1. Introducing Bayesian Approaches to Twin Data Analysis
Lindon Eaves,
VIPBG, Richmond.
Boulder, March 2001

2. Outline Why use a Bayesian approach? Basic concepts “BUGS”
Live Demo of simple application
Applications to twin data

3. Why Use Bayesian Approach?
Intellectually satisfying
Get more information out of existing problems (distributions of model parameters, individual“genetic” scores)
Tackle problems other methods find difficult (non-linear mixed models – growth curves; GxE interaction)

4. Some references
Gilks, W.R., Richardson, S., Spiegelhalter, D.J. (1996) Markov Chain Monte Carlo in Practice. Chapman & Hall, London.
Spiegelhalter, D., Thomas, A., Best, N. (2000) WinBUGS Version 1.3, User Manual, MRC BUGS Project: Cambridge.
Eaves, L.J., Erkanli, A. (In preparation) Markov Chain Monte Carlo Approaches to Analysis of Genetic and Environmental Components of Human Developmental Change and GxE Interaction. (For Behavior Genetics).

5. The Traditional Approach: Via Likelihood
Given Data D and parameters q:
The likelihood function, l, is
l=P(D|q).
We find q that maximizes l.

6. Typically Maximize likelihood numerically
Fairly easy for linear models and normal variables (“LISREL”)
Mx works well (best!)

7. Some things don’t work so well
BUT….
Some things don’t work so well

8. For example: Getting confidence intervals etc.
Non-linear models (require integration over latent variables – hard for large # of parameters)
Estimating large numbers of latent variables (e.g. “genetic factor scores”)

9. Markov Chain Monte Carlo Methods: (MCMC)
Allow more general models
Obtain confidence intervals and other summary statistics
Estimates missing values
Estimates latent trait values
All as part of the model-fitting process

10. Bayesian approach l=P(D|q). B=P(q|D). ML works with
Bayesian approach seeks distribution of parameters given data:
B=P(q|D).

11. Use Bayes theorem: P(q|D)=P( q & D)/P(D) = P(D|q).P(q)/P(D)
How do we get P(q|D)?
Use Bayes theorem:
P(q|D)=P( q & D)/P(D)
= P(D|q).P(q)/P(D)

12. A couple of problems
We don’t know P(q)
What is P(D)?

13. P(q) “Prior” distribution not known but
may know (guess?) its form, e.g.,
Means may be normal
Variances may be gamma

14. P(D)=SP(D|q).P(q)dq Where S=integral sign (!)

15. (“Monte Carlo” integration)
How do we get integral?
If we know P(q) we could sample q many times and evaluate function. Integral is approximated to desired accuracy by mean of k (=large) samples
(“Monte Carlo” integration)

16. We don’t know P(q) We only know its “shape”
We still have a problem..
We don’t know P(q)
We only know its “shape”

17. “Markov Chain” Monte Carlo…
Simulate a sequence of samples of q that ultimately converge to (non-independent) samples from the desired distribution, P(q).

18. If we succeed…
When the sequence has converged (“stationary distribution”, after “burn in” from trial q) we may construct P(q) from sequence of samples.

19. One algorithm that can generate chains in large number of cases…
…The “Gibbs Sampler”, hence:
“Bayesian Inference Using Gibbs Sampling”
“BUGS” for short
Spiegelhalter, D., Thomas, A., Best, N. (2000) WinBUGS Version 1.3, User Manual, MRC BUGS Project: Cambridge.

20. Obtaining WinBUGS Find MRC BUGS project on www (search on WinBUGS)
Download educational version (free)
Register by (at site)
Install educational version (Instructions at site)
Follow instructions in reply to convert to production version (free)

21. Preview of example: Using BUGS to estimate a mean and variance

22. Data and Initial Values for Mean-Variance Problem
list(n=50) y[] list(mu=10,tau=0.2)

23. “Doodle” for mean and variance model

24. BUGS Code for Mean and Variance

25. First 200 iterations of MCMC Algorithm
Values of Mean (mu):
First 200 iterations of MCMC Algorithm

26. Values of Variance (Sigma2): First 200 iterations of MCMC Algorithm

27. MCMC Estimates of Mean and Variance:
5000 iterations after 1000 iteration “burn in”.

28. Application to Twin Data
Fitting the AE model to bivariate twin data

29. Table 1: Population parameter values used in simulation of bivariate
twin data and values realized using Mx for ML estimation
(N=100 MZ and 100 DZ pairs).
Parameter
ML estimate
Population value
mu[1]
9.993
10.0
mu[2]
10.047
sigma2.g[1,1]
0.704
0.8
sigma2.g[1,2]
0.371
0.4
sigma2.g[2,2]
0.741
sigma2.e[1,1]
0.194
0.2
sigma2.e[1,2]
0.098
0.1
sigma2.e[2,2]
0.254

30. Doodle for Multivariate AE Model

31. Start of Data for Bivariate Twin Example
list(N=2,nmz=100,ndz=100,mean=c(0,0), precis =structure(.Data=c(0.0001,0, 0, ),.Dim=c(2,2)), omega.g=structure(.Data=c(0.0001,0,0,0.0001),.Dim=c(2,2)), omega.e=structure(.Data=c(0.0001,0,0,0.0001),.Dim=c(2,2))) ymz[,1,1] ymz[,1,2] ymz[,2,1] ymz[,2,2]

32. Iteration history for estimates of means

33. Iteration History for Genetic Covariances

34. Summary statistics for 5000 MCMC iterations of bivariate AE model after 2000 iteration "burn in"
node mean sd MC error 2.5% median 97.5%
deviance
mu[1]
mu[2] g[1,1] g[1,2] g[2,2]
e[1,1] e[1,2] E e[2,2]

35. Comparison of ML and MCMC Estimates for Bivariate AE model
Parameter ML
MCMC
Mu(1)
10.00
Mu(2)
10.05
G(1,1)
0.704
0.705
G(1,2)
0.371
0.372
G(2,2)
0.741
0.739
E(1,1)
0.194
0.197
E(1,2)
0.098
0.099
E(2,2)
0.254
0.258

36. Illustrative MCMC estimates of genetic effects:
first two DZ twin pairs on two variables
Observation Est S.e MC error % Median %
g1dz[1,1,1]
g1dz[1,1,2]
g1dz[1,2,1] g1dz[1,2,2] g1dz[2,1,1] g1dz[2,1,2] g1dz[2,2,1]
g1dz[2,2,2]