Summarizing Variation

Summarizing Variation Michael C Neale PhDVirginia Institute for Psychiatric and Behavioral GeneticsVirginia Commonwealth University

Overview • Mean • Variance • Covariance • Not always necessary/desirable

Computing Mean • Formula E(xi)/N • Can compute with • Pencil • Calculator • SAS • SPSS • Mx

One Coin toss 2 outcomes Probability 0.6 0.5 0.4 0.3 0.2 0.1 0 Heads Tails Outcome

Two Coin toss 3 outcomes Probability 0.6 0.5 0.4 0.3 0.2 0.1 0 HH HT/TH TT Outcome

Four Coin toss 5 outcomes Probability 0.4 0.3 0.2 0.1 0 HHHH HHHT HHTT HTTT TTTT Outcome

Ten Coin toss 9 outcomes Probability 0.3 0.25 0.2 0.15 0.1 0.05 0 Outcome

Pascal's Triangle Probability Frequency 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1/1 1/2 1/4 1/8 1/16 1/32 1/64 1/128 Pascal's friendChevalier de Mere 1654; Huygens 1657; Cardan 1501-1576

Fort Knox Toss Infinite outcomes 0.5 0.4 0.3 0.2 0.1 0 -4 -3 -2 -1 0 1 2 3 4 Heads-Tails Series 1 Gauss 1827

Variance • Measure of Spread • Easily calculated • Individual differences

Average squared deviation Normal distribution : xi di -3 -2 -1 0 1 2 3 Variance =Gdi2/N

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

Covariance • Measure of association between two variables • Closely related to variance • Useful to partition variance

Deviations in two dimensions :x + + + + + + + + + + + + + + :y + + + + + + + + + + + + + + + + + + +

Deviations in two dimensions :x dx + dy :y

Measuring Covariation Area of a rectangle • A square, perimeter 4 • Area 1 1 1

Measuring Covariation Area of a rectangle • A skinny rectangle, perimeter 4 • Area .25*1.75 = .4385 .25 1.75

Measuring Covariation Area of a rectangle • Points can contribute negatively • Area -.25*1.75 = -.4385 1.75 -.25

Measuring Covariation Covariance Formula F = E(xi - :x)(yi - :y) xy (N-1)

Correlation • Standardized covariance • Lies between -1 and 1 r = F xy xy 2 2 F * F y x

Summary Formulae : = (Exi)/N Fx= E(xi - :)/(N-1) 2 2 Fxy= E(xi-:x)(yi-:y)/(N-1) r = F xy xy 2 2 F * F y x

Variance covariance matrix Several variables Var(X) Cov(X,Y) Cov(X,Z) Cov(X,Y) Var(Y) Cov(Y,Z) Cov(X,Z) Cov(Y,Z) Var(Z)

Conclusion • Means and covariances • Conceptual underpinning • Easy to compute • Can use raw data instead

Biometrical Model of QTL m - a +a d

Biometrical model for QTL Diallelic locus A/a with p as frequency of a

Classical Twin Studies Information and analysis • Summary: rmz & rdz • Basic model: A C E • rmz = A + C • rdz = .5A + C • var = A + C + E • Solve equations

Contributions to Variance Single genetic locus • Additive QTL variance • VA = 2p(1-p) [ a - d(2p-1) ]2 • Dominance QTL variance • VD= 4p2 (1-p)2 d2 • Total Genetic Variance due to locus • VQ = VA + VD

Origin of Expectations Regression model • P = aA + cC + eE • Standardize A C E • VP = a2 + c2 + e2 • Assumes A C E independent

Path analysis Elements of a path diagram • Two sorts of variable • Observed, in boxes • Latent, in circles • Two sorts of path • Causal (regression), one-headed • Correlational, two-headed

Rules of path analysis • Trace path chains between variables • Chains are traced backwards, then forwards, with one change of direction at a double headed arrow • Predicted covariance due to a chain is the product of its paths • Predicted total covariance is sum of covariance due to all possible chains

ACE model MZ twins reared together

ACE model DZ twins reared together

ACE model DZ twins reared apart

Model fitting • Takes care of replicate statistics • Maximum likelihood estimates • Confidence intervals on parameters • Overall fit of model • Comparison of nested models

Fitting models to covariance matrices • MZ covariances • 3 statistics V1 CMZ V2 • DZ covariances • 3 statistics V1 CDZ V2 • Parameters: a c e • Df = nstat - npar = 6 - 3 = 3

Model fitting to covariance matrices • Inherently compares fit to saturated model • Difference in fit between A C E model and A E model gives likelihood ratio test with df = difference in number of parameters

Confidence intervals • Two basic forms • covariance matrix of parameters • likelihood curve • Likelihood-based has some nice properties; squares of CIs on a give CI's on a2 Meeker & Escobar 1995; Neale & Miller, Behav Genet 1997

Multivariate analysis • Comorbidity • Partition into relevant components • Explicit models • One disorder or two or three • Longitudinal data analysis • Partition into new/old • Explicit models • Markov • Growth curves

Cholesky Decomposition Not a model • Provides a way to model covariance matrices • Always fits perfectly • Doesn't predict much else

Perverse Universe A E .7 .7 P NOT!

Perverse Universe A E .7 .7 .7 -.7 X Y r(X,Y)=0; Problem for almost any multivariate method

Analysis of raw data • Awesome treatment of missing values • More flexible modeling • Moderator variables • Correction for ascertainment • Modeling of means • QTL analysis

Technicolor Likelihood Function For raw data in Mx m ln Li=fi3ln [wjg(xi,:ij,Gij)] j=1 xi- vector of observed scores onn subjects :ij - vector of predicted means Gij - matrix of predicted covariances - functions of parameters

Pihat Linkage Model for Siblings Each sib pair i has different COVARIANCE

Mixture distribution model Each sib pair i has different set of WEIGHTS rQ=.0 rQ=1 rQ=.5 weightj x Likelihood under model j p(IBD=2) x P(LDL1 & LDL2 | rQ = 1 ) p(IBD=1) x P(LDL1 & LDL2 | rQ = .5 ) p(IBD=0) x P(LDL1 & LDL2 | rQ = 0 ) Total likelihood is product of weighted likelihoods

Conclusion • Model fitting has a number of advantages • Raw data can be analysed with greater flexibility • Not limited to continuous normally distributed variables

Conclusion II • Data analysis requires creative application of methods • Canned analyses are of limited use • Try to answer the question!

Summarizing Variation