Bayesian Inference

Biostatistics (2010), 11, 3, pp. 397412 doi10. 1093/biostatistics/kxp053 Advance Access publication on December 4, 2009 verbaliseian conclusion for planetaryise bi unidimensional motley nonpluss YOUYI FONG Downloaded from http//biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 Department of Biostatistics, University of Washington, Seattle, WA 98112, USA ? HAVARD RUE Department of Mathematical Sciences, The Norse University for Science and Techno poundy, N-7491 Trondheim, Norway JON WAKEFI years? Departments of Statistics and Biostatistics, University of Washington, Seattle, WA 98112, USA emailprotected ashington. edu S UMMARY Generalized depictar conglomerate bewilders (GLMMs) continue to grow in popularity due to their ability to directly realise multiple levels of dependency and impersonate different entropy types. For sm all in all test sizes especially, likelihood-based inference can be unreliable with unevenness components wor ldness especially difficult to estimate. A Bayesian speak to is charitable just now has been hampered by the lack of a fast implementation, and the difficulty in particularizeing forward dispersions with variability components again being particularly problematic.Here, we briefly review previous get alonges to computing in Bayesian implementations of GLMMs and illust array in detail, the employment of structured nested Laplace approximations in this context. We conceptualize a weigh of examples, cargonfully finding previous distributions on meaning(prenominal) quantities in each courtship. The examples c everywhere a wide orbit of selective information types including those requiring smoothing over time and a relatively complicated slat precedent for which we psychoanalyze our anterior judicial admission in destinations of the implied degrees of immunity.We conclude that Bayesian inference is now pragmatically feasible for GLMMs and provides an attractive al ternative to likelihood-based approaches practically(prenominal) as penalized quasi-likelihood. As with likelihood-based approaches, great c be is necessary in the digest of clustered binary entropy since approximation strategies whitethorn be less(prenominal) consummate for such selective information. Keywords Integrated nested Laplace approximations longitudinal data Penalized quasi-likelihood anterior specification Spline works. 1.I NTRODUCTION Generalized linear change integrity impersonates (GLMMs) combine a generalized linear sit down with natural hit-or-miss effect on the linear predictor plateful, to bequeath a rich family of toughies that convey been personad in a wide variety of applications ( understand, e. g. Diggle and others, 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 isotropy should be addressed. c The findning 2009 . Published by Oxford University Press. All rights reserved. For permissions, please e-mail journals. emailprotected rg. 398 Y. F ONG AND OTHERS b disclose of implications including calculational complexity, and an unknown degree to which inference is bloodsucking on moulding assumptions. Likelihood-based inference may be carried out relatively easily inwardly many parcel system platforms (except perhaps for binary responses), but inference is dependent on asymptotic sampling distributions of estimators, with few guidelines available as to when such theory will produce accurate inference. A Bayesian approach is attractive, but requires the specification of antecedent distributions which is non naive, in particular for variance components.Computation is also an issue since the usual implementation is via Markov chain monte Carlo (MCMC), which carries a considerable computational overhead. The seminal article of Breslow and Clayton (1993) helped to popularize GLMMs and placed an fierceness on likelihood-based inference via penalized quasi-likelihood (PQL). It is the aim of this article to describe, through a serial 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 precedent specification. The social structure of this article is as follows.In office 2, we define notation for the GLMM, and in particle 3, we describe the integrated nested Laplace approximation (INLA) that has recently been proposed as a computationally convenient alternative to MCMC. parting 4 moots a pattern of prescriptions for front specification. Three examples are considered in subsection 5 (with additional examples being reported in the supplementary material available at Biostatistics online, along with a simulation study that reports the performance of INLA in the binary response situation). We conclude the paper with a discussion in function 6. 2.T HE G ENERALIZED LINEAR MIXED MODEL GLMMs extend the generalized linear model, as proposed by Nelder and Wedderburn (1972) and comprehensively expound in McCullagh and Nelder (1989), by adding comm entirely distributed hit-or-miss personal do on the linear predictor scale. Suppose Yi j is of exponential family form Yi j ? i j , ? 1 ? p(), where 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 wholes (clusters) and j = 1, . . , n i , measurements per unit and where ? i j is the (scalar) ? canonical parameter. permit ? 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 connective 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 hit-o r-miss effects, hence ? i j = ? i j (? , b i ). Assume b i Q ? N (0, Q ? 1 ), where ? the precision matrix Q = Q (? 2 ) depends on parameters ? 2 . For some plectrons of model, the matrix Q is homophile(a) examples include stochastic fling models (as considered in Section 5. ) and intrinsic conditional ? autoregressive models. We further assume that ? is assigned a everyday former 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 . allow ? = (? 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, in that location were few examples of the applications of Bayesian GLMMs since, away(p) of the linear mixed model, the models are analytically intractable.Kass and Steffey (1989) describe the occasion of Laplace approximations in Bayesian hierarchical mo dels, while Skene and Wakefield Bayesian GLMMs 399 (1990) used mathematical 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 normal conditional distributions being approximated by normal distributions.More general MetropolisHastings 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. in that respect are now a variety of additional software platforms for satisfactory GLMMs via MCMC including JAGS (Plummer, 2009) and BayesX (Fahrmeir and others, 2004). A large practical impediment to data abbreviation development MCMC is the large computational burden. For this reason, we now briefly review t he INLA computational approach upon which we concentrate.The order combines Laplace approximations and quantitative integration in a very efficient sort (see experience and others, 2009, for a more(prenominal)(prenominal) across-the-board preaching). For the GLMM described in Section 2, the goat is given(p) 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 wish to pose the stinker y y bare(a)s ? (? 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 product numerical integration, which does not scale well with dimension). We write y ? (? g y ) = which may be labeld 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 promontorys ? 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. Importantly for model comparison, 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 aberrancy information criterion (Spiegelhalter, Best, and others, 1998) is popular as a model alternative tool, but in haphazard-effects models, the implicit approximation in its use is sound only when the impelling number of parameters is much smaller than the number of free-lance observations (see Plummer, 2008). 400 Y. F ONG AND OTHERS 4. P RIOR DISTRIBUTIONS 4. 1 Fixed effects repeat that we assume ? is normally distributed. Often there will be ample information in the data for ? o be well estimated with a normal prior with a large variance (of course there will be circumstances under which we would like to specify more informative priors, e. g. when there are many agree covariates). The use of an untimely prior for ? will often lead to a fit posterior though care should be taken. For example, Wakefield (2007) shows that a Poisson likelihood with a linear link can lead to an im fit posterior if an inappropriate prior is used. Hobert and Casella (1996) discuss the use of improper priors in linear mixed effects models.If we wish to use informative pr iors, we may specify independent normal priors with the parameters for each 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, enjoyively). If ? 1 and ? 2 are the quantiles on the exponentiated scale and p1 and p2 are the associated probabilities, then 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 monetary 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 of 1 and a 95% point of 3 (if we regard 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 componentsWe begin by describing an approach for choosing a prior for a unity random effect, based on Wakefield (2009). The basic idea is to specify a range for the more interpretable marginal distribution of bi and use this to drive back specification of prior parameters. We state a trivial lemma upon which prior specification is based, but scratch line define some notation. We write ? ? Ga(a1 , a2 ) for the da Gamma distribution with un? normalized density ? a1 ? 1 exp(? a2 ? ). For q-dimensional x , we write x ? Tq (? , , d) for the school- days childs 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 wine random effect b and specify the degrees of freev d dom, d, and then solve for a1 and a2 . For the range (? R, R), we use the descent t1? (1? q)/2 a2 /a1 = d R, where tq is the 100 ? qth quantile of a schoolchild 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 remove to compute in terms of the marginal distribution of exp(b), the residual odds and rate ratio, respectively, and this distribution is log scholars 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 GLMM 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 nothing 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 justify in any practical curryting.We now specify another trivial lemma, but first prime 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 ? Wi shartq (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 multivariate Students t are t also, which allows r and S to be elect as in the univariate case.Specifically, the kth segment 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 relieve oneself no reason to believe that elements of S kk = 1/Skk , to re even up 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 then we may specify the correlations and solve for the off- prejudice elements of S . To ensure propriety of the posterior, proper priors are r equired for Zeger and Karim (1991) use an improper prior for , so that the posterior is improper also. 4. 3 Effective degrees of freedom variance components 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 representation of a slat 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 basis, 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. Ruppert and others (2003, p. 77) domiciliate concerns, about the instability of automatic smoothing parameter selection even for single predictor models, and c ontinue, Although we are attracted by the automatic nature of the mixed model-REML approach to fitting additive models, we discourage blind acceptance of whatever firmness 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 founder 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 Cornell University Library on April 20, 2013 (see, e. g. Ruppert and others, 2003, Section 8. 3). The add up 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 have 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 have 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 tally to set j. Then the degre es of freedom associated with this set is E C df j = trE j (C T C + )? 1 C T C . Note that the in force(p) degrees of freedom changes 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 assume 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 simple nonspline proof of the derived impressive 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 classifys and j = 1, . . . , n = 5 observa? 2 tions per group. For illustration, we assume ? ? Ga(0. 5, 0. 005). frame of graphic symbol 1 di splays 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 exclude a small number of points beyond ? 2. 5 (4% of points). In panel (c), we have placed hasten 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 substantial 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 wake 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 inve rse 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 parameters, which may not be practically possible.Fitting the model utilize 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 spatial applications (see, e. g. Besag and others, 1995 feel 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 localize Q m ? r , whose form is determined by the application at hand, and Q is a generalized determinant which is the product over the m ? r nonzero eigenvalues of Q . pickax a prior for ? u is not straightforward because ? u has an interlingual rendition as the conditional standard deviation, 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 deficiency, (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. indemnity u = z m? r +1 e n? r +1 + z 3 e 3 + + z n e m = E z , where e j a re 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 simulation 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 useful 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. T ogether these cover all the examples in Breslow and Clayton (1993), along with an additional spline example. In the first example, results apply 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, lo g ? i j = x i j ? + b1i + b0i j , where x i j is a 1 ? 6 vector containing a 1 (representing the intercept), an indicator for baseline measurement, a treatment indicator, the baseline by treatment interaction, which is the parameter of interest, age, and either an indicator of the fourthly visit (models (5. 1) and (5. 2) and denoted V4 ) or visit number work outd ? 3, ? 1, +1, +3 (model (5. 3) and denoted V j /10) and ? is the associated fixed effect. All 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 Table 1. PQL and INLA summaries for the epilepsy data Variable subject 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. 23 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 physical process outlined in Section 4. 2, we obtain r = 5 and S = diag(0. 43 9, 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 equivalent 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, development 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 deviations under the last mentioned, which probably reflects that with m = 59 clusters, a little verity is lost when using asymptotic inference.There are some differences in the point estimates which is at leas t 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 virtually y similar 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 powerfully preferred. 5. Smoothing of birth age group 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 following 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 set forth 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 boilers suit 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 decided on the choice ? u ? Ga(0. 5, 0. 001) ascertain 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 corporate as a linear term and as a factor. We see that the smoothing model provides a smooth fit in birth cohort, as we would bank. 5. 3 B-Spline nonparametric infantile fixation 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 female subjects aged between 8 and 27, and of 1 of 4 heathenish groups Asian, Black, Hispanic, and clean. 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, wher e x i is a 1 ? vector containing an indicator for the ethnicity of several(prenominal) i, with ? 1 the associated 4 ? 1 vector of fixed effects, z i jk is the kth basis associated with age, with associated parameter 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 bir th 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 scuttlebutt 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 . After 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 hie 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% separation for b2i of 0. 3 which, ? 2 with 1 degree of freedom for the marginal distribution, leads to ? 2 ? Ga(0. 5, 0. 00113). Figure 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 go into shows that there is considerable variability between the 230 women here. This is confirmed in Figure 3 where we observe large vertical differences between the profiles. This figure also shows the fitted spline, which appears to mimic 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 iterations, 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 million 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 degrees 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. bug 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 Fo r 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 that the approximation strategy utilise 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 swag when INLA is inaccurate. Conceptually, computation for nonlinear mixed effects models (Davidian and Giltinan, 1995 Pinheiro and Bates, 2000) can also be handled b y 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 software to implement INLA can also be found at this site. Recently, Breslow (2005) revisited PQL and reason that, PQL still performs remarkably well in comparison with more expatiate 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 involution 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 Caucasian youth. A longitudinal study. The Journal of clinical Endocrinology and Metabolism 84, 47024712. 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