Bayesian Inference

Biostatistics (2010), 11, 3, pp. 397412 inside10. 1093/biostatistics/kxp053 Advance Access publication on December 4, 2009 Bayesian inference for generalized additive obscure feignings YOUYI FONG Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University program library on April 20, 2013 Department of Biostatistics, University of Washington, Seattle, WA 98112, USA ? HAVARD grieve Department of Mathematical Sciences, The Norwegian University for Science and Technology, N-7491 Trondheim, Norway JON WAKEFIELD? Departments of Statistics and Biostatistics, University of Washington, Seattle, WA 98112, USA emailprotected ashington. edu S UMMARY generalised linear obscure mystifys (GLMMs) continue to grow in popularity due to their ability to straight acknowledge multiple levels of dependency and dumbfound different entropy types. For smooth sample sizes especi on the wholey, likelihood-based inference smoke be unreliable with unevenness components being par ticularly difficult to estimate. A Bayesian arise is benevolent however has been hampered by the lack of a fast writ of execution, and the difficulty in specifying previous distrisolelyions with variance components again being particularly problematic.Here, we briefly review previous glide pathes to computation in Bayesian implementations of GLMMs and illust aim in detail, the social occasion of integrated nested Laplace approximations in this context. We consider a sum of examples, conservatively specifying antecedent statistical distributions on cogitateingful quantities in separately slip. The examples cover a wide range of selective information types including those requiring smoothing over time and a relation entirelyy complicated spline specimen for which we witness our earlier judicial admission in terms of the implied degrees of emancipation.We conclude that Bayesian inference is now practically feasible for GLMMs and provides an attractive selection to likelihood-based approaches such as penalized quasi-likelihood. As with likelihood-based approaches, great c be is postulate in the epitome of clustered binary star data since approximation strategies may be less(prenominal) accurate for such data. Keywords Integrated nested Laplace approximations Longitudinal data Penalized quasi-likelihood front spec Spline gets. 1.I NTRODUCTION Generalized linear interracial models (GLMMs) combine a generalized linear model with normal ergodic effects on the linear predictor shell, to apportion a bountiful family of models that mother been dropd in a wide variety of applications ( cypher, e. g. Diggle and otherwises, 2002 Verbeke and Molenberghs, 2000, 2005 McCulloch and others, 2008). This flexibility comes at a price, however, in terms of analytical tractability, which has a ? To whom concord should be addressed. c The Author 2009. Published by Oxford University Press. all in all rights reserved. For permissions, please e-mail j ournals. emailprotected rg. 398 Y. F ONG AND OTHERS number of implications including computational complexity, and an unknown degree to which inference is parasitical on modeling assumptions. Likelihood-based inference may be carried out relatively easily within many softw ar system platforms (except perhaps for binary responses), but inference is dependent on asymptotic sampling distributions of estimators, with few guidelines usable as to when such theory will produce accurate inference. A Bayesian approach is attractive, but requires the specification of forward distributions which is not straightforward, in particular for variance components.Computation is also an issue since the usual implementation is via Markov chain Monte Carlo (MCMC), which carries a freehanded computational overhead. The seminal article of Breslow and Clayton (1993) helped to popularize GLMMs and placed an fury on likelihood-based inference via penalized quasi-likelihood (PQL). It is the aim of this article to describe, through a series of examples (including all of those considered in Breslow and Clayton, 1993), how Bayesian inference may be performed with computation via a fast implementation and with guidance on prior specification. The structure of this article is as follows.In Section 2, we define notation for the GLMM, and in Section 3, we describe the integrated nested Laplace approximation (INLA) that has recently been proposed as a computationally convenient alternative to MCMC. Section 4 gives a number of prescriptions for prior specification. Three examples are considered in Section 5 (with additional examples being report in the supplementary material lendable at Biostatistics online, along with a cloak study that reports the cognitive operation of INLA in the binary response situation). We conclude the account with a discussion in Section 6. 2.T HE G ENERALIZED LINEAR MIXED MODEL GLMMs extend the generalized linear model, as proposed by Nelder and Wedderburn ( 1972) and comprehensively exposit in McCullagh and Nelder (1989), by adding normally distributed random effects on the linear predictor scale. recollect Yi j is of exponential function family form Yi j ? i j , ? 1 ? p(), whither p() is a member of the exponential family, that is, p(yi j ? i j , ? 1 ) = exp yi j ? i j ? b(? i j ) + c(yi j , ? 1 ) , a(? 1 ) Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 for i = 1, . . . , m building blocks (clusters) and j = 1, . . , n i , taxments per unit and where ? i j is the (scalar) ? crapperonical parameter. allow ? i j = EYi j ? , b i , ? 1 = b (? i j ) with g(? i j ) = ? i j = x i j ? + z i j b i , where g() is a monotonic link function, x i j is 1 ? p, and z i j is 1 ? q, with ? a p ? 1 vector of fixed ? Q effects and b i a q ? 1 vector of random effects, thus ? i j = ? i j (? , b i ). Assume b i Q ? N (0, Q ? 1 ), where ? the precision matrix Q = Q (? 2 ) depends on paramete rs ? 2 . For some plectrums of model, the matrix Q is singular examples entangle random passing game models (as considered in Section 5. ) and intrinsic qualified ? autoregressive models. We further pay that ? is assigned a normal prior distribution. Let ? = (? , b ) denote the G ? 1 vector of parameters assigned Gaussian priors. We also require priors for ? 1 (if not a constant) and for ? 2 . Let ? = (? 1 , ? 2 ) be the variance components for which non-Gaussian priors are ? assigned, with V = dim(? ). 3. I NTEGRATED NESTED L APLACE APPROXIMATION Before the MCMC revolution, on that point were few examples of the applications of Bayesian GLMMs since, outside of the linear obscure model, the models are analytically intractable.Kass and Steffey (1989) describe the physical exertion of Laplace approximations in Bayesian class-conscious models, while Skene and Wakefield Bayesian GLMMs 399 (1990) social occasiond numerical integration in the context of a binary GLMM. The use of MCMC for GLMMs is particularly appealing since the conditional independencies of the model may be exploited when the required conditional distributions are calculated. Zeger and Karim (1991) described approximate Gibbs sampling for GLMMs, with non example conditional distributions being approximated by normal distributions. more(prenominal) general cityHastings algorithms are straightforward to construct (see, e. g. Clayton, 1996 Gamerman, 1997). The winBUGS (Spiegelhalter, Thomas, and Best, 1998) software example manuals contain many GLMM examples. there are now a variety of additional software platforms for fitting GLMMs via MCMC including JAGS (Plummer, 2009) and BayesX (Fahrmeir and others, 2004). A large practical curb to data compend victimization MCMC is the large computational burden. For this reason, we now briefly review the INLA computational approach upon which we concentrate.The order combines Laplace approximations and numerical integration in a very efficient ma nner (see Rue and others, 2009, for a more than extensive treatment). For the GLMM described in Section 2, the back(prenominal) is given by m Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 ? y ? ? ? ?(? , ? y ) ? ?(? ? )? (? ) i=1 y ? p(y i ? , ? ) m i=1 1 ? ? Q ? ? b ? ?(? )? (? )Q (? 2 )1/2 exp ? b T Q (? 2 )b + 2 y ? log p(y i ? , ? 1 ) , where y i = (yi1 , . . . , yin i ) is the vector of observations on unit/cluster i.We attentiveness to become the posterior y y peripherals ? (? g y ), g = 1, . . . , G, and ? (? v y ), v = 1, . . . , V . The number of variance components, V , should not be too large for accurate inference (since these components are integrated out via Cartesian increase numerical integration, which does not scale well with dimension). We print y ? (? g y ) = which may be tryd via the approximation y ? (? g y ) = K ? ? y ? ?(? g ? , y ) ? ?(? y )d? , ? ? y ? ?(? g ? , y ) ? ? (? y )d? ? y ? ? (? g ? k , y ) ? ? (? k y ) ? k, ? (3. 1) k=1 here Laplace (or other related analytical approximations) are applied to carry out the integrations required ? ? for evaluation of ? (? g ? , y ). To produce the grid of points ? k , k = 1, . . . , K over which numerical inte? y gration is performed, the mode of ? (? y ) is located, and the Hessian is approximated, from which the grid is created and exploited in (3. 1). The output of INLA consists of posterior marginal distributions, which can be summarized via means, variances, and quantiles. significantly for model proportion, the normaly izing constant p(y ) is calculated.The evaluation of this quantity is not straightforward employ MCMC (DiCiccio and others, 1997 Meng and Wong, 1996). The deviance information measuring (Spiegelhalter, Best, and others, 1998) is popular as a model selection tool, but in random-effects models, the implicit approximation in its use is valid only when the rough-and-ready number of parameters is much sm aller than the number of self-supporting observations (see Plummer, 2008). four hundred Y. F ONG AND OTHERS 4. P RIOR DISTRIBUTIONS 4. 1 Fixed effects Recall that we embrace ? is normally distributed. Often there will be sufficient information in the data for ? o be well estimated with a normal prior with a large variance (of course there will be good deal under which we would like to specify more informative priors, e. g. when there are many check covariates). The use of an improper prior for ? will lots lead to a proper posterior though make do should be taken. For example, Wakefield (2007) shows that a Poisson likelihood with a linear link can lead to an improper posterior if an improper prior is used. Hobert and Casella (1996) discuss the use of improper priors in linear mixed effects models.If we invite to use informative priors, we may specify independent normal priors with the parameters for from each one component being obtained via specification of 2 quantiles with associated probabilities. For logistic and log-linear models, these quantiles may be given on the exponentiated scale since these are more interpretable (as the odds ratio and rate ratio, observeively). If ? 1 and ? 2 are the quantiles on the exponentiated scale and p1 and p2 are the associated probabilities, because the parameters of the normal prior are given by ? = ? = z 2 log(? 1 ) ? z 1 log(? 2 ) , z2 ? 1 Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 log(? 2 ) ? log(? 1 ) , z2 ? z1 where z 1 and z 2 are the p1 and p2 quantiles of a standard normal random variable. For example, in an epidemiological context, we may wish to specify a prior on a relative risk parameter, exp(? 1 ), which has a median(prenominal) of 1 and a 95% point of 3 (if we think it is unlikely that the relative risk associated with a unit increase in exposure exceeds 3). These specifications lead to ? 1 ? N (0, 0. 6682 ). 4. 2 Variance componentsW e arrive by describing an approach for choosing a prior for a single random effect, based on Wakefield (2009). The staple fiber idea is to specify a range for the more interpretable marginal distribution of bi and use this to drive specification of prior parameters. We state a trivial lemma upon which prior specification is based, but first define some notation. We write ? ? Ga(a1 , a2 ) for the gamma distribution with un? normalized density ? a1 ? 1 exp(? a2 ? ). For q-dimensional x , we write x ? Tq (? , , d) for the Students x x t distribution with unnormalized density 1 + (x ? ? )T ? 1 (x ? )/d? (d+q)/2 . This distribution has location ? , scale matrix , and degrees of freedom d. L EMMA 1 Let b? ? N (0, ? ?1 ) and ? ? Ga(a1 , a2 ). Integration over ? gives the marginal distribution of b as T1 (0, a2 /a1 , 2a1 ). To decide upon a prior, we give a range for a generic random effect b and specify the degrees of freev d dom, d, and therefore solve for a1 and a2 . For the range (? R, R), we use the relationship t1? (1? q)/2 a2 /a1 = d R, where tq is the 100 ? qth quantile of a Student t random variable with d degrees of freedom, to give d a1 = d/2 and a2 = R 2 d/2(t1? (1? q)/2 )2 .In the linear mixed effects model, b is directly interpretable, while for binomial or Poisson models, it is more appropriate to think in terms of the marginal distribution of exp(b), the residual odds and rate ratio, respectively, and this distribution is log Students t. For example, if we choose d = 1 (to give a Cauchy marginal) and a 95% range of 0. 1, 10, we take R = log 10 and obtain a = 0. 5 and b = 0. 0164. Bayesian GLMMs 401 ?1 Another convenient choice is d = 2 to give the exponential distribution with mean a2 for ? ?2 . This leads to closed-form expressions for the more interpretable quantiles of ? o that, for example, if we 2 specify the median for ? as ? m , we obtain a2 = ? m log 2. Unfortunately, the use of Ga( , ) priors has become popular as a prior for ? ?2 in a GLM M context, arising from their use in the winBUGS examples manual. As has been pointed out many times (e. g. Kelsall and Wakefield, 1999 Gelman, 2006 Crainiceanu and others, 2008), this choice places the majority of the prior mass away from zero and leads to a marginal prior for the random effects which is Students t with 2 degrees of freedom (so that the tails are much heavier than even a Cauchy) and difficult to rationalise in any practical facilityting.We now specify another trivial lemma, but first run aground notation for the Wishart distribution. For the q ? q nonsingular matrix z , we write z ? Wishartq (r, S ) for the Wishart distribution with unnormalized Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 Q Lemma Let b = (b1 , . . . , bq ), with b Q ? iid Nq (0, Q ? 1 ), Q ? Wishartq (r, S ). Integration over Q b as Tq (0, (r ? q + 1)S ? 1 , r ? q + 1). S gives the marginal distribution of The margins of a multivariat e Students t are t also, which allows r and S to be chosen as in the univariate case.Specifically, the kth element of a generic random effect, bk , follows a univariate Student t distribution with location 0, scale S kk /(r ? q + 1), and degrees of freedom d = r ? q + 1, where S kk d is element (k, k) of the inverse of S . We obtain r = d + q ? 1 and S kk = (t1? (1? q)/2 )2 /(d R 2 ). If a priori b are correlated we may specify S jk = 0 for j = k and we accept no reason to accept that elements of S kk = 1/Skk , to recover the univariate specification, recognizing that with q = 1, the univariate Wishart has parameters a1 = r/2 and a2 = 1/(2S).If we believe that elements of b are dependent and so we may specify the correlations and solve for the off-diagonal elements of S . To ensure propriety of the posterior, proper priors are required for Zeger and Karim (1991) use an improper prior for , so that the posterior is improper also. 4. 3 Effective degrees of freedom variance compone nts prior z z z z density z (r ? q? 1)/2 exp ? 1 tr(z S ? 1 ) . This distribution has Ez = r S and Ez ? 1 = S ? 1 /(r ? q ? 1), 2 and we require r q ? 1 for a proper distribution.In Section 5. 3, we describe the GLMM arrayation of a spline model. A generic linear spline model is given by K yi = x i ? + k=1 z ik bk + i , where x i is a p ? 1 vector of covariates with p ? 1 associated fixed effects ? , z ik denote the spline 2 base of operations, bk ? iid N (0, ? b ), and i ? iid N (0, ? 2 ), with bk and i independent. Specification of a prior for 2 is not straightforward, but may be of great importance since it contributes to determining the amount ? b of smoothing that is applied. R upper bertht and others (2003, p. 77) raise concerns, most the instability of reflex(a) smoothing parameter selection even for single predictor models, and continue, Although we are attracted by the automatic nature of the mixed model-REML approach to fitting additive models, we discour time blind acceptance of whatever outcome it provides and recommend looking at other amounts of smoothing. While we would echo this general advice, we believe that a Bayesian mixed model approach, with carefully chosen priors, can increase the stability of the mixed model representation. There has been 2 some discussion of choice of prior for ? in a spline context (Crainiceanu and others, 2005, 2008). More general discussion can be found in Natarajan and Kass (2000) and Gelman (2006). In practice (e. g. Hastie and Tibshirani, 1990), smoothers are often applied with a fixed degrees of freedom. We extend this rationale by examining the prior degrees of freedom that is implied by the choice 402 Y. F ONG AND OTHERS ?2 ? b ? Ga(a1 , a2 ). For the general linear mixed model y = x ? + zb + , we strike x z where C = x z is n ? ( p + K ) and C y = x ? + z b = C (C T C + 0 p? p 0K ? p )? 1 C T y , = 0 p? K 2 cov(b )? 1 b ? )? 1 C T C , Downloaded from http//biostatistics. xfordjournals. org/ at Corn ell University Library on April 20, 2013 (see, e. g. Ruppert and others, 2003, Section 8. 3). The total degrees of freedom associated with the model is C df = tr(C T C + which may be decomposed into the degrees of freedom associated with ? and b , and extends easily to situations in which we nominate additional random effects, beyond those associated with the spline basis (such an example is considered in Section 5. 3). In each of these situations, the degrees of freedom associated C with the respective parameter is obtained by summing the appropriate diagonal elements of (C T C + )? C T C . Specifically, if we bedevil j = 1, . . . , d sets of random-effect parameters (there are d = 2 in the model considered in Section 5. 3) then let E j be the ( p + K ) ? ( p + K ) diagonal matrix with ones in the diagonal positions corresponding to set j. Then the degrees of freedom associated with this set is E C df j = trE j (C T C + )? 1 C T C . Note that the effective degrees of freedom chan ges as a function of K , as expected. To evaluate , ? 2 is required. If we specify a proper prior for ? 2 , then we may specify the 2 2 joint prior as ? (? b , ? 2 ) = ? (? 2 )? (? b ? 2 ).Often, however, we don the improper prior ? (? 2 ) ? 1/? 2 since the data provide sufficient information with respect to ? 2 . Hence, we have found the substitution of an estimate for ? 2 (for example, from the fitting of a spline model in a likelihood implementation) to be a practically reasonable strategy. As a fair nonspline demonstration of the derived effective degrees of freedom, consider a 1-way analysis of variance model Yi j = ? 0 + bi + i j 2 with bi ? iid N (0, ? b ), i j ? iid N (0, ? 2 ) for i = 1, . . . , m = 10 groups and j = 1, . . . , n = 5 observa? 2 tions per group. For illustration, we assume ? ? Ga(0. 5, 0. 005). Figure 1 displays the prior distribution for ? , the implied prior distribution on the effective degrees of freedom, and the bivariate plot of these quantities. For clarity of plotting, we pretermit a small number of points beyond ? 2. 5 (4% of points). In panel (c), we have placed stippled horizontal lines at effective degrees of freedom equal to 1 (complete smoothing) and 10 (no smoothing). From panel (b), we conclude that here the prior choice favors quite strong smoothing. This may be contrasted with the gamma prior with parameters (0. 001, 0. 001), which, in this example, gives reater than 99% of the prior mass on an effective degrees of freedom greater than 9. 9, again showing the inappropriateness of this prior. It is appealing to extend the above argument to nonlinear models but unfortunately this is not straightforward. For a nonlinear model, the degrees of freedom may be approximated by C df = tr(C T W C + where W = diag Vi? 1 d? i dh 2 )? 1 C T W C , and h = g ? 1 denotes the inverse link function. Unfortunately, this quantity depends on ? and b , which means that in practice, we would have to use prior estimates for all of the p arameters, which may not be practically possible.Fitting the model using likelihood and then substituting in estimates for ? and b seems philosophically dubious. Bayesian GLMMs 403 Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 Fig. 1. Gamma prior for ? ?2 with parameters 0. 5 and 0. 005, (a) implied prior for ? , (b) implied prior for the effective degrees of freedom, and (c) effective degrees of freedom versus ? . 4. 4 Random walk models Conditionally represented smoothing models are popular for random effects in both temporal and spacial applications (see, e. g. Besag and others, 1995 Rue and Held, 2005).For illustration, consider models of the form ? (m? r ) Q u 2 exp ? p(u ? u ) = (2? )? (m? r )/2 Q 1/2 ? u 1 T u Qu , 2 2? u (4. 1) 404 Y. F ONG AND OTHERS where u = (u 1 , . . . , u m ) is the collection of random effects, Q is a (scaled) precision matrix of association Q m ? r , whose form is determined by the applica tion at hand, and Q is a generalized determinant which is the product over the m ? r nonzero eigenvalues of Q . Picking a prior for ? u is not straightforward because ? u has an interpretation as the conditional standard difference, where the elements that are conditioned upon depends on the application.We may simulate realizations from (4. 1) to examine candidate prior distributions. Due to the rank lack, (4. 1) does not define a probability density, and so we cannot directly simulate from this prior. However, Rue and Held (2005) give an algorithm for generating samples from (4. 1) 1. Simulate z j ? N (0, 1 ), for j = m ? r + 1, . . . , m, where ? j are the eigenvalues of Q (there are j m ? r nonzero eigenvalues as Q has rank m ? r ). 2. Return u = z m? r +1 e n? r +1 + z 3 e 3 + + z n e m = E z , where e j are the corresponding eigenvectors of Q , E is the m ? (m ? ) matrix with these eigenvectors as columns, and z is the (m ? r ) ? 1 vector containing z j , j = m ? r + 1, . . . , m. The disguise algorithm is conditioned so that samples are zero in the null-space of Q if u is a sample and the null-space is spanned by v 1 and v 2 , then u T v 1 = u T v 2 = 0. For example, suppose Q 1 = 0 so that the null-space is spanned by 1, and the rank deficiency is 1. Then Q is improper since the eigenvalue corresponding to 1 is zero, and samples u produced by the algorithm are such that u T 1 = 0. In Section 5. 2, we use this algorithm to evaluate different priors via simulation.It is also efficacious to note that if we wish to compute the marginal variances only, simulation is not required, as they are available as the diagonal elements of the matrix j 1 e j e T . j j 5. E XAMPLES Here, we report 3 examples, with 4 others described in the supplementary material available at Biostatistics online. Together these cover all the examples in Breslow and Clayton (1993), along with an additional spline example. In the first example, results using the INLA numerical/ analytical approximation described in Section 3 were compared with MCMC as implemented in the JAGS software (Plummer, 2009) and found to be accurate.For the models considered in the second and third examples, the approximation was compared with the MCMC implementation contained in the INLA software. 5. 1 Longitudinal data We consider the much analyzed epilepsy data set of Thall and Vail (1990). These data concern the number ? of seizures, Yi j for patient i on visit j, with Yi j ? , b i ? ind Poisson(? i j ), i = 1, . . . , 59, j = 1, . . . , 4. We concentrate on the 3 random-effects models fitted by Breslow and Clayton (1993) log ? i j = x i j ? + b1i , (5. 1) (5. 2) (5. 3) Downloaded from http//biostatistics. oxfordjournals. rg/ at Cornell University Library on April 20, 2013 log ? i j = x i j ? + b1i + b2i V j /10, log ? i j = x i j ? + b1i + b0i j , where x i j is a 1 ? 6 vector containing a 1 (representing the intercept), an forefinger for baseline measurement, a treatment ind icator, the baseline by treatment interaction, which is the parameter of interest, age, and either an indicator of the 4th visit (models (5. 1) and (5. 2) and denoted V4 ) or visit number coded ? 3, ? 1, +1, +3 (model (5. 3) and denoted V j /10) and ? is the associated fixed effect. solely 3 models 2 include patient-specific random effects b1i ? N 0, ? , while in model (5. 2), we introduce independent 2 ). Model (5. 3) includes random effects on the slope associated with measurement errors, b0i j ? N (0, ? 0 Bayesian GLMMs 405 put back 1. PQL and INLA summaries for the epilepsy data unsettled Base Trt Base ? Trt Age V4 or V/10 ? 0 ? 1 ? 2 Model (5. 1) PQL 0. 87 0. 14 ? 0. 91 0. 41 0. 33 0. 21 0. 47 0. 36 ? 0. 16 0. 05 0. 53 0. 06 INLA 0. 88 0. 15 ? 0. 94 0. 44 0. 34 0. 22 0. 47 0. 38 ? 0. 16 0. 05 0. 56 0. 08 Model (5. 2) PQL 0. 86 0. 13 ? 0. 93 0. 40 0. 34 0. 21 0. 47 0. 35 ? 0. 10 0. 09 0. 36 0. 04 0. 48 0. 06 INLA 0. 8 0. 15 ? 0. 96 0. 44 0. 35 0. 2 3 0. 48 0. 39 ? 0. 10 0. 09 0. 41 0. 04 0. 53 0. 07 Model (5. 3) PQL 0. 87 0. 14 ? 0. 91 0. 41 0. 33 0. 21 0. 46 0. 36 ? 0. 26 0. 16 0. 52 0. 06 0. 74 0. 16 INLA 0. 88 0. 14 ? 0. 94 0. 44 0. 34 0. 22 0. 47 0. 38 ? 0. 27 0. 16 0. 56 0. 06 0. 70 0. 14 Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 visit, b2i with b1i b2i ? N (0, Q ? 1 ). (5. 4) We assume Q ? Wishart(r, S ) with S = S11 S12 . For prior specification, we begin with the bivariate S21 S22 model and assume that S is diagonal.We assume the upper 95% point of the priors for exp(b1i ) and exp(b2i ) are 5 and 4, respectively, and that the marginal distributions are t with 4 degrees of freedom. Following the map outlined in Section 4. 2, we obtain r = 5 and S = diag(0. 439, 0. 591). We take ? 2 the prior for ? 1 in model (5. 1) to be Ga(a1 , a2 ) with a1 = (r ? 1)/2 = 2 and a2 = 1/2S11 = 1. 140 (so that this prior coincides with the marginal prior obtained from the bivariate specification). In model (5. 2), ? 2 ? 2 we assume b1i and b0i j are independent, and that ? 0 follows the same prior as ? , that is, Ga(2, 1. 140). We assume a flat prior on the intercept, and assume that the rate ratios, exp(? j ), j = 1, . . . , 5, lie between 0. 1 and 10 with probability 0. 95 which gives, using the approach described in Section 4. 1, a normal prior with mean 0 and variance 1. 172 . Table 1 gives PQL and INLA summaries for models (5. 15. 3). There are some differences between the PQL and Bayesian analyses, with slightly larger standard differences under the last mentioned, which probably reflects that with m = 59 clusters, a little accuracy is lost when using asymptotic inference.There are some differences in the point estimates which is at least partly due to the nonflat priors usedthe priors have relatively large variances, but here the data are not so abundant so there is sensitivity to the prior. Reassuringly under all 3 models inference for the baseline-treatment interaction of interest is intimately y identical and suggests no significant treatment effect. We may compare models using log p(y ) for 3 models, we obtain values of ? 674. 8, ? 638. 9, and ? 665. 5, so that the second model is strongly preferred. 5. Smoothing of birth cohort effects in an age-cohort model We analyze data from Breslow and Day (1975) on breast cancer rates in Iceland. Let Y jk be the number of breast cancer of cases in age group j (2024,. . . , 8084) and birth cohort k (18401849,. . . ,19401949) with j = 1, . . . , J = 13 and k = 1, . . . , K = 11. Following Breslow and Clayton (1993), we assume Y jk ? jk ? ind Poisson(? jk ) with log ? jk = log n jk + ? j + ? k + vk + u k (5. 5) and where n jk is the person-years denominator, exp(? j ), j = 1, . . . , J , represent fixed effects for age relative risks, exp(? is the relative risk associated with a one group increase in cohort group, vk ? iid 406 Y. F ONG AND OTHERS 2 N (0, ? v ) represent unstructured random effects associated with cohort k, with smooth cohort terms u k by-line a second-order random-effects model with Eu k u i i k = 2u k? 1 ? u k? 2 and Var(u k u i 2 i k) = ? u . This latter model is to allow the rates to change smoothly with cohort. An equivalent representation of this model is, for 2 k K ? 1, 1 Eu k u l l = k = (4u k? 1 + 4u k+1 ? u k? 2 ? u k+2 ), 6 Var(u k u l l = k) = 2 ? . 6 Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 The rank of Q in the (4. 1) representation of this model is K ? 2 reflecting that both the overall level and the overall trend are aliased (hence the appearance of ? in (5. 5)). The term exp(vk ) reflects the unstructured residual relative risk and, following the argument in Section 4. 2, we specify that this quantity should lie in 0. 5, 2. 0 with probability 0. 95, with a marginal log Cauchy ? 2 distribution, to obtain the gamma prior ? v ? Ga(0. 5 , 0. 00149).The term exp(u k ) reflects the smooth component of the residual relative risk, and the specification of a 2 prior for the associated variance component ? u is more difficult, given its conditional interpretation. Using the algorithm described in Section 4. 2, we examined simulations of u for different choices of gamma ? 2 hyperparameters and obdurate on the choice ? u ? Ga(0. 5, 0. 001) Figure 2 shows 10 realizations from the prior. The rationale here is to examine realizations to see if they conform to our prior expectations and in particular exhibit the required amount of smoothing.All but one of the realizations vary smoothly across the 11 cohorts, as is desirable. Due to the tail of the gamma distribution, we will always have some extreme realizations. The INLA results, summarized in graphical form, are presented in Figure 2(b), alongside likelihood fits in which the birth cohort effect is incorporated as a linear term and as a factor. We see that the smoothing mod el provides a smooth fit in birth cohort, as we would hope. 5. 3 B-Spline nonparametric regression We demonstrate the use of INLA for nonparametric smoothing using OSullivan splines, which are based on a B-spline basis.We illustrate using data from Bachrach and others (1999) that concerns longitudinal measurements of spinal bone mineral density (SBMD) on 230 egg-producing(prenominal) subjects aged between 8 and 27, and of 1 of 4 ethnic groups Asian, Black, Hispanic, and White. Let yi j denote the SBMD measure for subject i at occasion j, for i = 1, . . . , 230 and j = 1, . . . , n i with n i being between 1 and 4. Figure 3 shows these data, with the gray lines indicating measurements on the same woman. We assume the model K Yi j = x i ? 1 + agei j ? 2 + k=1 z i jk b1k + b2i + ij, where x i is a 1 ? vector containing an indicator for the ethnicity of one-on-one i, with ? 1 the associated 4 ? 1 vector of fixed effects, z i jk is the kth basis associated with age, with associated par ameter b1k ? 2 2 N (0, ? 1 ), and b2i ? N (0, ? 2 ) are woman-specific random effects, finally, i j ? iid N (0, ? 2 ). All random terms are assumed independent. Note that the spline model is assumed common to all ethnic groups and all women, though it would be straightforward to allow a different spline for each ethnicity. Writing this model in the form y = x ? + z 1b1 + z 2b 2 + = C ? + . Bayesian GLMMs 407Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 Fig. 2. (a) Ten realizations (on the relative risk scale) from the random effects second-order random walk model in which the prior on the random-effects precision is Ga(0. 5,0. 001), (b) summaries of fitted models the solid line corresponds to a log-linear model in birth cohort, the circles to birth cohort as a factor, and + to the Bayesian smoothing model. we use the method described in Section 4. 3 to examine the effective number of parameters implied by the ? 2 ? 2 priors ? 1 ? Ga(a1 , a2 ) and ? 2 ? Ga(a3 , a4 ).To fit the model, we first use the R code provided in Wand and Ormerod (2008) to construct the basis functions, which are then input to the INLA program. Running the REML version of the model, we obtain 2 ? = 0. 033 which we use to evaluate the effective degrees of freedoms associated with priors for ? 1 and 2 . We assume the usual improper prior, ? (? 2 ) ? 1/? 2 for ? 2 . by and by some experimentation, we settled ? 2 408 Y. F ONG AND OTHERS Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 Fig. 3. SBMD versus age by ethnicity. Measurements on the same woman are joined with gray lines.The solid curve corresponds to the fitted spline and the dashed lines to the individual fits. ?2 2 on the prior ? 1 ? Ga(0. 5, 5 ? 10? 6 ). For ? 2 , we wished to have a 90% interval for b2i of 0. 3 which, ? 2 with 1 degree of freedom for the marginal distribution, leads to ? 2 ? Ga(0. 5, 0. 00113). F igure 4 shows the priors for ? 1 and ? 2 , along with the implied effective degrees of freedom under the assumed priors. For the spline component, the 90% prior interval for the effective degrees of freedom is 2. 4,10. Table 2 compares estimates from REML and INLA implementations of the model, and we see close correspondence between the 2.Figure 4 also shows the posterior medians for ? 1 and ? 2 and for the 2 effective degrees of freedom. For the spline and random effects these correspond to 8 and 214, respectively. The latter figure shows that there is considerable variability between the 230 women here. This is substantiate in Figure 3 where we observe large vertical differences between the profiles. This figure also shows the fitted spline, which appears to imitate the trend in the data well. 5. 4 Timings For the 3 models in the longitudinal data example, INLA takes 1 to 2 s to run, using a single CPU.To get estimates with similar precision with MCMC, we ran JAGS for 100 000 it erations, which took 4 to 6 min. For the model in the temporal smoothing example, INLA takes 45 s to run, using 1 CPU. Part of the INLA procedure can be executed in a parallel manner. If there are 2 CPUs available, as is the case with todays prevalent INTEL Core 2 Duo processors, INLA only takes 27 s to run. It is not currently possible to implement this model in JAGS. We ran the MCMC utility built into the INLA software for 3. 6 meg iterations, to obtain estimates of comparable accuracy, which took 15 h.For the model in the B-spline nonparametric regression example, INLA took 5 s to run, using a single CPU. We ran the MCMC utility built into the INLA software for 2. 5 million iterations to obtain estimates of comparable accuracy, the analysis taking 40 h. Bayesian GLMMs 409 Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 Fig. 4. Prior summaries (a) ? 1 , the standard deviation of the spline coefficients, (b) effective degre es of freedom associated with the prior for the spline coefficients, (c) effective degrees of freedom versus ? , (d) ? 2 , the standard deviation of the between-individual random effects, (e) effective degrees of freedom associated with the individual random effects, and (f) effective degrees of freedom versus ? 2 . The vertical dashed lines on panels (a), (b), (d), and (e) correspond to the posterior medians. Table 2. REML and INLA summaries for spinal bone data. Intercept corresponds to Asian group Variable Intercept Black Hispanic White Age ? 1 ? 2 ? REML 0. 560 0. 029 0. 106 0. 021 0. 013 0. 022 0. 026 0. 022 0. 021 0. 002 0. 018 0. 109 0. 033 INLA 0. 563 0. 031 0. 106 0. 021 0. 13 0. 022 0. 026 0. 022 0. 021 0. 002 0. 024 0. 006 0. 109 0. 006 0. 033 0. 002 Note For the entries marked with a standard errors were unavailable. 410 Y. F ONG AND OTHERS 6. D ISCUSSION In this paper, we have demonstrated the use of the INLA computational method for GLMMs. We have found t hat the approximation strategy employed by INLA is accurate in general, but less accurate for binomial data with small denominators. The supplementary material available at Biostatistics online contains an extensive simulation study, replicating that presented in Breslow and Clayton (1993).There are some suggestions in the discussion of Rue and others (2009) on how to construct an improved Gaussian approximation that does not use the mode and the curvature at the mode. It is likely that these suggestions will improve the results for binomial data with small denominators. There is an urgent need for diagnosis tools to flag when INLA is inaccurate. Conceptually, computation for nonlinear mixed effects models (Davidian and Giltinan, 1995 Pinheiro and Bates, 2000) can also be handled by INLA but this capability is not currently available. The website www. r-inla. rg contains all the data and R scripts to perform the analyses and simulations reported in the paper. The latest release of s oftware to implement INLA can also be found at this site. Recently, Breslow (2005) revisited PQL and concluded that, PQL still performs remarkably well in comparison with more elaborate procedures in many practical situations. We believe that INLA provides an attractive alternative to PQL for GLMMs, and we hope that this paper stimulates the greater use of Bayesian methods for this class. Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013S UPPLEMENTARY MATERIAL Supplementary material is available at http//biostatistics. oxfordjournals. org. ACKNOWLEDGMENT Conflict of Interest None declared. F UNDING National Institutes of Health (R01 CA095994) to J. W. Statistics for Innovation (sfi. nr. no) to H. R. R EFERENCES BACHRACH , L. K. , H ASTIE , T. , WANG , M. C. , NARASIMHAN , B. AND M ARCUS , R. (1999). Bone mineral acquisition in healthy Asian, Hispanic, Black and Caucasic youth. A longitudinal study. The diary of Clinical Endo crinology and Metabolism 84, 47024712. 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