F e m1 1 f T1 E E e m3 1 F e 1 m4 1 1 1 Q 1 1 F q f T1 E E e m5 1 F e 1 1 m6 f T2 Q q • • 1 Variance = Q + F + E Covariance = πQ + F + E f T2 Q q Extensions to More Complex Applications • Endophenotypes • Linkage Analysis • Association Analysis Basic Linkage (QTL) Model = p(IBD=2) + .5 p(IBD=1) 1 1 1 1 E1 F1 Q1 e f P1 q Pihat 1 1 1 Q2 F2 E2 q f e P2 Q: QTL Additive Genetic F: Family Environment E: Random Environment 3 estimated parameters: q, f and e Every sibship may have different model Measurement Linkage (QTL) Model = p(IBD=2) + .5 p(IBD=1) 1 1 1 1 E1 F1 Q1 e f Pihat 1 1 1 Q2 F2 E2 q q P1 l6 M6 M5 M4 M2 e P2 l1 M3 f l1 M1 M1 l6 M2 M3 M4 M5 M6 q f e F1 Q 1 1 1 E 1 1 Q: QTL Additive Genetic F: Family Environment E: Random Environment 3 estimated parameters: q, f and e Every sibship may have different model Fulker Association Model M Geno1 Geno2 G1 G2 0.50 0.50 -0.50 0.50 S D m m b Multilevel model for the means w B W 1.00 1.00 -1.00 1.00 LDL1 LDL2 R R C 0.75 Measurement Fulker Association Model (SM) M Geno1 Geno2 G1 G2 0.50 0.50 0.50 0.50 S D m m b w B W 1.00 1.00 1.00 1.00 F1 M1 M1 w 0. 50 b m D m S 0. 50 0. 50 0. 50 G 2 G 1 Gen o2 Gen o1 M M2 B M3 W M4 1. 00 M5 l1 1. 00 1. 00 M6 l1 1. 00 l6 F2 l6 M2 M3 M4 M5 M6 Multivariate Linkage & Association Analyses • Computationally burdensome • Distribution of test statistics questionable • Permutation testing possible – Even heavier burden – Sarah Medland’s rapid approach • Potential to refine both assessment and genetic models • Lots of long & wide datasets on the way – Dense repeated measures: EMA; fMRI(!) – Need to improve software! Open source Mx