Introducing Bayesian Approaches to Twin Data Analysis

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Introducing Bayesian Approaches to Twin Data
Analysis

• Lindon Eaves,
• VIPBG, Richmond.
• Boulder, March 2001

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Outline

• Why use a Bayesian approach?
• Basic concepts
• BUGS
• Live Demo of simple application
• Applications to twin data

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Why Use Bayesian Approach?

• Intellectually satisfying
• Get more information out of existing problems
  (distributions of model parameters,
  individualgenetic 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).
•  

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The Traditional ApproachVia Likelihood

• Given Data D and parameters q
• The likelihood function, l, is
• lP(Dq).
• We find q that maximizes l.

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Typically

• Maximize likelihood numerically
• Fairly easy for linear models and normal
  variables (LISREL)
• Mx works well (best!)

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BUT.

• Some things dont work so well

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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)

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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

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Bayesian approach

• ML works with
• lP(Dq).
• Bayesian approach seeks distribution of
  parameters given data
• BP(qD).

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How do we get P(qD)?

• Use Bayes theorem
• P(qD)P( q D)/P(D)
• P(Dq).P(q)/P(D)

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A couple of problems

• We dont know P(q)
• What is P(D)?

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P(q)

• Prior distribution not known but
• may know (guess?) its form, e.g.,
• Means may be normal
• Variances may be gamma

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P(D)

• P(D)SP(Dq).P(q)dq
• Where Sintegral sign (!)

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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)

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We still have a problem..

• We dont know P(q)
• We only know its shape

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Markov Chain Monte Carlo

• Simulate a sequence of samples of q that
  ultimately converge to (non-independent) samples
  from the desired distribution, P(q).

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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.

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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.

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Obtaining WinBUGS

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

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Preview of example Using BUGS to estimate a
mean and variance
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list(n50)y 14.1110 9.5125 13.2752
10.5952 8.6699... 10.0673 10.7618
8.2337 9.2170 7.3803 8.9194
4.9589list(mu10,tau0.2)
Data and Initial Values for Mean-Variance Problem
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Doodle for mean and variance model
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BUGS Code for Mean and Variance
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Values of Mean (mu) First 200 iterations of MCMC
Algorithm
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Values of Variance (Sigma2) First 200 iterations
of MCMC Algorithm
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MCMC Estimates of Mean and Variance 5000
iterations after 1000 iteration burn in.
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Application to Twin Data

• Fitting the AE model to bivariate twin data

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Table 1 Population parameter values used in
simulation of bivariate twin data and values
realized using Mx for ML estimation (N100 MZ
and 100 DZ pairs).    
 
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Doodle for Multivariate AE Model
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list(N2,nmz100,ndz100,meanc(0,0),precis
structure(.Datac(0.0001,0, 0,
0.0001),.Dimc(2,2)),omega.gstructure(.Datac(0.
0001,0,0,0.0001),.Dimc(2,2)),omega.estructure(.
Datac(0.0001,0,0,0.0001),.Dimc(2,2)))ymz,1,1
ymz,1,2 ymz,2,1 ymz,2,2 9.9648 9.4397
10.1008 9.6549 8.9251 10.5722 9.5299 10.5583
10.7032 9.9130 11.1373 10.2855 10.8231 11.5187
11.0396 10.7342 11.3261 12.4088 11.4504 11.9600
9.4009 10.7828 9.5653 11.8201
Start of Data for Bivariate Twin Example
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Iteration history for estimates of means
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Iteration History for Genetic Covariances
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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 de
viance 985.7 67.7 2.91 856.7 985.4 1118.0
mu1 9.991 0.05906 0.001452 9.877 9.992 10.11
mu2 10.05 0.06184 0.001871 9.924 10.05 10.17
g1,1 0.705 0.08122 0.003302 0.5567 0.7018 0.8
731 g1,2 0.3719 0.06418 0.0024 0.252 0.3711 0.
5065 g2,2 0.7394 0.08337 0.002786 0.5885 0.736
7 0.9113 e1,1 0.197 0.02872 0.001223 0.1476
0.1943 0.2606 e1,2 0.09904 0.02398 9.144E-4 0.
05537 0.09788 0.1486 e2,2 0.2581 0.03528 0.0012
3 0.1977 0.2555 0.3337
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Comparison of ML and MCMC Estimates for Bivariate
AE model
Parameter ML MCMC
Mu(1) 10.00 10.00
Mu(2) 10.05 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
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Illustrative MCMC estimates of genetic effects
first two DZ twin pairs on two variables
Observation Est S.e.
MC error 2.5 Median 97.5
g1dz1,1,1 11.53 0.3756 0.007243 10.77 11.53 12
.27 g1dz1,1,2 10.94 0.4263 0.007021 10.1 10.9
4 11.78 g1dz1,2,1 11.44 0.3763 0.006501 10.7
11.43 12.18 g1dz1,2,2 10.93 0.4286 0.007602 10
.08 10.95 11.76 g1dz2,1,1 9.231 0.3776 0.00786
6 8.499 9.228 9.992 g1dz2,1,2 10.25 0.4248 0.0
07331 9.409 10.25 11.09 g1dz2,2,1 9.331 0.373
0.005969 8.606 9.329 10.07 g1dz2,2,2 9.5
05 0.4188 0.007975 8.694 9.5 10.31