Euthanasia should be legalised. Agree or Disagree? Essay

Euthanasia is inducing a painless death, by agreement and with compassion, to ease suffering. There are also four different kind of euthanasia; active, passive, voluntary and involuntary. Active euthanasia means carrying out some action to help someone to die, whereas passive euthanasia is to not carry out actions which would prolong life. Thus with regards to the above, voluntary euthanasia is helping a person who wishes to die to do so and involuntary euthanasia is helping a person to die when they are unable to request this for themselves. It is argued on a yearly basis as to whether euthanasia should be legalised in the United Kingdom. There are several arguments in favour for the legalisation of euthanasia. In voluntary euthanasia, it’s argued that it shows mercy for those suffering with pain and a disease with no cure, a view which Thomas More (1478-1535) supports. In his book Utopia (1516), More argued that when a patient suffers ‘a torturing and lingering pain, so that there is no hope, either of recovery or ease, they may choose rather to die, since they cannot live but in much misery’. It is an opportunity to end needless suffering, one that we already offer to animals, thus should be offered to humans. Other advocates of voluntary euthanasia argue that it should be an option for an adult who is able and willing to make such a decision (autonomy). They argue that it should be on offer as one option among many, along with the kind of care of patients with a terminal illness is offered by hospitals and hospices. This argument is maintained by John Stuart Mill who, in his book On Liberty (1859), argued that in matters that do not concern others, individuals should have full autonomy: ‘The only part of the conduct of any one, for which (a citizen) is amenable to society, is that which concerns others. In the part which merely concerns himself, his independence is, of right, absolute. Over himself, over his body and mind, this individual is sovereign.’ The VES (www.dignityindying.org.uk) also argues that every human being deserves respect and has the right to choose his or her own destiny, including how he or she lives and dies. American doctor Jack Kervorkian has said (Gula, 1988): ‘In my view the highest principle in medical ethics – in any kind of ethics – is personal autonomy, self-determination. What counts is what the patient wants and judges to be a benefit or a value in his or her own life. That’s primary’. We have autonomy over our bodies in matters of life, and it should be the same in matters of death. Thus, voluntary euthanasia gives people full autonomy and should be legalised. Other believers of voluntary euthanasia claim that it maintains quality of life. They say that human beings should be able to maintain their dignity up until the end of their lives. Thus, not only is it a matter of pain, but of self respect. If someone’s standard of living is such that they no longer want to live, then they should be able to end their life and, if necessary, be assisted in doing so. However, the quality of life worth living is one that only the person in question can define. Having control over their life is a way of enhancing their human dignity. Thus, as euthanasia maintains this quality of life and human dignity it should be legalised. A further point arguing that euthanasia is acceptable claims that the act is not in fact murder and should therefore be legalised, as it doesn’t go against any other laws. This is sustained by Gregory E. Pence in his article ‘Why physicians should aid the dying’ (1997). Pence argues that killing humans who don’t want to live is not wrong. He continues to explain that it isn’t wrong to help the dying to die, because they are actually dying. There are also several arguments against voluntary euthanasia. One difficulty with euthanasia being legalised is a person’s motives. It is questionable as to whether we can be sure that when a person asks for death, that the person isn’t crying out in despair, rather than making a definitive decision. When a person is desperate, they may feel that they want to end their life and therefore deduce that the pain is too great and life too agonising. However perhaps these moments of desperation will pass and they will be glad that no one acted on their pleas. It is also questionable as to whether doctors can be sure that they know and understand all the facts. It could also be possible that they may fear a future which will not be realised. Thus any euthanasia process would have to establish, beyond any doubt, the true intentions of the patient who is requesting euthanasia and that the patient is fully aware of the situation. Thus from this view point euthanasia shouldn’t be legalised due to the risk of misinformation or a failure to comprehend the situation which would leave the patient vulnerable to a decision that he or she might not truly want to make. There are also arguments against the legalisation of euthanasia due to the risk of mistake that may occur, as we can’t be certain that they would be avoided. For example, someone chooses death because they have been diagnosed with a fatal, incurable and painful illness. Then, after the person has died, it is discovered that the diagnosis was incorrect. Therefore, in the legalisation of euthanasia, the diagnosis would have to be beyond a doubt and it is questionable about whether there can always be medical certainty about what the condition will entail and how long it will take to develop. Thus, being an area of doubt that could lead to irreversible mistakes, euthanasia shouldn’t be legalised to safeguard people against this. Glover (1977) noted that people who feel they are burdens on their families sometimes commit suicide. Thus it may be possible that elderly relatives who think they are burdens to their families ask for voluntary euthanasia out of a sense of duty to the family. It’s also questionable as to whether, on the other hand, they could be pressured into asking for voluntary euthanasia by their relatives. As an example, the conviction of Harold Shipman who, as a doctor, murdered elderly patients over a period of years shows the power of doctors. Thus, due to possible abuse of the system, euthanasia should not be legalised as the existence of such a system could allow such people even more capacity for murder by manipulating patients and documentation. There are also arguments against the legalisation of euthanasia due to its’ possible negative impact on the community. It is argued that the legalisation of voluntary euthanasia might lead to other forms of euthanasia being supported, for example, involuntary euthanasia may start to be carried out (like the Nazis did) on the sick, the elderly and the disabled. However, Glover (1977) says that this argument is unconvincing and thus rejects it, whilst Helga Kuhse (1991) has observed that this has not happened in the Netherlands, where voluntary euthanasia is legal. It is further argued that its’ negative effects on the community might include the damage of the care of patients who are dying. While oppressing voluntary euthanasia, people have developed caring and sensitive environments for the terminally ill within the hospice movement. Therefore it is possible that legalisation of voluntary euthanasia would effect the culture in which that approach to care has been developed. For example, it is questionable as to whether, if voluntary euthanasia was legalised, people would be concerned about visiting hospitals, fearful of what might happen such as an unwanted assisted death. There are other cases where a patient cannot let their wishes be known, such as a person who is in a coma in which recovery is very unlikely or impossible. There are also cases of babies who have severe, permanent and possibly deteriorating health conditions that cause suffering. The withdrawal of treatment or use of certain medicines may lead to involuntary euthanasia. The principle of this is uncontroversial. However, the question of taking away food and water is. Tony Bland (1989) was in a coma from which doctors believed he would never recover. He was classed as in a vegetative state and could open his eyes but he did not respond to anything around him. He couldn’t feed but could digest food and needed to have food and water provided to him through a feeding tube. He wasn’t dying, yet there was no cure. There ended in being a court case over whether or not it was right to remove artificial feeding, which would lead to his death. The court allowed Bland to die through starvation and dehydration, which would be painful if he was able to sense the pain, though is was presumed that he couldn’t. Thus this takes steps towards active involuntary euthanasia or even non-voluntary euthanasia as The 2005 Mental Capacity Act for England and Wales preserves in law the view that assisted food and fluids is a medical treatment that could be withdrawn. With there being instances where doctors are convinced a person will never wake up from a coma, or has no capacity for normal function, and yet can be kept alive, there is the question over whether it shows much or less respect for the value of a person to withdraw life saving measures and thus whether or not this should be legal. Other areas of controversy surround the care of disabled babies. It is possible to keep alive more and more physically disabled babies. However, some argue that allowing a disabled baby to live is to disable a family. The Royal College of Obstetricians and Gynaecologists (November 2006) urged health professionals to consider euthanasia for seriously disabled babies to spare the emotional burden of families bringing them up. Critics of this are concerned that the example of actively killing a baby or withdrawing treatment to bring about death develops a culture in which all disabled people are considered to be of less value and thus dispute as to whether or not this should be legal. Answers of these questions are also sought through religion. Questions such as what do we do for the person who is in a coma with no hope for recovery? How do we care for the terminally ill who is in a lot of pain? These questions can be answered by Christianity and Islam. In Christianity, biblical teachings forbid killing (Sixth commandment). They also say that life should not be violated and there is also a powerful message of the importance of healing and care for the sick. However, there are exceptions for warfare and self-defence. There are also examples in the bible where the sacrifice of life is considered moral (‘greater love has no man than this: That a man lay down his life for his friends’ John 15:13). The bible does not prohibit all taking of life in all circumstances, although Christians have traditionally considered taking one’s own life to be wrong. Thus is can be seen that Christians would accept euthanasia in certain circumstances.

Bayesian Inference

Biostatistics (2010), 11, 3, pp. 397–412 doi:10. 1093/biostatistics/kxp053 Advance Access publication on December 4, 2009 Bayesian inference for generalized linear mixed models 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 Norwegian University for Science and Technology, N-7491 Trondheim, Norway JON WAKEFIELD? Departments of Statistics and Biostatistics, University of Washington, Seattle, WA 98112, USA [email  protected] ashington. edu S UMMARY Generalized linear mixed models (GLMMs) continue to grow in popularity due to their ability to directly acknowledge multiple levels of dependency and model different data types. For small sample sizes especially, likelihood-based inference can be unreliable with variance components being particularly difficult to estimate. A Bayesian approach is appealing but has been hampered by the lack of a fast implementation, and the difficulty in specifying prior distributions with variance components again being particularly problematic.Here, we briefly review previous approaches to computation in Bayesian implementations of GLMMs and illustrate in detail, the use of integrated nested Laplace approximations in this context. We consider a number of examples, carefully specifying prior distributions on meaningful quantities in each case. The examples cover a wide range of data types including those requiring smoothing over time and a relatively complicated spline model for which we examine our prior specification in terms of the implied degrees of freedom.We conclude that Bayesian inference is now practically feasible for GLMMs and provides an attractive alternative to likelihood-based approaches such as penalized quasi-likelihood. As with likelihood-based approaches, great care is required in the analysis of clustered bina ry data since approximation strategies may be less accurate for such data. Keywords: Integrated nested Laplace approximations; Longitudinal data; Penalized quasi-likelihood; Prior specification; Spline models. 1.I NTRODUCTION Generalized linear mixed models (GLMMs) combine a generalized linear model with normal random effects on the linear predictor scale, to give a rich family of models that have been used in a wide variety of applications (see, 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 correspondence should be addressed. c The Author 2009. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals. [email  protected] rg. 398 Y. F ONG AND OTHERS number of implications including computational complexity, and an unknown degree to which inference is dependent on modeling assumptions. Lik elihood-based inference may be carried out relatively easily within many software 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 prior 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 large computational overhead. The seminal article of Breslow and Clayton (1993) helped to popularize GLMMs and placed an emphasis 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 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 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 described in McCullagh and Nelder (1989), by adding normally distributed random effects on the linear predictor scale. Suppose Yi j is of exponential family form: Yi j |? i j , ? 1 ? p(â₠¬ ¢), 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 units (clusters) and j = 1, . . , n i , measurements per unit and where ? i j is the (scalar) ? canonical parameter. Let ? i j = E[Yi 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 â€Å"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, 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 choices of model, the matrix Q is singular; examples include random walk models (as considered in Section 5. ) and intrinsic conditional ? autoregressive models. We further assume tha t ? 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, there were few examples of the applications of Bayesian GLMMs since, outside of the linear mixed model, the models are analytically intractable.Kass and Steffey (1989) describe the use of Laplace approximations in Bayesian hierarchical models, while Skene and Wakefield Bayesian GLMMs 399 (1990) used 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 nonstandar d conditional distributions being approximated by normal distributions.More general Metropolis–Hastings 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 impediment to data analysis using MCMC is the large computational burden. For this reason, we now briefly review the INLA computational approach upon which we concentrate.The method combines Laplace approximations and numerical integration in a very efficient manner (see Rue and others, 2009, for a more extensive treatment). For the GLMM described in Section 2, the posterior 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 wish to obtain the posterior y y marginals ? (? 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 evaluated 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. Importantly for model comparison, the normaly izing constant p(y ) is calculated.The evaluation of this quantity is not straightforward using MCMC (DiCiccio and others, 1997; Meng and Wong, 1996). The deviance information criterion (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 effective number of parameters is much smaller than the number of independent observations (see Plummer, 2008). 400 Y. F ONG AND OTHERS 4. P RIOR DISTRIBUTIONS 4. 1 Fixed effects Recall that we assume ? is normally distributed. Often there will be sufficient information in the data for ? o be well estimated with a n ormal 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 correlated covariates). The use of an improper prior for ? will often lead to a proper posterior though care 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 wish to use informative priors, 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, respectively). If ? 1 and ? 2 are the quantiles on the exponentiated scale and p1 and p2 are the associated probab ilities, 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 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 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 componentsWe begin by describing an approach for choosing a prior for a single 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 specification of prior parameters. We state a trivial lemma upon which prior specification is ba sed, 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 Student’s 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 then 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 Student’s 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 zero and leads to a marginal prior for the random effects which is Student’s t with 2 degrees of freedom (so that the tails are much heavier than even a Cauchy) and difficult to justify in any practical setting.We now specify another trivial lemma, but first establish 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 multivariate Student’s 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 have no reason to believe 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 then 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 components prior z z z z density |z |(r ? q? 1)/2 exp ? 1 tr(z S ? 1 ) . This distribution has E[z ] = r S and E[z ? 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 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 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) raise concerns, â€Å"about the instability of automatic smoothing parameter selection even for single predictor models†, and continue, â€Å"Although we are attracted by the automatic nature of the mixed model-REML approach to fitting additive models, we discourage blind acceptance of whatever answer 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 be en 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 have 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 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 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 corresponding to set j. Then the degrees of freedom associated with this set is E C df j = tr{E j (C T C + )? 1 C T C . Note that the effective 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 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 exclude a small number of points beyond ? > 2. 5 (4% of points). In panel (c), we have placed dashed 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 q uite 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 parameters, 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 spatial 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) â€Å"precision† matrix of rank 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 . Picking a prior for ? u is not straightforward because ? u has an interpretation as the conditional standard deviation, where the elements that are conditioned upon depend s 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. 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 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. 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 implement ation 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 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 fourth 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. 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 â€Å"measurement errors,† b0i j ? N (0, ? 0 Bayesian GLMMs 405 Table 1. PQL and INLA summaries for the epilepsy data Variable 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. 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 procedure 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 bivariat e 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. 1–5. 3). There are some differences between the PQL and Bayesian analyses, with slightly larger standard deviations under the latter, 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 used—the 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 bas eline-treatment interaction of interest is virtually 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 (20–24,. . . , 80–84) and birth cohort k (1840–1849,. . . ,1940–1949) 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 E[u 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 vary smoothly with cohort. An equivalent representation of this model is, for 2 < k < K ? 1, 1 E[u 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 decided 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 f actor. We see that the smoothing model 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 O’Sullivan 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 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 individual 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 birth cohort, the circles to birth cohort as a factor, and â€Å"+† 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 . 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 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 distributio n, 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 figure 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 wit h 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 today’s 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. 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 t his paper, we have demonstrated the use of the INLA computational method for GLMMs. We have found that 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 software to implement INLA can also be found at this site. Recently, Breslow (2005) revisited PQL and concluded that, â€Å"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 Caucasian youth. A longitudinal study. The Journal of Clinical Endocrinology and Metabolism 84, 4702–4712. B ESAG , J. , G REEN , P. J. , H IGDON , D. AND M ENGERSEN , K. 1995). Bayesian computation and stochastic systems (with discussion). Statistical Science 10, 3–66. B RESLOW, N. E. (2005). Whither PQL? In: Lin, D. and Heagerty, P. J. (editors), Proceedings of the Second Seattle Symposium. New York: Springer, pp. 1–22. B RESLOW, N. E. AND C LAYTON , D. G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association 88, 9–25. B RESLOW, N. E. AND DAY, N. E. (1975). Indirect standardization and multiplicative models for rates, with reference to the age adjustment of cancer incidence and relative frequency data.Journal of Chronic Diseases 28, 289–301. C LAYTON , D. G. (1996). Generalized linear mixed models. In: Gilks, W. R. , Richardson, S. and S piegelhalter, D. J. (editors), Markov Chain Monte Carlo in Practice. London: Chapman and Hall, pp. 275–301. Bayesian GLMMs 411 C RAINICEANU , C. M. , D IGGLE , P. J. AND ROWLINGSON , B. (2008). Bayesian analysis for penalized spline regression using winBUGS. Journal of the American Statistical Association 102, 21–37. C RAINICEANU , C. M. , RUPPERT, D. AND WAND , M. P. (2005). Bayesian analysis for penalized spline regression using winBUGS. Journal of Statistical Software 14.DAVIDIAN , M. AND G ILTINAN , D. M. (1995). Nonlinear Models for Repeated Measurement Data. London: Chapman and Hall. D I C ICCIO , T. J. , K ASS , R. E. , R AFTERY, A. AND WASSERMAN , L. (1997). Computing Bayes factors by combining simulation and asymptotic approximations. Journal of the American Statistical Association 92, 903–915. Downloaded from http://biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 D IGGLE , P. , H EAGERTY, P. , L IANG , K. -Y. Oxford: Oxford University Press. AND Z EGER , S. (2002). Analysis of Longitudinal Data, 2nd edition. FAHRMEIR , L. , K NEIB , T.AND L ANG , S. (2004). Penalized structured additive regression for space-time data: a Bayesian perspective. Statistica Sinica 14, 715–745. G AMERMAN , D. (1997). Sampling from the posterior distribution in generalized linear mixed models. Statistics and Computing 7, 57–68. G ELMAN , A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian Analysis 1, 515–534. H ASTIE , T. J. AND T IBSHIRANI , R. J. (1990). Generalized Additive Models. London: Chapman and Hall. H OBERT, J. P. AND C ASELLA , G. (1996). The effect of improper priors on Gibbs sampling in hierarchical linear mixed models.Journal of the American Statistical Association 91, 1461–1473. K ASS , R. E. AND S TEFFEY, D. (1989). Approximate Bayesian inference in conditionally independent hierarchical models (parametric empirical Bayes models). Journal of the American Statistical Association 84, 717–726. K ELSALL , J. E. AND WAKEFIELD , J. C. (1999). Discussion of â€Å"Bayesian models for spatially correlated disease and exposure data† by N. Best, I. Waller, A. Thomas, E. Conlon and R. Arnold. In: Bernardo, J. M. , Berger, J. O. , Dawid, A. P. and Smith, A. F. M. (editors), Sixth Valencia International Meeting on Bayesian Statistics. London: Oxford University Press.M C C ULLAGH , P. AND N ELDER , J. A. (1989). Generalized Linear Models, 2nd edition. London: Chapman and Hall. M C C ULLOCH , C. E. , S EARLE , S. R. AND N EUHAUS , J. M. (2008). Generalized, Linear, and Mixed Models, 2nd edition. New York: John Wiley and Sons. M ENG , X. AND W ONG , W. (1996). Simulating ratios of normalizing constants via a simple identity. Statistical Sinica 6, 831–860. NATARAJAN , R. AND K ASS , R. E. (2000). Reference Bayesian methods for generalized linear mixed models. Journal of the American Statistical Association 95, 22 7–237. N ELDER , J. AND W EDDERBURN , R. (1972). Generalized linear models.Journal of the Royal Statistical Society, Series A 135, 370–384. P INHEIRO , J. C. AND BATES , D. M. (2000). Mixed-Effects Models in S and S-plus. New York: Springer. P LUMMER , M. (2008). Penalized loss functions for Bayesian model comparison. Biostatistics 9, 523–539. P LUMMER , M. (2009). Jags version 1. 0. 3 manual. Technical Report. RUE , H. AND H ELD , L. (2005). Gaussian Markov Random Fields: Thoery and Application. Boca Raton: Chapman and Hall/CRC. RUE , H. , M ARTINO , S. AND C HOPIN , N. (2009). Approximate Bayesian inference for latent Gaussian models using integrated nested laplace approximations (with discussion).Journal of the Royal Statistical Society, Series B 71, 319–392. 412 RUPPERT, D. R. , WAND , M. P. University Press. AND Y. F ONG AND OTHERS C ARROLL , R. J. (2003). Semiparametric Regression. New York: Cambridge S KENE , A. M. AND WAKEFIELD , J. C. (1990). Hie rarchical models for multi-centre binary response studies. Statistics in Medicine 9, 919–929. S PIEGELHALTER , D. , B EST, N. , C ARLIN , B. AND VAN DER L INDE , A. (1998). Bayesian measures of model complexity and fit (with discussion). Journal of the Royal Statistical Society, Series B 64, 583–639. S PIEGELHALTER , D. J. , T HOMAS , A.AND B EST, N. G. (1998). WinBUGS User Manual. Version 1. 1. 1. Cambridge. T HALL , P. F. AND VAIL , S. C. (1990). Some covariance models for longitudinal count data with overdispersion. Biometrics 46, 657–671. V ERBEKE , G. V ERBEKE , G. AND AND Downloaded from http://biostatistics. oxfordjournals. org/ at Cornell University Library on April 20, 2013 M OLENBERGHS , G. (2000). Linear Mixed Models for Longitudinal Data. New York: Springer. M OLENBERGHS , G. (2005). Models for Discrete Longitudinal Data. New York: Springer. WAKEFIELD , J. C. (2007). Disease mapping and spatial regression with count data.Biostatistics 8, 158–1 83. WAKEFIELD , J. C. (2009). Multi-level modelling, the ecologic fallacy, and hybrid study designs. International Journal of Epidemiology 38, 330–336. WAND , M. P. AND O RMEROD , J. T. (2008). On semiparametric regression with O’Sullivan penalised splines. Australian and New Zealand Journal of Statistics 50, 179–198. Z EGER , S. L. AND K ARIM , M. R. (1991). Generalized linear models with random effects: a Gibbs sampling approach. Journal of the American Statistical Association 86, 79–86. [Received September 4, 2009; revised November 4, 2009; accepted for publication November 6, 2009]

Factors Impacting Spouse of a Chronically Ill Patient Essay

Factors Impacting Spouse of a Chronically Ill Patient - Essay Example For instance, a spouse might be compelled to set aside personal needs or reorganize private lives in preparation for the unpredictability of the condition of the patient (Kapteinm, et al., 2007). Such unexpected occurrences, as claimed by most spouses of the chronically ill patient may hinder individual achievements and create relationship changes. In support of the change in relationship argument, theory asserts that relationship changes emanate from the increased thoughts about the future on the part of the healthy spouse (Rees, et al., 2001). Most of the healthy spouses, if not encouraged by the ill partner to be free, end up suffering psychologically due to increased stress stemming from the worry of the pain experienced by the sick partner. Further research indicates that most ill partners tend to suffer as they try to cope with their new lifestyles. For instance, a blog shows the quality of chronically ill patients suffers in a myriad of ways (sickmomma, 2013). They have to deal with fear, malaise, and apathy. All these stated aspects have negating impacts on their sexuality and social perception. They often feel powerless due to the pity and constant attention they receive from the public, an aspect that tends to worsen the agony of the pain. In concurrence with these sentiments, Kaptein et al., (2007) claim that the lack of information on the public on how to treat terminally ill patients is a significant challenge. They argue that since their spouses and public perceives them with pity, the chronically ill patients end up viewing themselves as powerless in the society. In conclusion, chronically ill patients though suffer from the ailment; it is evident that their suffering spreads to their healthy spouses. The sudden change in lifestyle leads to unexpected shifts in the relationship as both partners try to adjust to the new developments. Kaptein, A. A., Scharloo, M., Helder, D. I., Snoei, L., van Kempen, G. M., , J., . . . C. (2007).

The Growing Middle Class in Emerging Markets in Asia Research Paper

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Patient narrative - Essay Example However, some exceptional charges are levied on optical services, dental services, and prescriptions. Services offered by this health organization includes; pathological services, emergency and urgent care, hospital services, dental services, GP services, pharmacy services, eye care services, sexual health services, mental health services, and social care services. In the United Kingdom, the infant mortality rate has reduced significantly; and the life expectancy levels have continuously risen (Baille, 2008); this has been noticed since the establishment of the NHS. Health surveys, most importantly, illustrate that patients are usually satisfied with services received from the National Health Services. Patient satisfaction is directly determined by the patient’s experience when dealing with the health services provider. This interaction is perceived in patient’s conscious and also subconscious mind. Patient experience is mainly about three main issues; delivery of the organization as a whole in the healthcare, the emotional and rational experience during the service delivery, and the intuitive perceptions of patients (Tschudin, 2003). Therapeutic effect involves the consequences of any medical treatment. The results of the therapeutic effect are usually seen to be beneficial and also desirable to the patient. The Nursing and Midwifery Council (NMC) code of conduct; is responsible for ensuring acceptable performance, ethics and conduct; for professional nurses and midwives. Nursing is a profession found in the health care sector. It deals with care on people, families and communities; so as to maintain, recover or attain quality health and quality life. Nurses are responsible for developing plans for health care, working in teams with therapists, physicians, the patient’s family, the patient, and other staffs in the team (Chin, 2008). The healthcare plan leads to treating of illness, so as

The Library Organisational Project Case Study Example | Topics and Well Written Essays - 1500 words

The Library Organisational Project - Case Study Example Different data-gathering methods were used, such as conducting interviews, distributing questionnaires and searching information from various sources, to gather as much information as the project requires. Team members were assigned to conduct interviews with each senior manager. I was responsible for interviewing Mr. Antony Brewerton, Head of Academic Support. Initially, I asked Mr. Brewerton to fill out questionnaires with the intention to analyse his perspective against his employee's perspective. Afterwards, I proceeded to interview him on leadership and management so as to analyse his leadership style and how he manages his department. Organisational Diagnosis Questionnaire (ODQ) designed by Robert Preziosi (1980) was used and the questionnaires were distributed to cover the five departments in the library. Using this questionnaire, a variety of dimensions of an organisation could be studied-such as purpose, structure, leadership, relationships, rewards, helpful mechanisms and attitude toward change. The questionnaire was used the in the analysis because of its benefits, as enumerated below- With the data gathered from the questionnaires, I conducted the qualitative analysis. ... b) Questionnaires: Organisational Diagnosis Questionnaire (ODQ) designed by Robert Preziosi (1980) was used and the questionnaires were distributed to cover the five departments in the library. Using this questionnaire, a variety of dimensions of an organisation could be studied-such as purpose, structure, leadership, relationships, rewards, helpful mechanisms and attitude toward change. The questionnaire was used the in the analysis because of its benefits, as enumerated below- Relatively easy to analyse, People familiar with the questionnaire, A large sample of the given population can be conducted, Information is collected in a standardised way, The questionnaires can be used for sensitive topics, which users may feel uncomfortable speaking about to an interviewer, and Respondents have time to think about their answers because they are not usually required to reply immediately. After finishing the interview with Mr. Brewerton, I asked for his cooperation to distribute the questionnaires among his staff through emails and to kindly forward the responses to me. c) Various sources: To get acquaintance with the organisation, I searched the information from different sources such as library websites, articles and documents provided by the Librarian. 2. Data Analysis The findings were divided into two categories, which are qualitative analysis and quantitative analysis- a) Qualitative analysis With the data gathered from the questionnaires, I conducted the qualitative analysis. The ODQ questionnaires were grouped under the seven dimensions and I calculated the average ranking for each of the dimensions. I plotted graphs in MS

Coursework 1 Essay Example | Topics and Well Written Essays - 750 words

Coursework 1 - Essay Example Benchmarking in the Egyptian hotel sector involved administering questionnaires to staff members of hotels who acted as respondents. 128 managers from selected star hotels located in Cairo, Alexandrian, Aswan and sharmel- sheik cities took part as respondents (Nassar 2012). The questionnaires contained three sections that sought information on respondents and their hotels, their attitude towards benchmarking and their perceptions drawn from their understanding of advantages and barriers of benchmarking. The methodology used in Taiwan was conduction of a case study. Relevant statistical data and information was obtained from reports on international tourists’ hotels operation. This data was provided by the Tourism Bureau, an entity in the Ministry of Transport and Communication (Wei-Wen, Lan & Yu-Ting 2013). A number of 80 hotels were used for the case study. There was an application of DEA (data envelopment analysis) approach purposed to identify the periods that sustained hig h performance by the hotel industry. The benchmarking process carried out in Egypt was an empirical case. Questionnaires developed obtained feedback from a given number of respondents that provided information used to draw conclusions about the hotels in the country. The process utilized an average number of 128 managers in 5 star hotels in Egypt (Nassar 2012). The type of benchmarking applied in Taiwan was performance benchmarking. The activity was empirical as well involving use of statistical data. The data was obtained from 80 international tourists hotels in the country. It was further used to draw out conclusions on performance of the mentioned hotels. Findings and conclusions were made after successful benchmarking. In Egypt, participants in the benchmarking displayed positive attitude towards the process. They clearly understood that the initiative served the purpose enhancing quality and providing a platform for connecting different or