Module 4: Bayesian Methods Lecture 7: Generalized Linear Modeling

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1 Module 4: Bayesian Methods Lecture 7: Generalized Linear Modeling Jon Wakefield Departments of Statistics and Biostatistics University of Washington Outline Introduction and Motivating Examples Generalized Linear Models Definition Bayes Linear Model Bayes Logistic Regression Generalized Linear Mixed Models Approximate Bayes Inference The Approximation Case-Control Example Conclusions

2 Introduction In this lecture we will discuss Bayesian modeling in the context of Generalized Linear Models (GLMs). This discussion will include the addition of random effects, to give Generalized Linear Mixed Models (GLMMs). Estimation via a quick and R based technique (INLA) will be demonstrated. An approximation technique that is useful in the context of Genome Wide Association Studies (GWAS) (in which the number of tests is large) will also be introduced. Motivating Example I: Logistic Regression We consider case control data for the disease Leber Hereditary Optic Neuropathy (LHON) disease with genotype data for marker rs : CC CT TT Total Cases Controls Total Let x =0, 1, 2 represent the number of T alleles, and p(x) the probability of being a case, given x alleles. Within a likelihood framework one may fit the multiplicative odds model: p(x) 1 p(x) = exp(α) exp(θx). Interpretation: exp(α) is of little use given the case-control sampling. exp(θ) is the log odds ration describing the multiplicative change in risk given an additional T allele.

3 R code for Logistic Regression Estimation via Likelihood x < c(0,1,2) y < c(6,8,75) z < c(10,66,163) logitmod < glm( cbind (y, z ) x, family= binomial ) thetahat < logitmod$coeff [2] # Log odds ratio thetahat x exp( thetahat ) # Odds ratio x # Odds of disease are associated with an increase # of 61% for each extra T V < vcov( logitmod )[2,2] # standard error ˆ2 # Asymptotic confidence interval for odds ratio > exp( thetahat 1.96 sqrt(v)) x > exp( thetahat sqrt(v)) x # So 95% interval ( just ) contains 1. R code for Logistic Regression Hypothesis Testing via a LRT #Now let s look at a likelihood ratio test > logitmod Call : glm(formula = cbind(y, z) x, family = binomial ) Coefficients : (Intercept) x Degrees of Freedom : 2 Total ( i. e. Null ); 1 Residual Null Deviance : Residual Deviance : AIC: > pchisq( ,1, lower. tail=f) [1] # So just significant at the 5% level. So for these data both estimation and testing point towards borderline significance. Under Hardy Weinberg Equilibrium (HWE) we would estimate (in the controls) the minor allele (C) frequency (MAF) as /2 239 which helps to explain the lack of power. =0.18,

4 Motivating Example II: FTO Data Revisited y = weight x g = fto heterozygote {0, 1} x a = age in weeks {1, 2, 3, 4, 5} We examine the fit of the model Linear Model Example E[Y x g, x a ]=β 0 + β g x g + β a x a + β int x g x a. > fto < read. table ( http ://www. stat. washington. edu/ hoff/sisg/fto data. txt header=true) > liny < fto$y > linxg < fto$xg > linxa < fto$xa > linxint < fto$xgxa > ftodf < list(liny=liny,linxg=linxg,linxa=linxa, linxint=linxint) > ols. fit < lm( l i n y l i n x g+l i n x a+l i n x i n t, data=ftodf ) > summary( ols. f i t ) Coefficients : Estimate Std. Error t value Pr(> t ) (Intercept) linxg linxa e 06 linxint Motivating Example III: Salamander Data McCullagh and Nelder (1989) describe data on the success of matings between male and female salamanders of two population types (roughbutts, RB, and whitesides, WS). The experimental design is complex but involves three experiments having multiple pairings, with each salamander being involved in multiple matings, so that the data are not independent. One way of modeling the dependence is through crossed random effects. The first lab experiment was in the summer of 1986, and the second and third were in the fall of the same year.

5 Motivating Example III: Salamander Data There are 20 females and 20 males, with half of each gender being RB and half being WS. In each experiment each female is paired with 3 males (to give 30 matings in total), and each male is paired with 3 females (to give 30 matings in total). The data are: Experiment 1 Experiment 2 Experiment 3 Y % Y % Y % RR RW WR WW The Observed number (Y ) and percentage (%) of successes out of 30 matings in three experiments. Motivating Example III: Salamander Data Mixed effects models consist of fixed effects and random effects. Fixed effects are population characteristics, while random effects are unit-specific quantities that are assigned a distribution. Mixed effects models allow dependencies in responses to be modeled. Example: The simplest case is the one-way ANOVA model with n observations in each of m groups: i =1,..., m; j =1,..., n. Y ij = β 0 + b i + ij ij iid N(0,σ 2 ) b i iid N(0,σ 2 b) In this model, β 0 is the fixed effect and b 1,..., b m are random effects.

6 Crossed versus Nested Designs Suppose we have two factors, A and B with a and b levels respectively. If each level of A is crossed with each level of B we have a factorial design. In a nested situation j = 1 in level 1 of factor A has no meaningful connection with j = 1 in level 2 of factor A. Treatment Treatment Subject Subject Left: Crossed design. Right: Nested design. The symbols show where observations are measured. Motivating Example III: Salamander Data Let Y ijk denote the response (failure/success) for female i and male j in experiment k. There are 360 binary responses in total. For illustration we fit model C that was previously considered by Karim and Zeger (1992) and Breslow and Clayton (1993): logit Pr(Y ijk =1 β, b f ik, b m jk) =x ijk β k + b f ik + b m jk where x ijk is a 1 4 vector representing the intercept and indicators for female WS, male WS, and male and female both WS, and β k is the corresponding fixed effect (so that this model allows the fixed effects to vary by experiment). The model contains six random effects: b f ik iid N(0,σ 2 fk), b m ik iid N(0,σ 2 mk), k =1, 2, 3 one for each of males and females, and in each experiment.

7 Generalized Linear Models Generalized Linear Models (GLMs) provide a very useful extension to the linear model class. GLMs have three elements: 1. The responses follow an exponential family. 2. The mean model is linear in the covariates on some scale. 3. A link function relates the mean of the data to the covariates. In a GLM the response y i are independently distributed and follow an exponential family so that the distribution is of the form where θ i and α are scalars. p(y i θ i,α) = exp({y i θ i b(θ i )}/α + c(y i,α)), Examples: Normal, Poisson, binomial. It is straightforward to show that E[Y i θ i,α]=µ i = b (θ i ), var(y i θ i,α)=b (θ i )α, for i =1,..., n, with cov(y i, Y j θ i,θ j,α) = 0 for i = j. Generalized Linear Models The link function g( ) provides the connection between µ = E[Y θ, α] and the linear predictor xβ, via g(µ) =xβ, where x is a vector of explanatory variables and β is a vector of regression parameters. For normal data, the usual link is the identity g(µ) =µ. For binary data, a common link is the logistic «µ g(µ) = log. 1 µ For Poisson data, a common link is the log g(µ) = log (µ).

8 Bayesian Modeling with GLMs For a generic GLM, with regression parameters β and a scale parameter α, the posterior is p(β,α y) p(y β,α) p(β,α). Two problems immediately arise: How to specify a prior distribution p(β,α)? How to perform the computations required to summarize the posterior distribution (including the calculation of Bayes factors). Bayesian Computation Various approaches are available: Conjugate analysis the prior combines with likelihood in such a way as to provide analytic tractability (at least for some parameters). Analytical Approximations asymptotic arguments used (e.g. Laplace). Numerical integration. Direct (Monte Carlo) sampling from the posterior, as we have already seen. Markov chain Monte Carlo very complex models can be implemented, for example within the free software WinBUGS. Integrated nested Laplace approximation (INLA). Cleverly combines analytical approximations and numerical integration.

9 Integrated Nested Laplace Approximation (INLA) To download INLA: install.packages(c( fields, numderiv, pixmap, mvtnorm, R. utils )) source( http :// ntnu.no/inla/givemeinla.r ) library(inla) inla.upgrade(testing=true) The homepage of the software is here: The fitting of many common models is described here: There are also lots of examples. INLA can fit GLMs, GLMMs and many other classes. INLA for the Linear Model We first fit a linear model to the FTO data with the default prior settings. > liny < fto$y > linxg < fto$x [,2] > linxa < fto$x [,3] > linxint < fto$x [,4] > ftodf < list(liny=liny,linxg=linxg,linxa=linxa, linxint=linxint) > lin.mod < inla(liny linxg+linxa+linxint,data=ftodf, family= gaussian ) > summary( l i n.mod) Fixed effects : mean sd quant 0.5 quant quant (Intercept) linxg linxa linxint Model hyperparameters : mean sd quant 0.5 quant Precision for the Gaus obsers quant Precision for the Gaus observations The posterior means and standard deviations are in very close agreement with the OLS fits presented earlier.

10 R Code for Marginal Distributions betaxggrid < lin.mod$marginals. fixed$linxg [,1] betaxgmarg < lin.mod$marginals. fixed$linxg [,2] betaxagrid < lin.mod$marginals. fixed$linxa [,1] betaxamarg < lin.mod$marginals. fixed$linxa [,2] betaxintgrid < lin.mod$marginals. fixed$linxint [,1] betaxintmarg < lin.mod$marginals. fixed$linxint [,2] precgrid < lin.mod$marginals.hyperpar$ Precision for the Gaussian observations [,1] precmarg < lin.mod$marginals.hyperpar$ Precision for the Gaussian observations [,2] par(mfrow=c(2,2)) plot(betaxgmarg betaxggrid, type= l,xlab=expression(beta [g]), ylab= Marginal Density,cex. lab=1.5) abline(v=0,col= red ) plot(betaxamarg betaxagrid, type= l,xlab=expression(beta [a]), ylab= Marginal Density,cex. lab=1.5) abline(v=0,col= red ) plot(betaxintmarg betaxintgrid, type= l,xlab=expression(beta [ int ]), ylab= Marginal Density,cex. lab=1.5) abline(v=0,col= red ) plot(precmarg precgrid, type= l,xlab=expression(sigmaˆ{ 2}), ylab= Marginal Density,cex. lab=1.5) FTO Posterior Marginal Distributions Marginal Density Marginal Density β g β a Marginal Density Marginal Density β int σ 2 Marginal distributions of the three regression coefficients corresponding to xg, xa and the interaction, and the precision.

11 FTO Extended Analysis In order to carry out model checking we rerun the analysis, but now switch on a flag to obtain fitted values. lin.mod < inla(liny linxg+linxa+linxint,data=ftodf, family= gaussian,control. predictor=list (compute=true)) fitted < lin.mod$summary. fitted.values [,1] sigmamed < 1/ sqrt (.2924) With the fitted values we can examine the fit of the model. In particular: Normality of the errors (sample size is relatively small). Errors have constant variance (and are uncorrelated). Linear model is adequate. Assessing the Model residuals < (liny fitted)/sigmamed par(mfrow=c(2,2)) qqnorm( residuals, main= ) title ( (a) ) abline (0,1, lty=2,col= red ) plot( residuals linxa, ylab= Residuals,xlab= Age ) title ( (b) ) abline(h=0,lty=2,col= red ) plot( residuals fitted, ylab= Residuals,xlab= Fitted ) title ( (c) ) abline(h=0,lty=2,col= red ) plot( fitted liny, xlab= Observed,ylab= Fitted ) title ( (d) ) abline (0,1, lty=2,col= red )

12 (a) FTO Diagnostic Plots (b) Sample Quantiles Residuals Theoretical Quantiles Age (c) (d) Residuals Fitted Fitted Observed Plots to assess model adequacy: (a) Normal QQ plot, (b) residuals versus age, (c) residuals versus fitted, (d) fitted versus observed. Bayes Logistic Regression The likelihood is Logistic link: The prior is Y (x) p(x) Binomial(N(x), p(x)), x =0, 1, 2. «p(x) log = α + θx 1 p(x) p(α, θ) =p(α) p(θ) with α N(µ α,σ α ) and θ N(µ θ,σ θ ). The first analysis uses the default priors in INLA (which are relatively flat). In the second analysis we specify α N(0, 1/0.1) and θ N(0, W ) where W is such that the 97.5% point of the prior is log(1.5), i.e. we believe the odds ratio lies between 2/3 and 3/2 with probability 0.95.

13 R Code for Logistic Regression Example > x < c(0,1,2) > y < c(6,8,75) > z < c(10,66,163) > cc. dat < list(x=x,y=y,z=z) > cc. dat < as. data. frame( rbind (y,z,x)) > cc.mod < inla(y x, family= binomial,data=cc.dat, Ntrials=y+z) > summary( cc.mod) mean sd quant 0.5 quant quant (Intercept) x #Now with informative priors > Upper975 < log (1.5) > W < (log(upper975)/1.96)ˆ2 > cc.mod2 < inla(y x, family= binomial,data=cc.dat, Ntrials=y+z, control. fixed=list (mean. intercept=c(0),prec. intercept=c(.1), mean=c (0), prec=c (1/W) ) ) > summary( cc. mod2) mean sd quant 0.5 quant quant (Intercept) x The quantiles for θ can be translated to odds ratios by exponentiating. R Code For Extracting the Marginal Densities We obtain plots of the marginal distributions of α and θ, from the model with informative priors. Alternatively, can use plot.inla. alphagrid < cc. mod2$marginals. fixed$ ( Intercept ) [,1] alphamarg < cc. mod2$marginals. fixed$ ( Intercept ) [,2] thetagrid < cc. mod2$marginals. fixed$x [,1] thetamarg < cc. mod2$marginals. fixed$x [,2] par(mfrow=c(1,2)) plot(alphamarg alphagrid, type= l,xlab=expression(alpha), ylab= Marginal Density,cex. lab=1.5) plot(thetamarg thetagrid, type= l,xlab=expression(theta ), ylab= Marginal Density,cex. lab=1.5) abline(v=0,col= red )

14 Logistic Marginal Plots Marginal Density Marginal Density α θ Posterior marginals for the intercept α and the log odds ratio θ. A Simple ANOVA Example We begin with simulated data from the simple one-way ANOVA model example: Y ij = β 0 + b i + ij ij iid N(0,σ 2 ) b i iid N(0,σ 2 b) i =1,..., 10; j =1,..., 5, with β 0 =0.5, σ 2 =0.2 2 and σ 2 b = Simulation: sigma.b < 0.3 sigma. e < 0.2 m < 10 ni < 5 beta0 < 0.5 b < rnorm(m, mean=0, sd=sigma. b) e < rnorm(m ni, mean=0, sd=sigma. e) Yvec < beta0 + rep(b, each=ni ) + e simdata < data. frame(y=yvec, ind=rep (1:m, each=ni ))

15 A Simple ANOVA Example > result < inla(y f(ind, model= iid ), data = simdata) > summary( r e s u l t ) Fixed effects : mean sd quant 0.5 quant quant (Intercept) Model hyperparameters : mean sd quant 0.5 quant quant Precision for Gaus obs Precision for ind > sigma. est < 1/ sqrt (summary( hyper ) $hyperpar [,4]) > sigma. est Gaus obs ind # Extract the posterior medians # of the precision and invert The Salamander Data: INLA Analysis We assume that for each of the random effects the residual odds lies between 0.1 and 10 with probability 0.9, so that Ga(1,0.622) priors are used for each of the six precisions, σ 2 fk,σ 2 mk for k =1, 2, 3. See Fong et al. (2010) for more details. We present results for the fixed effects and variance components, with the model fitted using both restricted ML (REML) and INLA. We see the reduction in standard errors in the REML analysis. There is some attenuation of the Bayesian results here due to the INLA approximation strategy. Again, see Fong et al. (2010) for more details. Experiment 1 Experiment 2 Experiment 3 Variable REML INLA REML INLA REML INLA Intercept 1.34± ± ± ± ± ±0.73 WS F -2.94± ± ± ± ± ±0.92 WS M -0.42± ± ± ± ± ±0.94 WS F WS M 3.18± ± ± ± ± ±1.05 σ f ± ± ±0.28 σm ± ± ±0.48 REML and INLA summaries for Salamander data. For the entries marked with a standard errors were unavailable.

16 INLA Code for Salamander Data: Data Setup ## C r o s s e d Random E f f e c t s Salamander load( salam.rdata ) ## o r g a n i z e d a t a i n t o a f o r m s u i t a b l e f o r l o g i s t i c r e g r e s s i o n dat0 < data. frame( y =c(salam$y ), f W =as. integer(salam$x[, W/R ]==1 salam$x [, W/W ]==1), m W =a s. i n t e g e r ( s a l a m $ x [, R/W ]==1 salam$x [, W/W ]==1), W W = a s. i n t e g e r ( s a l a m $ x [, W/W ]==1 ) ) ## a d d s a l a m a n d e r i d id < t( apply(salam$z, 1, function(x) { tmp = which ( x==1) tmp [ 2 ] = tmp [ 2 ] 20 tmp }) ) ## i d s a r e s u i t a b l e f o r m o d e l A a n d C, b u t n o t B id.moda < rbind(id, id+40, id+20) colnames ( id.moda) < c( f.moda, m.moda ) dat0 < cbind (dat0, id.moda, group=1) dat0$experiment < as. factor (rep (1:3, each=120)) dat0$group < as. factor (dat0$group) # salamander < dat0 salamander. e1 < subset (dat0, dat0$experiment==1) salamander. e2 < subset (dat0, dat0$experiment==2) salamander. e3 < subset (dat0, dat0$experiment==3) INLA Code for Salamander Data: Model Fitting Below is the code for fitting to the data in experiment 1, along with the output. # salamander. e1 > salamander. e1. inla. fit < inla(y f.w + m.w + W.W + f(f.moda, model= iid, param=c(1,.622)) + f(m.moda, model= iid, param=c(1,.622)), family= binomial, data=salamander.e1, Ntrials=rep(1,nrow(salamander.e1))) > salamander. e1. hyperpar < inla.hyperpar(salamander.e1. inla. fit ) > summary( salamander. e1. i n l a. f i t ) Fixed effects : mean sd quant 0.5 quant quant (Intercept) f.w m.w W.W Random e f f e c t s : Model hyperparameters : mean sd quant 0.5 quant quant Precision for f.moda Precision for m.moda

17 Approximate Bayes Inference Particularly in the context of a large number of experiments, a quick and accurate model is desirable. We describe such a model in the context of a GWAS. We first recap the normal-normal Bayes model. Subsequently, we describe the approximation and provide an example. Recall: The Normal Posterior Distribution For the model Prior: θ normal(µ 0,τ0 2 ) and Likelihood: Y 1,..., Y n θ normal(θ, σ 2 ). Posterior θ y 1,..., y n } normal(µ, τ 2 ) where and var(θ y 1,..., y n )=τ 2 = [1/τ n/σ 2 ] 1 Precision = 1/τ 2 = 1/τ n/σ 2 E[θ y 1,..., y n ]=µ = µ 0/τ0 2 +ȳn/σ 2 1/τ0 2 + n/σ2 ««1/τ0 2 n/σ 2 = µ 0 1/τ ȳ n/σ2 1/τ0 2 + n/σ2

18 A Normal-Normal Approximate Bayes Model Consider again the logistic regression model with interest focusing on θ. logit p i = α + x i θ We require priors for α, θ, and some numerical/analytical technique for estimation/bayes factor calculation. Wakefield (2007, 2009) considered replacing the likelihood by the approximation p(θ b θ) p( b θ θ)p(θ) where b θ θ N(θ, V ) the asymptotic distribution of the MLE, θ N(0, W ) the prior on the log RR. Can choose W so that 95% of relative risks lie in some range, e.g. [2/3,1.5]. Posterior Distribution Under the alternative the posterior distribution for the log odds ratio θ is where θ b θ N(r b θ, rv ) r = W V + W. Hence, we have shrinkage to the prior mean of 0. The posterior median for the odds ratio is exp(r b θ) and a 95% credible interval is exp(r b θ ± 1.96 rv ). Note that as W the non-bayesian point and interval estimates are recovered.

19 A Normal-Normal Approximate Bayes Model We are interested in the hypotheses: H 0 : θ =0, evaluation of the Bayes factor H 1 : θ = 0 and BF = p(b θ H 0 ) p( b θ H 1 ). Using the approximate likelihood and normal prior we obtain: 1 Approximate Bayes Factor = exp Z «2 1 r 2 r, with Z = b θ V, r = W V +W. A Normal-Normal Approximate Bayes Model The approximation can be combined with a Prior Odds = (1 π 0 )/π 0 to give Posterior Odds on H 0 = BFDP = ABF Prior Odds 1 BFDP where BFDP is the Bayesian False Discovery Probability. BFDP depends on the power, through r. For implementation, all that we need from the data is the Z-score and the standard error V, or a confidence interval. Hence, published results that report confidence intervals can be converted into Bayes factors for interpretation (see Lecture 8). The approximation relies on large sample sizes, so the normal distribution of the estimator provides a good summary of the information in the data.

20 Bayesian Logistic Regression: Estimation > source( http :// faculty. washington.edu/jonno/bfdp.r ) # # 97.5 point of prior is log (1.5) so that we with prob # 0.95 we think theta lies in (2/3,1.5) > Upper975 < log (1.5) > W < (log(upper975)/1.96)ˆ2 > x < c(0,1,2) > y < c(6,8,75) > z < c(10,66,163) > logitmod < glm( cbind (y, z ) x, family= binomial ) > thetahat < logitmod$coef [2] > V < vcov( logitmod )[2,2] > r < W/ (V+W) > r [1] # Not so much data here, so weight on prior is high. # Bayesian posterior median > exp( r thetahat) x # Shrunk towards prior median of 1 # Bayesian approximate 95% credible interval > exp( r thetahat 1.96 sqrt(r V)) x > exp( r thetahat+1.96 sqrt(r V)) x Bayesian Logistic Regression: Hypothesis Testing Now we turn to testing using Bayes factors. We examine the sensitivity to the prior on the alternative, π 1. > pi1 < c(1/2,1/100,1/1000,1/10000,1/100000) > BFcall < BFDPfunV( thetahat,v,w, pi1 ) > BFcall $BF x $ph0 x $ph1 x $BFDP [1] So data are twice as likely under the alternative as compared to the null. Apart from the 0.5 prior, under these priors the overall evidence is of no association.

21 Combination of data across studies Suppose we wish to combine data from two studies where we assume a common log odds ratio θ.. The estimates from the two studies are b θ 1, b θ 2 with standard errors V 1 and V 2. The Bayes factor is The approximate Bayes factor is p( b θ 1, b θ 2 H 0 ) p( b θ 1, b θ 2 H 1 ). ABF( b θ 1, b θ 2 ) = ABF( b θ 1 ) ABF( b θ 2 b θ 1 ) (1) where and ABF( b θ 2 b θ 1 )= p(b θ 2 H 0 ) p( b θ 2 b θ 1, H 1 ) p( θ b 2 θ b 1, H 1 )=E θ θ1 b hp( θ b i 2 θ) so that the density is averaged with respect to the posterior for θ. Important Point: The Bayes factors are not independent. Combination of data across studies This leads to an approximate Bayes factor (which summarizes the data from the two studies) of r j ABF( θ b 1, θ b W 2 )= exp 1 Z1 2 RV 2 +2Z 1 Z 2 R ff V 1 V 2 + Z2 2 RV 1 RV 1 V 2 2 where R = W /(V 1 W + V 2 W + V 1 V 2 ) θ Z 1 = b 1 V1 and θ Z 2 = b 2 V2 are the usual Z statistics. The ABF will be small (evidence for H 1 ) when the absolute values of Z 1 and Z 2 are large and they are of the same sign.

22 Example of Combination of Studies in a GWAS Frayling et al. (2007) report a GWAS for Type II diabetes. For SNP rs : Pr(H 0 data) with prior: Stage Estimate (CI) p-value log 10 BF 1/5,000 1/50,000 1st 1.27 ( ) nd 1.15 ( ) Combined Combined evidence is stronger than each separately since the point estimates are in agreement. For summarizing inference the (5%, 50%, 95%) points for the RR are: Prior 1.00 ( ) First Stage 1.26 ( ) Combined 1.21 ( ) Conclusions Computationally GLMs and GLMMs can now be fitted in a relatively straightforward way. INLA is very convenient and is being constantly improved. As with all analyses, it is crucial to check modeling assumptions (and there are usually more in a Bayesian analysis). For binary observations INLA can produce inaccurate estimates. Markov chain Monte Carlo provides an alternative for computation. WinBUGS is one popular implementation. Other MCMC possibilities include: JAGS, BayesX.

23 References Breslow, N. and Clayton, D. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88, Fong, Y., H.Rue, and Wakefield, J. (2010). Bayesian inference for generalized linear mixed models. Biostatistics, 11, Frayling, T., Timpson, N., Weedon, M., Zeggini, E., Freathy, R., and et al., C. L. (2007). A common variant in the FTO gene is associated with body mass index and predisposes to childhood and adult obesity. Science, 316, Karim, M. and Zeger, S. (1992). Generalized linear models with random effects: Salamander mating revisited. Biometrics, 48, McCullagh, P. and Nelder, J. (1989). Generalized Linear Models, Second Edition. Chapman and Hall, London. Wakefield, J. (2007). A Bayesian measure of the probability of false discovery in genetic epidemiology studies. American Journal of Human Genetics, 81, Wakefield, J. (2009). Bayes factors for genome-wide association studies: comparison with p-values. Genetic Epidemiology, 33,