Financial Analysis Of British American Tobacco Plc Finance Essay

Financial Analysis Of British American Tobacco Plc Finance Essay Introduction The purpose of this report is to provide information Interpretation of British American Tobacco plc (BAT) in terms of historical record, comparative financial indicators, and position in the market along with key value drivers for the company and performance indicators. This is done by analysing the information provided in the historical and present financial statements. About British American Tobacco Revenue Analysis British American Tobacco is a public company listed on the London Stock Exchange, which has direct and indirect stakes in several companies together constituting to be the British American Tobacco Group of companies. The group achieved gross turnover of GBP 40,713 million with revenues amounting to GBP 14,208 Million in the year 2009. There are more than 250 brands in their portfolio with Dunhill, Kent, Lucky Strike Paul Mall being their flagship cigarette brands which are together sold in more than 120 countries with total sales constituting 196 Billion cigarette units (Annual Report, 2009). The company follows the accounting cycle starting 1st January of the calendar year ending on 31st December of the same calendar year. The company currently has 95,710 employees on its payrolls. It sold 724 Billion cigarette units in the same year and has production capabilities in 50 cigarette factories based in 41 countries. British American Tobacco is the second largest tobacco company in the world (Excluding China), while it has 6.4% share of the UK market, dominant players being Imperial Tobacco Japan Tobacco International (Nielsen Data, Feb 2010). 74% of BATs sales come from the developing countries emerging economies (Annual Report, 2009). Revenue by Geography BATs largest market lies in Western Europe which accounted for 27.3% of its total revenues in FY 2009. BAT saw an increase of 20.7% from the revenues it had registered in the FY 2008. Asia Pacific accounted for 23% of its total revenues while Americas accounted for 22.2%. Eastern Europe accounted for 11.5% o the total revenues while the highest performed for BAT turns out to be Africa and Middle east which accounted for 22.2% but it saw an increase of 31.2% in terms of revenue over last year. Historical Performance Analysis British American Tobacco was established as a joint venture between the Imperial tobacco group of the United Kingdom the American Tobacco Group of the United States in the year 1902. Subsequently, In the year 1911 the company was listed on the London Stock Exchange. In 1913, the company looked overseas for expansion and entered the Argentinean markets. In the 1920s, British American Tobaccos capitalization had quadrupled since 1902 and sales grew by nearly a factor of 40. In 1923, The Companys worldwide sales had grown to 50 billion cigarettes per year. By 1962, British American Tobaccos capitalization allowed it to move towards diversification. It grew consistently, however, and was achieving turnover at the rate of 15% per annum by 1970. Diversifying and expanding at a rapid pace, British American Tobacco became a well known brand globally. With uncertainty about a long term market in tobacco, steps were taken to reduce BATs dependence on the tobacco industry. In 1986, only half o f its total pre tax profit came from tobacco group which was down from 57% pre tax profit BAT achieved in the year 1985. The 1990s were not a good time of the decade for tobacco sales as companies in the industry faced several lawsuits and litigations. The U.S. courts awarded verdicts which cost the tobacco companies millions of dollars as the consumer claimed tobacco related illness and relatives of smokers who claimed heavy compensation. The company continued with its strategy to take over small to midsized companies as it acquired Canadas dominant tobacco company, Imasco in the year 2000. The company has been consistently achieving Year on Year growth in the range of 8-12% in terms of revenue since 2005. While it has consistently maintained the operating margin percentage in the range of 26%-29% from 2005 to 2009, the Adjusted Diluted earnings per share has been rising from 56.9p in 2000 to 153p in the year 2009 (Annual Report, 2005-2009). Annual Report Analysis The goal of accounting information is to provide economic decision makers with useful information, according to Williams, Haka, Bettne Carcello (2006, p. 670). Financial statements analysis is not just important for the shareholders but various stakeholders as well. The Group has prepared its annual consolidated financial statements in accordance with International Financial Reporting Standards (IFRS), as endorsed by the European Union. Some of the highlights of the Annual Report include revenue increase of 10% at constant rates of exchange 17% at current rate, when compared to the financials for the year 2008. Adjusted profits from operations too increased by 10% at constant rates and 20% at current rates. The total benefit of the result amounted to GBP 355 Million, which resulted in adjusted diluted earnings per share grow by 19% to 153 pence. Over the past 5 years, BAT has achieved a compounded annual growth rate of 15% in earnings per share and 19 % in dividends per share. The total shareholder returns over the same 5 year has been 175% compared to the FTSE 100 index which gave 35% returns to the investors. The financial analysis views the group as a strong player in the global tobacco industry. The resilience in achieving profits wealth generation along with geographical diversification has positioned the group as a multi national company with strong fundamentals which makes it more susceptible to face risks and unforeseen events in the future. In the first half of the financial year 2009, Sales volumes had increased by 4% or declined by 2% when the benefit derived from acquisition was excluded. BATs free cash flow remained strong and resilient during the year 2009 and looks set to remain the same in the year 2010 despite volume pressures. Price increases sales improvements continue to offset the volume pressure faced by the company in the broad range of other markets. In the financial year 2009, FFO was 31% of fully adjusted debt which was 3% points higher than the FFO in the year 2008. The reason for the same was marginal decrease in debt along with the increase in earnings due to acquisitions. The absorption of operating cash flow in to discretionary spending has slowed due to the company suspending the offer to buy back shares until further notice given. BAT is likely to generate free cash flows despite various expenses like restructuring dividend payments to the shareholders as it has ample internal liquidity, cash flow characteristics access to capital markets. The liquidity of BAT was supported by a) USD 1.75 Billion revolving credit facility for a five year period which along with the cash balance of USD 1.3 billion exceeded gross debt maturing on June 2010; b) Bond maturities; Bond worth USD 2.5 billion were issued to extend the groups debt portfolio c) Finance accessibilities; It has ready access to credit facilities on offer by the financial institutions; d) Significant cash position in excess of USD 1billion. Based on analysis, BATs profitability margins are on par with its global peers. BAT intends to improve its sales by 2 % year on year by price increases and product mixes. Market share reached by BAT in specific markets determined its profitability. Operating margin in developing countries can be compared with the margins achieved in matured markets as shown by the margin achieved in excess of 30% in regions like Latin America, Africa and Middle East. EBIDTA margins for BAT in the year 2009 figured 36%, has had a significant rise of 2.5% percentage point from 33.5% margin it achieved in 2008 and 32.5% in 2007. One widely accepted method of assessing financial statements is ratio analysis which uses data from balance sheet and income statement to produce interpretation which have financial meaning to it. Assessment of the financial health of a business is quick and relatively simple when information is derived using the relevant financial ratios. Ratio Analysis A ratio is a simple mathematical expression of the relationship of one item to another, according to Williams, Haka, Bettner, and Carcello (2005, p. 674). Ratios can provide diverse information to diverse financial information users. The analysis of annual report suggests the following ratio analysis of the group. The relevant ratios have been grouped and presented in this paper under various heads. Profitability Assessment: Operating Margin: BAT achieved an operating margin of 31.12% in the year 2009 mainly due to savings it achieved in supply chain, general overheads and indirect costs. The impact of higher leaf prices and input costs were offset due to these savings. It allowed the overall operating margin to increase from 30.7% to 31.4% in the year which was much greater than the industry and sector average of 22.01% and 10.10% respectively. BAT also, had much better margins when compared to Japan tobacco which could achieve operating margin of 4.76%. Return on Equity: BAT has been a fundamentally sound company demonstrating consistency in giving return on equity to its shareholders. In the FY 2009, BATs Return on Equity was standing at 37.05% which was considerably higher than the industry and sector average of 11.57% and 8.01% respectively. While Japan Tobacco could manage to achieve little less than 10% Return on Equity, BAT has been since the past able to maintain consistent returns. In the past four years starting 2005 2008, BAT gave returns of 26.12%, 27.70%, 28.95% 30.44% respectively. These numbers give confidence to investors and allow BAT to be looked upon as a good company to place ones bet on. Return on Capital Employed (ROCE): Corporate Profitability can be determined by assessing the trading profit that the company has achieved over the capital employed by it to achieve the same. BAT achieved an ROCE of 20.82% in the FY 2009 which is slightly better than the Industry average and sector average. BAT fared approximately twice as better when compared to Japan Tobacco. Asset Utilisation: Asset Turnover: BAT was able to main asset utilisation which was on par with the industry average. The industry average for the FY 2009 was 0.58, while BAT registered an asset utlisation ratio of 0.73. The company is expected to register even better numbers in the near future as FY 2009 saw some acquisitions which resulted in BATs performance on par with the market in terms of making its asset sweat. Closure of the Soeborg factory in Denmark, Downsizing of manufacturing plant in Australia and impairment charges for certain software assets resulted in these assets having minimum and limited future economic benefit. But with consolidation resulting in greater savings and better utlisation of the assets of the companies acquired in the emerging markets, the asset turnover ratio is expected to fare better in the next financial year. Interest Cover: BATs interest cover remains steady at 8.6x in the FY 2009 compared to 8.5x which was reported in the FY 2008. It was offset as a result of the financial arrangements carried out for the acquisitions. Pre-Tax impact on adjusting item distorts the interest cover. Risk Tolerance By analysing the financial statements, BATs policy seems to be moderate as it focuses on maintaining EBIDTA coverage of gross interest payments between 5x and 9x at the same time maintaining cash balance exceeding USD 1 billion and five year maturity on its debt profile. BAT needs to strike a fine balance between acquisitions and share buy backs to avoid over stretching its debt capacity over a short time frame due its commitments to a 65% annual dividend payout ratio. In the year 2008, BAT reduced its annual buy back commitment from USD 750 Million to USD 400 Million to accommodate acquisitions. BAT also suspended its buy back program in 2009 until further notice. Cash Flow Adequacy Growing profitability would benefit the companies debt protection metrics to a greater extent. Strong conversion of profits into cash supports BATs financial metrics. The groups future working capital requirements will remain stable in proportion to its annual sales unless there are any significant large scale acquisitions. BATs capital expenditure of its net operating cash flows is very low when compared with the averages in food, beverage tobacco industries. In the year 2009, BATs capital expenditure accounted for 15%-20% of net operating cash flows. BATs future capital expenditure is most likely to grow giving the compounded annual growth rate of the company along with the industry. Key Performance Indicators The key performance indicators for BAT has been its consistent ability to maintain growth in its core competency. Revenue for the FY 2009 grew by 17% which is 3 4 per annum greater than the target growing revenue for the medium and long term. This was possible due to acquisitions it made and favourable exchange rate movements. One of the key strengths of the company in terms of its performance is its diversified global drive brands which constitute majority of the sales for the company. Though growth of 16% was achieved in the FY 2008 in this segment, FY 2009 volumes grew by 4% which is coherent with the companys strategy to achieved single digit growth over the long run. The adjust profit from operations achieved by the firm was well above the company target to achieve 6% profit from operations. BAT registered a growth of 20%. The net cash from operating activities in the FY 2009 was up by  £26 million to  £ 2630 million. Free cash flow per share increased by 2%, the ratio of free cash flow to adjusted diluted earnings was 86%. Adjusted Earnings Per Share (EPS) had grown at an average of 11% over the last ten years. This exceeds the companys target of growing at a single digit figure per annum on an average. Adjusted diluted EPS grew by 19% in the year 2009. S.W.O.T. Analysis Strengths: Diversified Global Brand Drive (GBD) portfolio One of the keys strengths of the group is its diversified portfolio of cigarette brands. In the FY 2009, overall volume of the GBD grew by 4%. Emerging Economies Currently, 74% of BATs revenues come from the emerging economies. Enhanced Internal Operations BAT saved GBP 239 Million in the FY 2009 by improving its supply chain, overheads and indirect costs. Weaknesses: Legal Issues Recoupment actions and Class actions are filed against the company and its subsidiaries which in turn impacts not only the brand image but also its cost structure. Poor Asset Utlization The companys Return on Assets (ROA) and Return on Equity (ROE) has been poor when compared to its competitors. Philip Morris International and Altria Group recorded an ROA of 29.7% and 20% respectively with BAT recorded an ROA of 15.1% only. Similarly, ROE for BAT was lower than Philip Morris International which recorded 99.2% while BAT could achieve 40.6% only. Opportunities Acquisitions BAT completed an acquisition of a Turkish state owned tobacco company in the year 2008 which elevated its market share from 7% to 36%. In the same year, BAT Bought Skandinavisk Tobakskompagni (ST) which allowed it to increase annual sales of approximately 30,000 million cigarettes. Recently BAT acquired PT Bentoel Internasional Investama, Indonesias fourth largest cigarette maker which had sales of around 250 million cigarettes a year. These acquisitions would increase the global presence of BAT across the globe and in turn enhance its topline and profitability. Growth of Tobacco Industry The tobacco industry is forecasted to witness growth. It is estimated that there would be 1.3 billion smokers in the world by 2020 up from 1.3 billion currently. According to the Datamonitor estimates, the global tobacco market generated total revenues of $429.3 billion in 2009, representing a compound annual growth rate (CAGR) of 3.1% for the period spanning 2005-2009. Cigarette sales generated total revenues of $394.2 billion in the FY 2009, equivalent to 91.8% of the markets overall value. The global tobacco market is forecasted to have a value of $490.2 billion, an increase of 14.2% since 2009. BAT is the second-largest global cigarette player. It tends to benefit from this positive outlook. Threats Illicit Trade Illicit trade is estimated up to 660 million cigarettes a year which represents 12% of world cigarette consumption. This results in losses upto GBP 4 Billion to GBP 7 billion a year. Increase in illicit trade would reduce revenues of the company. Advertising restrictions Brand Building, advertising and promotion are facing hindrances globally. The absence of marketing would effect introduction and promotion of new products in the markets. It could have a negative impact on BATs sales. Consumer focus and awareness on health issues Increasing health consciousness and introduction of substitutes to cigarettes into the market has led to decline of sales for the company. Pharma products and nicotine replacement patches along with chewing gums are the new source of harmless alternatives. What makes British American Tobacco work? The year 2009 was a challenging year for the Fast Moving Consumer Goods (FMCG) segment. Total market volumes declined by 2% for the BAT products. The overall performance for BAT was firm . It continued to invest in its marketing initiatives which resulted in it maintaining its market share in key markets. The Global Drive brands (GDB) grew by combined 4% in terms of volume. These accounted for 27% of the global volume sales for BAT. The overall brand mix for BAT is balanced between premium, mid-price low-price. Managing business to business relationships makes up for a large part of BATs trade marketing activities. BAT co-ordinate its business with its trading partners to ensure that it is able to meet the demands of the customer at the right place and at the right time. This has worked out well for the company as it helps it maintain the market share in a highly competitive tobacco industry. Understanding customer and their needs is of the core non financial activities of BAT. BAT regularly surveys their customer base internationally against its peers in the FMCG industry and particularly against its competitiors in the tobacco industry. Their efforts made them be recognised as the leading business in the tobacco category for customer relationship management by Dow Jones sustainability index for the third successive year in 2009. Apart from its own marketing initiatives, BAT makes efforts to develop marketing programmes jointly with its retail partners, who engage with consumers in market channels like Global Travel Retail and Global Convenience Retail. For BAT, the Direct Store Sales in the most preferred way of selling cigarettes to customers. It fecilitates greater access to consumer information and market. It has also helped them with a direct commericla link to their most strategic retail accounts. In the FY 2009, total sales volumes distributed through DSS reached 50%. Integrated Global Enterprise Apart from the key revenue generators for the company, BAT has been able to achieve growth by savings. BAT was successful in turning a multinational business operations in over 180 markets into an integrated global enterprise which take better advantage of its scale. It led to savings in supply chain, overheads and indirect costs amounting to  £239 million. The company has a target achieve  £800 million savings by 2012.

Mouse movement and keyboard skills Essay

This again applied to visual basic; having used it in my standard grade course I could see the potential. Better than UniComal, I could see I could do more with it, it was more user friendly, but just not what I was looking for. 3. Using macromedia Authorware was my ideal choice. Having browsed through the software I was surprised how user friendly it was. Although I could see it was going to take a lot of time to familiarise myself with it. If time had allowed me I would certainly have tried authorware. It had seemed perfect for my application; I could include sound, a scoreboard, links were automatic and its potential was amazing 4. For time allowance and teaching I have decided to go along with this idea. My suggestion is to carry out how I would like my software to go on a Power Point presentation. This will not only allow me to have a prototype of what I initially would have liked my final piece to look like but will also show links and documentation. In Power Point I will be able to use colour, sound and graphics, out of reach in the likes of UniComal and Visual Basic. This makes it more appealing to young users. As I already have an extensive knowledge of Power Point I will only have to refresh this. However, in reference to 1. , Users will have to have access to a multimedia computer. My suggestions to go along with the fourth option are feasible as there is sufficient technology available; it is economically feasible as everything that I need as already assessable. Software Design Opening Page The opening page to my project will be a general introduction. Basically its purpose is to give the links to the activities and also to the parental page. The main characters for the whole project will first appear here. In an ideal word I would have liked for him to speak as well as the words be on the screen as children may not be able to read them. Opening Page Design 1. Show Opening Screen (Time Hold 10 Seconds) 2. Enter choice 2. 1 IF choice = Mouse Movement Open Mouse Movement IF choice = Keyboard Skills Open Keyboard Skills IF choice = Parental Information Open Parental Information END IF END IF END IF Mouse Movement The mouse movement screen will be very colourful. As it is the first activity a good impression must be made to encourage the user to continue using the software. The first screen will be Daffy Duck describing the game and telling the user what to do. This will be on a time hold for ten seconds to allow the user to read the information before automatically progressing forward. This is also beneficial as the whole point in the exercise is to learn how to use the mouse, therefore I don’t wont the user to have to click to enter the page. The following exercise would therefore be pointless if the user already knew how to do this. The screen will only move on if the correct colour is selected. If the wrong colour is selected, a try again screen will appear and a time hold will apply before moving back to the previous screen. If the correct colour is selected then a well-done screen will appear, a time hold and then a progression on to the next colour. Once all the colours have been completed a closing screen will appear. This will have a small picture of the main character and then will automatically go back to the opening page. Mouse Movement Design 1. Show Mouse Movement Screen (Time Hold 10 Seconds) 2. Show Blue screen 3. Enter choice 3. 1 IF choice = Blue Show Well Done ELSE Show Try Again (Previous page) END IF 4. Show Yellow screen 5. Enter choice 5. 1 IF choice = Yellow Show Well Done ELSE Show Try Again (Previous page) END IF 6. Show Red screen 7. Enter choice 7. 1 IF choice = Red Show Well Done ELSE Show Try Again (Previous page) END IF 8. Show Green screen 9. Enter choice 9. 1 IF choice = Green Show Well Done ELSE Show Try Again (Previous page) END IF 10. Show Pink screen 11. Enter choice 11. 1 IF choice = Pink Show Well Done ELSE Show Try Again (Previous page) END IF 12. Show Closing screen (Time Hold 10 Seconds) 13. Show Opening Page Keyboard Skills Keyboard skills will take the same form as Mouse Movements only obviously it will describe the keyboard skills activity. The same process will take place; the screen will only move on if the correct word is typed. If the wrong word is typed, a try again screen will appear and a time hold will apply before moving back to the previous screen. If the correct word is typed then a well-done screen will appear, a time hold and then a progression on to the next word. Once all the words have been completed a closing screen will appear. This will have a small picture of the main character and then will automatically go back to the opening page. Keyboard Skills Design 1. Show Keyboard Skills Page (Time Hold 10 Seconds) 2. Show Cake Page 3. Enter word 3. 1 IF word = Cake Show Well Done ELSE Show Try Again (Previous page) END IF 4. Show Ball page 5. Enter word 5. 1 IF word = Ball Show Well Done ELSE Show Try Again (Previous page) END IF 6. Show Apple page 7. Enter word 7. 1 IF word = Apple Show Well Done ELSE Show Try Again (Previous page) END IF 8. Show Telephone page 9. Enter word 9. 1 IF word = Telephone Show Well Done ELSE Show Try Again (Previous page) END IF 10. Show Umbrella page 11. Enter word 11. 1 IF word = Umbrella Show Well Done ELSE Show Try Again (Previous page) END IF 12. Show Closing Screen (Time hold 10 seconds) 13. Show Opening Page Parental Page The parental page is a word document, basically the user guide. 1. Show Parental Page (time hold 10 seconds) 2. Open Word document â€Å"Parental Page† 3. IF closed show Opening Page END IF Project Plan The basic data flow diagram represents a logical plan for the project. Screen Demonstrations Mouse Movement opening page Mouse Movement (Yellow) Implementing and Testing Some time was spent at the beginning of the new school year looking into the project, since then, up until now this is it just being started. With only two teaching periods a week at was difficult to get the project off the ground. Initially I had plans to use Macromedia Authorware to carry out my project, but since I have been left with a mere two weeks to complete all the project and write up I have decided that a prototype presentation on Microsoft Power Point would basically â€Å"have to do†. Implementing Implementation of the solution started in mid April. Due to the time deflect shortcuts were taken to enable me to complete the project. Implementation was fairly straight forward as it merely consisted of creating the designed pages on the screen. I had big plans, and am gutted at having to reduce these for my final piece. I have a great imagination and love experimenting to see what looks and works best. I had already drawn out all my page layouts. I then basically had to enter this into the computer. I changed my design slightly and decided to have an opening page for every separate activity. I went through each activity systematically inserting slide after slide. After one activity’s slides were all designed on the computer I had to put all the links into the slides. These take the form of â€Å"Action Settings† and are available through right clicking on the object that you wish the link to go from. Sound was then recorded too using a computer microphone. This again was inserted using the action settings. The characters and pictures that appear on the screen were all uplifted from the Internet, and copy and pasted on to my presentation. All images were found from www. google. com in the image section. Link from the home page to each separate activity is going to be done through separate presentations that will be linked together. This will save confusion over one large single presentation. Once all the pages were entered and suitable images entered, sound recorded and links on; it was time to check if everything was running smoothly. Having checked the run of the presentation a few times I could see that this was indeed going to be the most difficult part of the project. As with everything nothing ever does run perfectly. My links were dysfunctional. When Well Done was displayed, instead of continuing to the next screen my presentation was jumping to Try Again then indeed stopping running at all. Due to the nature of the software, no test data as such was employed. However a dry run-through is needed from different circumstances and answers. EG correct answer chosen, wrong answer chosen, to ensure that I had fixed the presentation from wandering through unnecessary slides. Problems Encountered I have been lucky that I have not hit many major problems and although the presentation is far from finished I can see that it is a matter of following the routine I have been doing, testing as I go along so as not to have progressive mistakes. My main problem as you can tell has been time. Everything has been rather rushed. I had great plans at the beginning and was not prepared to concentrate on one area only but to try hard to do everything. This has been a disaster. For the time it has taken to write the report has left me nearly no time to neither concentrate on the actual presentation nor study for the exam. As I worked with Power Point it is fairly idiot proof any mistakes I made were easy to fix. My links will work with a little time and effort. Testing Once all link testing is complete I will a variety of users to try the software. These will include experienced computer users, older beginner users and also two five year olds (relatives). This will allow me to see how different people react to the software. Paying more attention to the five year olds I shall ask them to undertake the tasks above (1-3), to validate the criteria. There are three main questions I would ask to see if the solution is valid.

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. 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Should Testing Be Tested For Medical Progress - 991 Words

Every year, thousands of animals are forced to undergo multiple experiments and procedures in laboratories for the sake of testing new products or for scientific purposes. Some issues that are in support of testing on lab animals include its contribution to many life-saving cures and treatments, allows to test on a living body system, are appropriate research subjects due to their similarity to human subjects, animals do not have human rights, allows for human products to be tested for safety, religious beliefs that entitle humans to have dominion over all animals, and this method of testing is a small price to pay for advancing medical progress. According to a study done by the California Biomedical Research Association, â€Å"†¦nearly every medical breakthrough in the last 100 years has resulted directly from research using animals. 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