 Download Download Presentation Summarizing Variation

# Summarizing Variation

Télécharger la présentation ## Summarizing Variation

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
##### Presentation Transcript

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

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

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

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

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

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

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

8. 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

9. 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

10. Variance • Measure of Spread • Easily calculated • Individual differences

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

12. Measuring Variation Weighs & Means • Absolute differences? • Squared differences? • Absolute cubed? • Squared squared?

13. Measuring Variation Ways & Means • Squared differences Fisher (1922) Squared has minimum variance under normal distribution

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

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

16. Deviations in two dimensions :x dx + dy :y

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

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

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

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

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

22. 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

23. 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)

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

25. Biometrical Model of QTL m - a +a d

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

27. 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

28. 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

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

30. 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

31. 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

32. ACE model MZ twins reared together

33. ACE model DZ twins reared together

34. ACE model DZ twins reared apart

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

36. 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

37. 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

38. 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

39. 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

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

41. Perverse Universe A E .7 .7 P NOT!

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

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

44. 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

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

46. 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

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

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