Wells, H. G. Essays - The Time Machine, Morlock, Time Travel

Wells, H. G. Herbert George Wells was conceived in 1866 in Bromley, Kent, a couple of miles from London, the child of a house-servant and plant specialist. Wells passed on in 1946, a well off and celebrated creator, having seen sci-fi become a perceived artistic structure and having seen the world understand some of science fictions fondest dreams and most exceedingly awful fears. Wells mother endeavored to discover him a protected occupation as a draper or on the other hand scientist. Wells had a speedy psyche and a decent memory that empowered him to pass subjects by assessment and win a grant to the Normal School of Science, where he remained for a long time and, in particular, was presented to science under the acclaimed Thomas H. Huxley. Wells went into instructing and composing reading material what's more, articles for the magazines that were of that time. In 1894 he started to compose sci-fi stories. - James Gunn Wells vision of things to come, with its troglodytic Morlocks dropped from the common laborers of his day and the pretty yet vulnerable Eloi lapsed from the recreation class, may appear to be out of date political hypothesis. It rose out of the worry for social equity that attracted Wells to the Fabian Society and propelled quite a bit of his later composition, yet time has not darkened the interest of the circumstance and the awfulness of the symbolism. The Time Machine brought these worries into his fiction. It, as well, included the future, yet a future envisioned with more noteworthy authenticity and in more prominent detail than prior accounts of things to come. It likewise presented, for the first time in fiction, the idea of a machine for going in time. In this novel the Time Machine by H. G. Wells, begins with the time voyager attempting to convince his visitors the hypothesis of the fourth measurement also, even the development. He attempts to clarify the fourth measurement before he shows them the time machine so they dont consider him an entertainer. H. G. Wells utilizes subtleties about the fourth measurement to show the peruser the hypothesis about it to catch your consideration. Likewise Wells character the time traveler says Logical individuals, Realize very well that time is just a sort of room. In this statement he is unmistakably utilizing influence strategies. He attempts to assault there consious by saying that, logical individuals realize this is just a sort of room. He says this in trusts that they will accept what he says on the grounds that other savvy individuals accept the hypothesis. This is an exceptionally crude yet at the same time a compelling way to attempt to convince individuals. The thought is on the grounds that numerous individuals trust it, so it must be valid. The individuals he is attempting to convince are of nineteenth century thinking what's more, well to do individuals and they are serious among other wealthy individuals so on the off chance that other rich and shrewd individuals accept this fourth measurement hypothesis so the time traveler trusts this will spur them to find out about it. The Characters in the book Time Machine are The time traveler, Filby, the therapist, and the common civic chairman. Later the quiet man and the manager come in to play. Filby is depicted as a contentious individual with red hair. He has another mark that Wells puts on him; he consider him the youngster. The analyst likewise has another name; he is the clinical man. The time voyager is portrayed quickly when the gathering of minds head down the passage to the research facility. He utilizes his eccentric wide head in outline. When the show up at the machines area it is portrayed as Parts were made of nickel, portions of ivory, parts had positively been documented or sawn out of rock precious stone. He likely picked these characters as witnesses since they hold advanced education what's more, individuals would trust them from that point notorieties. The therapist would be recipient in persuading the other that it is anything but a lie since he is mindful of human conduct. The commonplace city hall leader is likewise a shrewd man and the individuals chosen him so on the off chance that he is to accept that this works, at that point numerous individuals would follow him. Filby is another character yet never discusses his remaining in society it could be his companion since he winked at the time traveler or possibly he isn't since he questioned the time travelers time machine in his face also, behind his back. H. G. Wells utilizes two different characters that come to supper to meet the time voyager. The primary character returns from what's to come. The clinical specialist and the common civic chairman

Public vs private management Research Paper Example | Topics and Well Written Essays - 500 words

Open versus private administration - Research Paper Example The principal distinction is on the responsibility for association where the open areas are run and financed by the legislature, and the private division is worked and claimed by the enterprise or the private proprietors. In the open part, all representatives including the administrators work for the legislatures while the representatives in the private division work for the association (Dresang, 2009). The other distinction between the two parts is that open segments are set up by law. This reality demoralizes chiefs from changing the laws relying upon the evolving conditions. Then again, chiefs in a private division can change the target when important. The yield of a private part can be esteemed by the paying clients. This is effectively caught on the organization’s targets, missions and choice standards. Be that as it may, it may not be simple for the market powers to direct the additions of an open segment element (Mgbeke, 2009). Hence, private parts have the duty to produce palatable profits for their speculation and private financial specialists get more inspiration to put more ventures. In opposition to this, open divisions despite everything have the presumption that their capital is free, and the need to increase huge benefits are overlooked (Dresang, 2009). Aptitudes One expertise an open supervisor ought to create is the capacity to speak with the general population. The correspondence with the pubic and different pariahs is a central and viable expertise for the open supervisor.

10 June New Releases To Put On Hold at the Library Right Now

10 June New Releases To Put On Hold at the Library Right Now Wishlist upcoming releases youre dying to read. Get exclusive podcasts and newsletters. Enter to win swag. Do it all when you join Insiders. Subscribe to Book Riot Insiders! The eternal quandary of readers everywhere: How to decide what upcoming books to request at the library before the list is eleventy-million people long. Ive got the cure for the common hold: Here are ten big books of note coming out in June to help you choose what to reserve now. Get to these babies before everyone else at your local library! (And as always, you can find me raving about a bunch more on social media. Only picking ten books is HARD.) The Book of M by Peng Shepherd (June 5) Fans of Station Eleven, listen up! In a dystopian near-future world, people have begun to lose their shadows. And not in a cutesy Peter Pan wayâ€"when someone loses their shadow in this book, it means they are destined to also lose their memory shortly after. Fear has caused mass chaos and struggles for power, and the world is a bleak place, and it is up to the remaining survivors to find a cure before they lose their own shadows. This one is g-r-e-a-t. Florida by Lauren Groff  (June 5) Groff, author of Fates and Furies (one of Obamas favorite books, NBD) is back with an electric collection of stories revolving around Florida and the trials and tribulations of living there, whether the issues are caused by nature or by humans. This is a fantastic showcase of Groffs amazing ability to capture human emotions and behaviors and describe them for our benefit. Sick: A Memoir by Porochista Khakpour  (June 5) Khakpur has two wonderful novels under her belt. Now she takes readers on a personal journey through her life and her struggles with late-stage Lymes disease, and what its like to live with a chronic illness. She explores her illness by way of the different places she has lived, and explains how she manages the impact her illness has on her mental and physical health, and the toll it has taken. Perfect for fans of Brain on Fire. There There by Tommy Orange  (June 5) A powerful, contemporary, multi-generational portrait of Native Americans in the United States, revolving around several people coming together at a powwow.  There  are people hoping to repair rifts, defeat addiction, participate in culture, and also cause grief. It’s a devastating and sad novel, but also filled to the brim with beauty and hope. This one will stay with you for a long time. Expect it to win awards. Social Creature by Tara Isabella Burton  (June 5) A debut thriller about an obsessive friendship. Lavinia is a gorgeous socialite who takes Louise under her wing and makes it her mission to bring her out of her cocoon. But as they party their way through endless nights of glitz and glamour, can their friendship last? What happens when you try to put a butterfly back in its cocoon? A Reaper at the Gates (An Ember in the Ashes) by Sabaa Tahir  (June 12) Its almost here!!!!!! This is the third book in the fantastic Ember quartet. Im not going to tell you what its about, in case you havent read the first two books. If youve read them, you are already excited for this, and if you havent read them, OMG do that right now. Ill wait here. The Great Believers by Rebecca Makkai  (June 19) A moving novel of friendship and loss set in Chicago during the height of the AIDS crisis in the 1980s and then in Paris 30 years later, about a gallery owner saying goodbye to his friends and a woman searching for her daughter. Rebecca Makkai is an unsung gem. Maybe this novel is the one to catapult her to new heights! Tango Lessons: A Memoir by Meghan Flaherty  (June 19) Not only is this Flahertys triumphant account of overcoming her fears after trauma and learning to follow her dreams and trust in herself, but its also a beautiful look at the history of tango itself. This is sure to be a big book club pick. A Thousand Beginnings and Endings by Ellen Oh and Elsie Chapman  (June 26) Fifteen wonderful stories reimagining the folklore and mythology of East and South Asia, written by such amazing authors as  Renée Ahdieh, Sona Charaipotra, Preeti Chhibber, Melissa de la Cruz, Julie Kagawa, Cindy Pon, and Alyssa Wong. This beauty was compiled by the team behind We Need Diverse Books. Dead Girls: Essays on Surviving an American Obsession by Alice Bolin (June 26) Bolin examines Americas national obsession with stories surrounding dead girls. Her essays include examinations of Twin Peaks, Serial, and works by Joan Didion and James Baldwin, as well as a discussion of the information and narratives surrounding dead girls that we absorb every day. This is wise, fascinating stuff.

Pride And Prejudice And Cat On A Hot Tin Roof - 2074 Words

Marriage is a consistent theme throughout the entirety of â€Å"Pride and Prejudice† and â€Å"Cat on a Hot Tin Roof†. Whilst Jane Austen uses a well- informed narrative subtly highlighting both pride and prejudice throughout the society at the time, Williams uses dialogue, ‘plastic theatre’ and stage directions that appear to be set in ‘real time’ to express themes such as love, marriage, power and respect. Despite the obvious contextual differences such as the difference in in which that the play and novel were both written and the cultural differences in marriage, I intend to look at the lack of respect in marriages between both participants in the relationship which features in both â€Å"Pride and Prejudice† and â€Å"Cat on a Hot Tin Roof† and which therefore makes Mr and Mrs Bennet and Big Daddy and Big Mama comparable. The marriages in both texts begin when Mr Bennet and Big Daddy marry women potentially for the wrong reasons: for their physical appearances. Subsequently, disdain towards both of the female characters is shown throughout, with both of the husbands failing to show respect for their wives, using humour and irony to ridicule the females intellectually and physically. This is particularly shown with Big Daddy as he outwardly insults Big Mama in front of everybody: Big Daddy: â€Å"All I ask... is that she leaves me alone†¦ she makes me sick†¦Ã¢â‚¬  The stage directions also support the failure of the marriage, where Big Daddy â€Å"regards her with†¦ annoyance†. This language is harsh andShow MoreRelatedIgbo Dictionary129408 Words   |  518 Pageschewing-sticks and for à  kà  loà ²gà ²là ¬ figures A. plant resembling aÅ‹Ä , used for its fibres B. the rope made from it wooden vessel used for carrying (usu. à ²Ã¯â‚¬ ¥ká » ¥kà ¹Ã¯â‚¬ ¥ abà ¹ke) kind of fowl which never grows to a large size but is tough (usually used for sacrifice) pus cat-like animal that sleeps by day, probably the Two-Spotted Palm Civet or genet armpit boil in armpit song; solo A. song (esp Song of Solomon) B. mourning song opening song dismissal song victory song memorial song old hymn book songs of praise new hymnRead MoreThe Ballad of the Sad Cafe46714 Words   |  187 Pagesestranged from all other places in the world. The nearest train stop is Society City, and the Greyhound and White Bus Lines use the Forks Falls Road which is three miles away. The winters here are short and raw, the summers white with glare and fiery hot. If you walk along the main street on an August afternoon there is nothing whatsoever to do. The largest building, in the very center of the town, is boarded up completely and leans so far to the right that it seems bound to collapse at any minute

An Issue Of Oil Fracking - Free Essay Example

Sample details Pages: 4 Words: 1337 Downloads: 4 Date added: 2019/08/16 Category Technology Essay Level High school Tags: Fracking Essay Did you like this example? Oil fracking has been a long practice undergoing the resource mining of oil to be purposed for our own benefits. The underlying history that is brought from the beginning of oil fracking relies in polices and the Yom Kippur War. These two aspects in oil fracking will help understand what the contemporary issues history is from then. To then help better understand the current state of the issue. Oil fracking polices are an important aspect the contemporary issue for specific business-related reasons during the twentieth century. One example of a company that was around during the twentieth century was the Hawaiian-Texas oil company LTD. This company is a larger forceful company that owns a multitude of oil properties. The company owns sixty different leased oil properties. Totaling six thousand acres. From these properties being leased not enough information was being disclosed to the investors of these oil producing spaces. For example, the policy states their will provide a fifty percent split between the investors and LTD. 50 of the net proceeds from Oil produced, and profits derived from the Companys Operations . Along with them drilling additional wells from the money being profited. The remaining percentage of money that was invested into the company was put towards the continued development of the business. LTDs business and mindset to expand and further develop their production of oil was going well. But investors werent disclosed were the oil fields were being produced. During this time the company had a total of sixty separate oil leases. One oil well in Okmulgee County, Oklahoma was being drilled about 600ft from the Hartfords producing oil well . This was leading to a conflict with their opposing business due to zoning restrictions not being established. Don’t waste time! Our writers will create an original "An Issue Of Oil Fracking" essay for you Create order George P. Mitchell is another example that involves oil fracking policies needing to be developed. George P. Mitchell was a significant oil business man that started in 1943. He was the known as the pioneer for oil fracking During his career in the oil fracking business. He participated in drilling some 10,000 wells, including more than 1,000 wildcats   wells drilled away from known fields . His company, Mitchell Energy Development, was credited with more than 200 oil and 350 natural gas discoveries . In the 1960s, Mr. Mitchell, looking to diversify, bought 66,000 acres of mostly undeveloped real estate within a 50-mile radius of Houston . In 1974 he created The Woodlands, 27,000-acre forested development 27 miles north of Houston, helped by a $50 million loan   from the Department of Housing and Urban Development. Many people were living there. This was another example of zoning restriction related to commercial land usage for the residential usage at the time. Exon Mobil at th e time was also involved with this commercial zoning issue with the land. After George P. Mitchells conflict with exon Mobil. This then later leads into the Right-to-Know Act. This was a legislation the United states wanted to pass to help disclose further activities that are being done. The establishment of this law sets a standard of disclosure for the toxic level exposure to the EPA. In the 1980s, the EPA and Congress agreed not to apply RCRA to oil and gas wastes, overriding objections from some officials at EPA after the agency had documented 62 cases in which oil and gas wastes had caused damage . Leading from this legislation being established was the need for many other categories to be implemented into the legislation. This is so the people can also be aware of the need of the Right-to-Know Act. During 1974 The Yom Kippur War had begun. Some background that lead to this war included the previous war in 1967 which was called the Six-Day war. This war had three distinctive battlefronts that were all tied together. There was a shared desire to ease Israel and redeem themselves for the defeat they had previously 19 years ago. When they failed to destroy the Jewish state. the Egyptians and Syrians launched their attacks at 1400 hours, 6 October 1973 . This attack was launched on the holiest day of the calendar for the Israeli population. There were to fronts in this war which were. The Southern Front and The Northern Front. The Southern front had the Egyptian Third Army in the south. Additionally, 8,000 assault infantrymen . Their objective was to storm across the canal. With the intent to take control of the eastern side. During the same time. The commando and infantry tank destroyer units were crossing the canal currently. The tanks were approaching the ramps for the preparatio n of ambushes that could await them from Israel. By 1600 hours, the crossing was more successful in the northern sector of the Canal than in the southern sector .   The Egyptians original plan of attack on the canal was to go through and infiltrate at one single point of the canal. Instead they had to divide up the attack into five different axis of infantry paths into the canal. They had limited air assets to help execute this attack in better manner.   The air assets were best believed to be used in the northern front of the attack. The Northern front had mirrored the circumstances of the southern fronts attack. Both fronts had built multiple defense positions on their sides. The Syrians had deployed three infantry divisions along the 45-mile front . This included the Syrians third armor division in the north and the Syrians first armored division in the center. The attack was executed in synchronization with the Egyptians. This attack method heavily relied on the penetration of the infantry troops to reach the high ground of the east Jordan river. The U.S didnt get involved until later when the war coming closer to the end. The United States and the United Nations had begun discussing the probable solutions to this conflict of war. The United States got involved in the behalf of the Israel forces. What was offered to Israel was a full-scale airlift of military equipment, the replenish of their forces, and a stronger offensive front. With this they both retook most of the territories that were previously lost before. The Resolution was in the need for the Israel and other Arab nations to conclude with a peace treaty deal. The United States also began to re-examine its policy in the Middle East when it faced the Arab oil Embargo at the end of the war . The need to settling this Israel-Arab conflict became the top priority for the United States. The oil Embargo was established by and Arab organization called OPEC. OPEC had set the oil embargo to stop the exportation of oil into the United States. The twelve members of the OPEC organization made that agreement on October 19, 1973. What set the growing attention amongst the members of OPEC was the depleting price of the dollar. The contract that was established between them and the United States was an agreement in oil pricing for the nation. The depleting value of the dollar was hurting them under the nations revenue. OPECs last straw was when the United States started to support the Israelis against Egypt in the war. The members of the OPEC organization halted the exportation of oil to the United States. This created an economical problem from the U.S due the war and the outcomes from the war. Oil fracking has been a long practice to obtain the resource of oil. The high demand for the oil underlies in the motives of business and economical power. LTD had a lot of properties to attract investors into their practice of oil fracking. But they werent fully disclosing all the oil fracking practices being to the investors. Since of them not doing this disclosure. Further regulations were implemented at the time do have better fracking practices for the business. The Yom Kippur War was motivated by economical around the embargo placed. This intervention the United States made was to help stop a conflict that later led to The United States taking an economic toll from that embargo in place. This understanding of the contemporary issue of oil fracking helps further understand the contemporary issue in our current time.

Bayesian Inference Free Essays

string(34) " in the context of a binary GLMM\." 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. We will write a custom essay sample on Bayesian Inference or any similar topic only for you Order Now 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 jonno@u. 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 binary 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. permissions@oxfordjournals. 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. Likelihood-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 estimato rs, 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(†¢), 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 ef fects, 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 that ? is assigned a normal prior distribution. Let ? = (? , b ) denote the G ? 1 vector of parameters assigned Gaussian priors. We also require priors for ? 1 (if not a constant) and for ? 2 . Let ? = (? 1 , ? 2 ) be the variance components for which non-Gaussian priors are ? assigned, with V = dim(? ). 3. I NTEGRATED NESTED L APLACE APPROXIMATION Before the MCMC revolution, 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. You read "Bayesian Inference" in category "Papers" 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 nonstandard 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 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 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 linea r 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 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 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 components We 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 based, but first define some notation. We write ? ? Ga(a1 , a2 ) for the gamma distribution with un? normalized density ? a1 ? 1 exp(? a2 ? ). For q-dimensional x , we write x ? Tq (? , , d) for the 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 pl aces 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 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 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 (c omplete smoothing) and 10 (no smoothing). From panel (b), we conclude that here the prior choice favors quite strong smoothing. This may be contrasted with the gamma prior with parameters (0. 001, 0. 001), which, in this example, gives reater than 99% of the prior mass on an effective degrees of freedom greater than 9. 9, again showing the inappropriateness of this prior. It is appealing to extend the above argument to nonlinear models but unfortunately this is not straightforward. For a nonlinear model, the degrees of freedom may be approximated by C df = tr{(C T W C + where W = diag Vi? 1 d? i dh 2 )? 1 C T W C }, and h = g ? 1 denotes the inverse link function. Unfortunately, this quantity depends on ? and b , which means that in practice, we would have to use prior estimates for all of the 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 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. 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 defici ency 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 implementation contained in the INLA software. 5. 1 Longitudinal data We consider the much analyzed epilepsy data set of Thall and Vail (1990). These data concern the number ? of seizures, Yi j for patient i on visit j, with Yi j |? , b i ? ind Poisson(? i j ), i = 1, . . . , 59, j = 1, . . . , 4. We concentrate on the 3 random-effects models fitted by Breslow and Clayton (1993): log ? i j = x i j ? + b1i , (5. 1) (5. 2) (5. 3) Downloaded from http://biostatistics. oxfordjournals. rg/ at Cornell University Library on April 20, 2013 log ? i j = x i j ? + b1i + b2i V j /10, log ? i j = x i j ? + b1i + b0i j , where x i j is a 1 ? 6 vector containing a 1 (representing the intercept), an 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) an d (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 bivariate specification). In model (5. 2), ? 2 ? 2 we assume b1i and b0i j are independent, and that ? 0 follows the same prior as ? , that is, Ga(2, 1. 140). We assume a flat prior on the intercept, and assume that the rate ratios, exp(? j ), j = 1, . . . , 5, lie between 0. 1 and 10 with probability 0. 95 which gives, using the approach described in Section 4. 1, a normal prior with mean 0 and variance 1. 172 . Table 1 gives PQL and INLA summaries for models (5. 1–5. 3). There are some differences between the PQL and Bayesian analyse s, 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 baseline-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 aliase d (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 factor. 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 407 Downloaded 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 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 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 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 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 Vari able Intercept Black Hispanic White Age ? 1 ? 2 ? REML 0. 560  ± 0. 029 0. 106  ± 0. 021 0. 013  ± 0. 022 0. 026  ± 0. 022 0. 021  ± 0. 002 0. 018 0. 109 0. 033 INLA 0. 563  ± 0. 031 0. 106  ± 0. 021 0. 13  ± 0. 022 0. 026  ± 0. 022 0. 021  ± 0. 002 0. 024  ± 0. 006 0. 109  ± 0. 006 0. 033  ± 0. 002 Note: For the entries marked with a standard errors were unavailable. 410 Y. F ONG AND OTHERS 6. D ISCUSSION In this paper, we have demonstrated the use of the INLA computational method for GLMMs. We have found 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 alter native 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, 2013 S 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. 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Kham Wa Essays (348 words) - , Term Papers

Kham Wa Dr. Robert J. Caputi HI-102-02 November 8, 2016 The Atomic Bombing of Hiroshima Although the government has very good reasons for using atomic bombs, it is my personal opinion that their reasons can never be good enough to justify the lives of millions of innocent people. The atomic bombing of Hiroshima was completely unjustified! The use of atomic bombs was totally unnecessary, inhumane, and irresponsible. However terrible the effect of atomic bombs on homes and buildings, what stands out most in my mind is the people. Although the immediate results were horrifying enough, people had not even begun to realize the great effects it would have on humans in the long term. The immediate effects on Hiroshima's people were just a foreshadowing of the ones to come. Thousands of people were killed instantly. Those who survived the initial blast died later from high doses of radiation, which burns off skin and hair, then eventually kills you. Without a doubt, the atomic bomb's effect on people was by far the worst. The development and usage of the first atomic bombs has caused a change in military, political, and public functionality of the world today. The bombings of Hiroshima and Nagasaki revolutionized warfare by killing large masses of civilian population with a single strike. The bombs created a temporary resolution that lead to another conflict. The Cold War was a political standoff between the Soviet Union and the United States that again created a new worldwide nuclear threat. The destructive potential of nuclear weapons had created a global sweep of fear as to what might happen if these terrible forces where unleashed again. The technology involved in building the first atomic bombs has grown into the creation of nuclear weapons that are potentially 40 times more powerful than the original bombs used. The fear of a potential nuclear attack had been heightened by the media and its release of movies impacting on public opinion and fear of nuclear devastation. The lives lost after the de tonation of the atomic bombs have become warning signs that changed global thinking and caused preventative actions.