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Health economics

Health economics is a branch of economics concerned with issues related to efficiency, effectiveness, value and behavior in the production and consumption of health and healthcare. Health economics is important in determining how to improve health outcomes and lifestyle patterns through interactions between individuals, healthcare providers and clinical settings. In broad terms, health economists study the functioning of healthcare systems and health-affecting behaviors such as smoking, diabetes, and obesity.

One of the biggest difficulties regarding healthcare economics is that it does not follow normal rules for economics. Price and quality are often hidden by the third-party payer system of insurance companies and employers. Additionally, QALYs (Quality Adjusted Life Years), one of the most commonly used measurements for treatments, is very difficult to measure and relies upon assumptions that are often unreasonable.

A seminal 1963 article by Kenneth Arrow is often credited with giving rise to health economics as a discipline. His theory drew conceptual distinctions between health and other goods. Factors that distinguish health economics from other areas include extensive government intervention, intractable uncertainty in several dimensions, asymmetric information, barriers to entry, externality and the presence of a third-party agent. In healthcare, the third-party agent is the patient's health insurer, who is financially responsible for the healthcare goods and services consumed by the insured patient.

Health economists evaluate multiple types of financial information: costs, charges and expenditures.

Uncertainty is intrinsic to health, both in patient outcomes and financial concerns. The knowledge gap that exists between a physician and a patient creates a situation of distinct advantage for the physician, which is called asymmetric information.

Externalities arise frequently when considering health and health care, notably in the context of the health impacts as with infectious disease or opioid abuse. For example, making an effort to avoid catching the common cold affects people other than the decision maker   or finding sustainable, humane and effective solutions to the opioid epidemic.

Scope
The scope of health economics is neatly encapsulated by Alan Williams' "plumbing diagram" dividing the discipline into eight distinct topics:
 * What influences health? (other than healthcare)
 * What is health and what is its value?
 * The demand for healthcare
 * The supply of healthcare
 * Micro-economic evaluation at treatment level
 * Market equilibrium
 * Evaluation at whole system level
 * Planning, budgeting and monitoring mechanisms.

History
In the third century BC, Aristotle, an ancient Greek thinker, once talked about the relationship between farmers and doctors in production and exchange. In the 17th century, William Petty, a British classical economist, pointed out that the medical and health expenses spent on workers would bring economic benefits.

Presently, contemporary health economics stands as a prominent interdisciplinary field, connecting economic theory with healthcare practice; its diverse sub-disciplines and research domains are evident. The academic roots of this knowledge are commonly traced back to the U.S. tradition.

The American Medical Association (AMA) was created in 1848, having as main goals scientific advancement, creation of standards for medical education, launching a program of medical ethics, and obtaining improved public health. Yet, it was only in 1931 that economic concerns came to the agenda, with the creation of the AMA Bureau of Medical Economics, established to study all economic matters affecting the medical profession.

After the Second World War, amid the rapid improvement of the level of medical research technology, the modernization of diagnosis and treatment means and health facilities and equipment, the aging of the population, the sharp increase of chronic diseases, and the improvement of people's demand for health care and other reasons, medical and health expenses increased significantly. For example, total U.S. health expenditures steadily increased as a share of gross domestic product (GDP), demonstrating the increased importance that society placed on health care relative to other non-health goods and services. Between 1960 and 2013, health spending as a share of GDP increased from 5.0 to 17.4 percent. Over the same period, the average annual growth in nominal national health expenditures was 9.2 percent compared to nominal GDP growth of 6.7 percent.

At the same time, the expenditure on health care in many European countries also increased, accounting for about 4% of GDP in the 1950s and 8% by the end of the 1970s. In terms of growth rate, the proportion of health care expenditure in GNP (gross national product) in many countries increased by 1% in the 1950s, 1.5% in the 1960s, and 2% in the 1970s. This high medical and health expenditure was a heavy economic burden on government, business owners, workers, and families, which required a way to restrain its growth.

In addition, the scale of health service increased, technical equipment became more advanced, and division of labor and specialization saw increases, too. The medical and health service developed into a "healthcare industry" which occupies a considerable amount of capital and labor and occupies an important position in social and economic life. The research on economic problems of the health sector became an important topic of economic research.

Selma Muskin published "Towards the definition of health economics" in 1958 and, four years later, the paper, "Health as an Investment". At that time, health was broadly regarded as rather a consumptive branch of the economy. Muhkin's analysis was the first understanding that health investment had long-term beneficial consequences for the community. Probably, the single most famous and cited contribution to the discipline was Kenneth Arrow's "Uncertainty and the welfare economics of medical care", published in 1963.

After the 1960s, research in health economics developed further, and a second academic seminar on health economics was held in the United States in 1962 followed by a third in 1968. In 1968, the World Health Organization held its first international health economics seminar in Moscow. The convening of the three meetings showed that health economics had boarded an academic forum as an independent discipline, which also marked the official formation of health economics.

After the 1970s, the health economy entered a period of rapid development and nursing economics gradually emerged. In 1979, Paul Feldstein, a famous American health economist, first used the principles of economics to discuss the long-term care market, registered market, and other nursing economy issues, laying the foundation for the emergence of nursing economics.

In 1983, Nursing Economic Magazine was founded in the United States, and its main research content included nursing market development, nursing cost accounting, policies related to nursing services, nursing economic management, etc. The magazine's publication was a mark of the formal formation of nursing economics. In 1993, The University of Iowa Cost Research Center conducted a systematic nursing cost study, simply the NIC System. The specific practice consisted of establishing a special research institution equipped with full-time researchers, sorting out the nursing cost accounting content, and, finally, identifying 433 items in 6 categories. At the same time, the Center adopted computer technology to carry out nursing cost management, including cost assessment, reasonable budget, decision making, etc., which played a crucial role in improving the efficiency of nursing management and alleviating the nursing management crisis.

Healthcare demand
The demand for healthcare is a derived demand from the demand for health. Healthcare is demanded as a means for consumers to achieve a larger stock of "health capital". The demand for health is unlike most other goods because individuals allocate resources in order to both consume and produce health.

The above description gives three roles of persons in health economics. The World Health Report (p. 52) states that people take four roles in healthcare:
 * 1) Contributors
 * 2) Citizens
 * 3) Provider
 * 4) Consumers

Michael Grossman's 1972 model of health production has been extremely influential in this field of study and has several unique elements that make it notable. Grossman's model views each individual as both a producer and a consumer of health. Health is treated as a stock which degrades over time in the absence of "investments" in health, so that health is viewed as a sort of capital. The model acknowledges that health is both a consumption good that yields direct satisfaction and utility, and an investment good, which yields satisfaction to consumers indirectly through fewer sick days. Investment in health is costly as consumers must trade off time and resources devoted to health, such as exercising at a local gym, against other goals. These factors are used to determine the optimal level of health that an individual will demand. The model makes predictions over the effects of changes in prices of healthcare and other goods, labour market outcomes such as employment and wages, and technological changes. These predictions and other predictions from models extending Grossman's 1972 paper form the basis of much of the econometric research conducted by health economists.

In Grossman's model, the optimal level of investment in health occurs where the marginal cost of health capital is equal to the marginal benefit. With the passing of time, health depreciates at some rate $$\delta$$. The interest rate faced by the consumer is denoted by $$r$$. The marginal cost of health capital can be found by adding these variables: $$MC_{HK}=r+\delta\,$$. The marginal benefit of health capital is the rate of return from this capital in both market and non-market sectors. In this model, the optimal health stock can be impacted by factors like age, wages and education. As an example, $$\delta\,$$ increases with age, so it becomes more and more costly to attain the same level of health capital or health stock as one ages. Age also decreases the marginal benefit of health stock. The optimal health stock will therefore decrease as one ages.

Beyond issues of the fundamental, "real" demand for medical care derived from the desire to have good health (and thus influenced by the production function for health) is the important distinction between the "marginal benefit" of medical care (which is always associated with this "real demand" curve based on derived demand), and a separate "effective demand" curve, which summarizes the amount of medical care demanded at particular market prices. Because most medical care is not purchased from providers directly, but is rather obtained at subsidized prices due to insurance, the out-of-pocket prices faced by consumers are typically much lower than the market price. The consumer sets $$MB=MC$$ out of pocket, and so the "effective demand" will have a separate relationship between price and quantity other than the "marginal benefit curve" or real demand relationship will have. This distinction is often described under the rubric of "ex-post moral hazard" (which is again distinct from ex-ante moral hazard, which is found in any type of market with insurance).

Health technology assessment
Economic evaluation, and in particular cost-effectiveness analysis, has become a fundamental part of technology appraisal processes for agencies in a number of countries. The Institute for Quality and Economy in Health Services (Institut für Qualität und Wirtschaftlichkeit im Gesundheitswesen – IQWiG) in Germany and the National Institute for Health and Care Excellence (NICE) in the United Kingdom, for example, both consider the cost-effectiveness of new pharmaceuticals entering the market.

Some agencies, including NICE, recommend the use of cost–utility analysis (CUA). This approach measures outcomes in a composite metric of both length and quality of life, the Quality-adjusted life year (QALY).

Healthcare markets
The five health markets typically analyzed are:
 * Healthcare financing market
 * Physician and nurses services market
 * Institutional services market
 * Input factors markets
 * Professional education market

Although assumptions of textbook models of economic markets apply reasonably well to healthcare markets, there are important deviations. Many states have created risk pools in which relatively healthy enrollees subsidize the care of the rest. Insurers must cope with adverse selection which occurs when they are unable to fully predict the medical expenses of enrollees; adverse selection can destroy the risk pool. Features of insurance market risk pools, such as group purchases, preferential selection ("cherry-picking"), and preexisting condition exclusions are meant to cope with adverse selection.

Insured patients are naturally less concerned about healthcare costs than they would if they paid the full price of care. The resulting moral hazard drives up costs, as shown by the famous RAND Health Insurance Experiment. Insurers use several techniques to limit the costs of moral hazard, including imposing copayments on patients and limiting physician incentives to provide costly care. Insurers often compete by their choice of service offerings, cost-sharing requirements, and limitations on physicians.

Consumers in healthcare markets often lack adequate information about what services they need to buy and which providers offer the best value proposition. Health economists have documented a problem with supplier induced demand, whereby providers base treatment recommendations on economic, rather than medical criteria. Researchers have also documented substantial "practice variations", whereby the treatment also on service availability to rein in inducement and practice variations.

Some economists argue that requiring doctors to have a medical license constrains inputs, inhibits innovation, and increases cost to consumers while largely only benefiting the doctors themselves.

Economic rationale for government intervention in the healthcare markets
Folland, Godman, and Stino the authors of the book, The Economics of Health and Health Care lists several separate and independent reasons for governments intervening in health-care systems rather than leaving it to the private market forces.

1. The first is to ensure that the optimum level of production and consumption of public goods (hospitals, vaccines) and goods with a partially public character are available.

2. Secondly, the rationale is to increase the quality and equity of insurance for those services that can be produced in the private sector but require risk-sharing due to the expense and uncertainties about needs. For example; investing in research and development for new cures and health care equipment. Governments usually subsidize for those who cannot afford insurance or, in certain situations, those low-cost activities and facilities that non-poor citizens can afford on their own. For example; the largest health insurance scheme in the world was launched in India by the name Ayushman Bharat in 2018.

3. The third reason for which the government might want to intervene is to prevent market failure. A classic example of market failure is Monopoly Power. Several health-care markets tend to have the potential for monopoly control to be exercised. Medical care in markets with few hospitals, patent-protected prescription products, and some health insurance markets is the major reason for higher costs and especially in cases where the providers are private companies.

4. Knowledge can be perceived as a public good with a strong economic value. The information provided by one user does not restrict the information available to another. While those who do not pay are often denied access to information and the marginal cost of providing information to another person is frequently low. As a result, one might argue that private markets would under-produce knowledge, necessitating government intervention to increase its availability. Government intervention, in this case, can be seen as assisting in the public distribution of established information, either directly or by subsidizing private sector operations.

5. The last point in this section is related to incomplete markets. Incomplete markets may arise when private markets struggle to satisfy existing demand. This situation can arise when the cure of disease is very expensive, such as cancer or a wide spread of new diseases such as HIV/AIDS or COVID-19. In such cases either private insurers require a high premium as the risk factor and costs are high or they may not insure the people for a particular case. This leads to a void in the market where a certain section of the population will not be able to afford healthcare. Certain insurance markets, such as those for patients with HIV/AIDS, cancer, or other pre-existing conditions who are searching for new coverage, may be incomplete in the sense that those patients may be unable to afford coverage at any price. In such cases, the government usually intervenes and provides health care for such cases. For example, during the COVID-19 pandemic, no private insurance company predicted (or could have predicted) that such an outbreak would occur; as a result, state intervention became necessary to treat people.

Public and private spending
Expand the OECD charts below to see the breakdown:
 * "Government/compulsory": Government spending and compulsory health insurance.
 * "Voluntary": Voluntary health insurance and private funds such as households' out-of-pocket payments, NGOs and private corporations.
 * They are represented by columns starting at zero. They are not stacked. The 2 are combined to get the total.
 * At the source you can run your cursor over the columns to get the year and the total for that country.
 * Click the table tab at the source to get 3 lists (one after another) of amounts by country: "Total", "Government/compulsory", and "Voluntary".

Medical economics
Often used synonymously with health economics, medical economics, according to Culyer, is the branch of economics concerned with the application of economic theory to phenomena and problems associated typically with the second and third health market outlined above: physician and institutional service providers. Typically, however, it pertains to cost–benefit analysis of pharmaceutical products and cost-effectiveness of various medical treatments. Medical economics often uses mathematical models to synthesise data from biostatistics and epidemiology for support of medical decision-making, both for individuals and for wider health policy.

Mental health economics
Mental health economics incorporates a vast array of subject matters, ranging from pharmacoeconomics to labor economics and welfare economics. Mental health can be directly related to economics by the potential of affected individuals to contribute as human capital. In 2009 Currie and Stabile published "Mental Health in Childhood and Human Capital" in which they assessed how common childhood mental health problems may alter the human capital accumulation of affected children. Externalities may include the influence that affected individuals have on surrounding human capital, such as at the workplace or in the home. In turn, the economy also affects the individual, particularly in light of globalization. For example, studies in India, where there is an increasingly high occurrence of western outsourcing, have demonstrated a growing hybrid identity in young professionals who face very different sociocultural expectations at the workplace and in at home.

Mental health economics presents a unique set of challenges to researchers. Individuals with cognitive disabilities may not be able to communicate preferences. These factors represent challenges in terms of placing value on the mental health status of an individual, especially in relation to the individual's potential as human capital. Further, employment statistics are often used in mental health economic studies as a means of evaluating individual productivity; however, these statistics do not capture "presenteeism", when an individual is at work with a lowered productivity level, quantify the loss of non-paid working time, or capture externalities such as having an affected family member. Also, considering the variation in global wage rates or in societal values, statistics used may be contextually, geographically confined, and study results may not be internationally applicable.

Though studies have demonstrated mental healthcare to reduce overall healthcare costs, demonstrate efficacy, and reduce employee absenteeism while improving employee functioning, the availability of comprehensive mental health services is in decline. Petrasek and Rapin (2002) cite the three main reasons for this decline as (1) stigma and privacy concerns, (2) the difficulty of quantifying medical savings and (3) physician incentive to medicate without specialist referral. Evers et al. (2009) have suggested that improvements could be made by promoting more active dissemination of mental health economic analysis, building partnerships through policy-makers and researchers, and employing greater use of knowledge brokers.

Health and utility
Generally, economists assume that individuals act rationally with the aim of maximizing their lifetime utility, while all are subject to the fact that they cannot buy more than their resources allow. However, this model gets complex as there exists the uncertainty over individuals' lifetime. As such, the issue is split into two parts: 1. How does health produce utility and 2. What affects health (e.g., medical care and life-style choices).

Probably the most fundamental thing in consumer demand theory is that the good increases an individual's utility. Health is not really a good in the traditional sense, but health in itself produces happiness. We can think of health as a durable good, much like for instance a car, a house or an education. Every person comes into the world with some inherent "stock" of health, and a healthy baby has a fairly high stock of health. Basically, every decision an individual take during their lifetime will affect their stock of health.

Let X be a bundle of other goods, and H a stock of health. With these variables the formula for an individual's utility is: Utility = U(X, H). For simplicity, the stock of health produces utility, but technically, is the flow of services created by the stock of health that produces utility. As the traditional fashion for goods, "more is better", in other words an increase in health leads to an increase in utility. From this assumption, X grows with health, for instance it is more enjoyable to visit the zoo when not experiencing a headache.

Like other durable goods, the stock of health wears out over time, much like other durable goods. This process can be called aging. When the stock of health has dropped low enough, a person will lose their ability to function. In economic terminology it can be said that the stock of health depreciates. Since life expectancy has risen a lot during this century, it implies that e.g., the depreciation rate has decreased during this time. Public healthcare efforts and individual medical care are in place to restore the stock of health or to decrease the depreciation rate on the stock of health. A plot graph of an individual's stock of health throughout their lifetime would be steadily increasing in the beginning during their childhood, and after that gradually decline because of aging, meanwhile having sudden drops created by random events, such as injury or illness.

There are many other things than "random" health care events, which individuals consume or do during their lives that affect the speed of aging and the severity and frequency of the drops. Lifestyle choices can deeply improve or worsen health of an individual. The variable X, the bundle of goods and services, can undertake numerous characteristics, some add value while others noticeably decrease the stock of health. Outstanding among such lifestyle choices are the decision to consume alcohol, smoke tobacco, use drugs, composition of diet, amount of exercise and so on. Not only can X and H work as substitutes for one another in producing utility, but X can also affect H in a production sense as well. X can then be split into different categories depending on which effect it has on H, for instance "good" types (e.g., moderate exercise), "bad" types (e.g., food with high cholesterol) and "neutral" types (e.g., concerts and books). Neutral goods do not have an apparent effect on individuals' health.

Journals

 * Journal of Health Economics
 * International Journal of Technology Assessment in Health Care
 * Health Economics
 * The European Journal of Health Economics
 * PharmacoEconomics (journal)
 * Applied Health Economics and Health Policy
 * Journal of Medical Economics
 * Value in Health
 * Cost Effectiveness and Resource Allocation
 * Health Economics, Policy and Law
 * Expert Review of Pharmacoeconomics and Outcomes Research
 * Medical Decision Making (journal)
 * Social Science and Medicine

title: Health economics


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Medical decision-making

Medical decision-making may refer to:

 * Decision-making, a cognitive process for selecting a course of action, in the context of health or medical diagnosis and treatment
 * Medical Decision Making (journal), a peer-reviewed academic journal that publishes papers on the study of medical decisions

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Medical decision-making


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User talk:Crown

Try Sandbox for experiments. --Menchi 12:15 12 Jul 2003 (UTC)

 * I even installed wikiwiki on my homepage. I think, I'm not goot at making wikipedia, but i do my best - Crown 21:03 15 Jul 2003 (UTC)

 * You haven't copyedited any Wikipedia page yet. :-) Start with the things you've learned about (and enjoyed). --Menchi 23:14 15 Jul 2003 (UTC)

title: User talk:Crown


text:
User talk:The Anomebot

The category project
I am adding category links to pages, as part of an experiment for supporting a soon-to-be-relased feature of the Wikipedia software. The category links are invisible, so will not be seen in articles, but are I hope self-explanatory.

Categories assigned so far (no particular pattern to this):
 * metrology (just the page "UTC" for now)
 * time period
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more to come

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Please note: this is an unofficial personal project. Please direct any complaints to User talk:The Anome.

Note: Category edits are currently stopped.

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I notice that you added a "Physics" category tag to How Archimedes used infinitesimals, and then removed it. I don't think that article is primarily about physics at all. It applies some ideas of physics to geometry; the latter subject is the goal. Michael Hardy 18:56, 7 Sep 2003 (UTC)

 * I noticed recently a large number of category tag removals; what is the rationale for this? Waveguy

They were causing annoyance to other users who had certain skin options enabled. So I took them out. When the software has been adjusted to stop junk appeating in the article body, I'll put them back in. -- The Anome 21:59, 12 Sep 2003 (UTC)

Article Licensing
Hi, I've started a drive to get users to multi-license all of their contributions that they've made to either (1) all U.S. state, county, and city articles or (2) all articles, using the Creative Commons Attribution-Share Alike (CC-by-sa) v1.0 and v2.0 Licenses or into the public domain if they prefer. The CC-by-sa license is a true free documentation license that is similar to Wikipedia's license, the GFDL, but it allows other projects, such as WikiTravel, to use our articles. Since you are among the top 2000 Wikipedians by edits, I was wondering if you would be willing to multi-license all of your contributions or at minimum those on the geographic articles. Over 90% of people asked have agreed. For More Information:
 * Multi-Licensing FAQ - Lots of questions answered
 * Multi-Licensing Guide
 * Free the Rambot Articles Project

To allow us to track those users who muli-license their contributions, many users copy and paste the " " template into their user page, but there are other options at Template messages/User namespace. The following examples could also copied and pasted into your user page:

 * Option 1
 * I agree to multi-license all my contributions, with the exception of my user pages, as described below:

OR
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Or if you wanted to place your work into the public domain, you could replace " " with "  ". If you only prefer using the GFDL, I would like to know that too. Please let me know what you think at my talk page. It's important to know either way so no one keeps asking. -- Ram-Man (comment| talk)

Source code for this bot?
I want to write and run a bot to perform simple tasks. I was gonna start with adding user pages to a user category. I figure if i can do that i can figure out how to do other stuff too. I've been looking around some bot pages, but not having any formal or extensive knowledge of programming, networks or the internet, I'm still not sure what to do. I can't seem to find source code for simple bots to modify, and even if i wrote a bot, I wouldn't be sure what to do with it. Do i have to have it installed on some kind of server, or can i just activate it on my computer at home when i want to run it?

Is there any chance of seeing the code for this bot as it was used for adding pages to categories? Any other help would be useful too. I've had a look around Bots, but it doesn't seem to give any help on how to run a bot. Cheers. Steveire 22:44, 10 November 2005 (UTC)

 * Wnen and then you could just go to your between the rest of the intended purposes for the future and you too will be able for the best Create and then 24.240.181.58 (talk) 18:32, 5 May 2023 (UTC)

This is a automated to all bot operators
Please take a few moments and fill in the data for your bot on Bots/Status Thank you Betacommand (talk • contribs • Bot) 19:50, 12 February 2007 (UTC)

Automated message to bot owners
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Please reconfigure the bot
Anomebot has been placing "coord missing" on a LOT of non-locational articles which implicity cannot have and will not have coordinates, e.g. North American Water and Power Alliance (which is a continent-wide scheme with no one specfic location and up to 150 actual ones proposed). I remove as many of these as I come across, but it's really a waste of my time; can't the bot be targeted only at articles with location-stubs (geo-stubs?).Skookum1 (talk) 15:43, 28 February 2009 (UTC)

title: User talk:The Anomebot


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Duke of Argyll

Duke of Argyll (Diùc Earraghàidheil) is a title created in the peerage of Scotland in 1701 and in the peerage of the United Kingdom in 1892. The earls, marquesses, and dukes of Argyll were for several centuries among the most powerful noble families in Scotland. As such, they played a major role in Scottish history throughout the 16th, 17th, and 18th centuries. The Duke of Argyll also holds the hereditary titles of chief of Clan Campbell and Master of the Household of Scotland.

Since 2001, Torquhil Campbell has been Duke of Argyll and is the thirteenth man to hold the title.

History
Sir Colin Campbell of Lochow was knighted in 1280. In 1445 James II of Scotland raised Sir Colin's descendant Sir Duncan Campbell to the peerage to become Duncan Campbell of Lochow, Lord of Argyll, Knight, 1st Lord Campbell. Colin Campbell (c. 1433–1493) succeeded his grandfather as the 2nd Lord Campbell in 1453 and was created Earl of Argyll in 1457.

The 8th Earl of Argyll was created a marquess in 1641, when Charles I visited Scotland and attempted to quell the rising political crisis (and the fall-out from the event known as The Incident). With Oliver Cromwell's victory in England, the marquess became the effective ruler of Scotland. Upon the restoration, the marquess offered his services to King Charles II but was charged with treason and executed in 1661. His lands and titles were forfeited but in 1663, they were restored to his son, Archibald, who became the 9th Earl of Argyll. In 1685 the 9th Earl was executed for his part in the Monmouth rebellion.

On 21 June 1701 the 9th Earl's son was created Duke of Argyll, Marquess of Kintyre and Lorne, Earl of Campbell and Cowal, Viscount of Lochow and Glenyla, Lord Inveraray, Mull, Morvern, and Tiree for his services to William of Orange. His son, the 2nd Duke, was created Baron Chatham and Earl of Greenwich in 1705 as a reward for his support for the Act of Union and further elevated to the title Duke of Greenwich in 1719. Upon his death his Scottish titles passed to his brother and the English titles became extinct.

The 5th Duke sat as a member of parliament for Glasgow Burghs until his father's accession to the Dukedom in 1761 disqualified him from representing a Scottish seat. He then became the member for Dover until 1766, when he was created Baron Sundridge and obtained the right to sit in the House of Lords.

On 17 April 1892, the 8th Duke was created Duke of Argyll in the Peerage of the United Kingdom. Thus, the Duke is one of only five people to hold two or more different dukedoms, the others being the Duke of Cornwall and Rothesay, the Duke of Buccleuch and Queensberry, the Duke of Hamilton and Brandon, and the Duke of Richmond, Lennox, and Gordon.

During the 19th century, a distant Prussian descendant of the family, Jenny von Westphalen, became the wife of the philosopher Karl Marx. In a famous story, when exiled to Paris and reduced to poverty, Marx was nearly arrested for attempting to pawn a part of Jenny's dowry: a silver dinner service bearing the coat of arms of the House of Argyll. Of the incident Marx wrote to Engels, possibly in an attempt to solicit another loan from his wealthy friend: "My wife cried all night". However, the silver was eventually sold to pay off long-standing debts incurred by the Neue Rheinische Zeitung.

In the late 19th century the then current Duke of Argyll visited America. While there, he stayed at the American Hotel situated in the main square of the village of Babylon, New York. The townspeople took a liking to the duke, and festivals and parades took place while he visited there. Just before the turn of the century (1900) the township of Babylon renamed the Bythbourne Lake/Park to Argyle Lake/Park (Argyll evolved to the currently accepted Argyle) in memory of the duke's visit.

Family seats and abodes
The family seat is Inveraray Castle beside Loch Fyne, Inveraray, Argyll. The estate, 75,000 acres, is a mixture of commercial forestry, residential property, sources of renewable energy, and a caravan park.

The principal burial place of the Dukes and Duchesses is St Munn's Parish Church, Kilmun. The 11th and the 12th Dukes chose to be buried on the island of Inishail in Loch Awe.

In 1706 John Campbell, second Duke of Argyll, became the inhabitant of a house on the east side of King Street, St James (Soho end), Westminster, London which stood on a site occupied by the western end of Little Argyll Street which in 1735 or 1736 he vacated for redevelopment. A succession of Argyll Houses followed in the same block of streets "A water-colour drawing of Argyll House by T. H. Shepherd and two plans (at Inveraray Castle, engraved and undated) [depicted in this source] all suggest a house of little beauty and less convenience. Harriette Wilson, the sixth Duke's mistress, called it a 'dismal chateau' and described it, along with the Duke's meagre personal possessions, as 'old'." In 1808 the 6th Duke sold the latter-day House to the 4th Earl of Aberdeen.

In and before 1764 the family had a house near to London at Ham which was then in the county of Surrey, a parish historically associated with Richmond, and a nearby second holding, see map above left.

Subsidiary titles
The Duke holds several subsidiary titles, including: Marquess of Kintyre and Lorne (created 1701), Earl of Argyll (created 1457), Earl Campbell and Cowall and Viscount Lochow and Glenyla (created 1701), Lord Campbell (created 1445), Lord Lorne (created 1470), Lord Kintyre (created 1626), Lord Inveraray, Mull, Mover and Tiry (created 1701), Baron Hamilton of Hameldon (created 1776) and Baron Sundridge (created 1766). They are in the Peerage of Scotland, except the last two, which are in the Peerage of Great Britain. The Duke is also a Baronet of Lundie (created 1627) in the Baronetage of Nova Scotia. The courtesy title for the Duke's eldest son and heir is Marquess of Lorne, shortened from Marquess of Kintyre and Lorne.

Hereditary offices

 * Hereditary Master of the Royal Household of Scotland
 * Hereditary High Justiciar of Argyll
 * Admiral of the Western Coasts and Isles
 * Hereditary Keeper of the Royal Castles of: Carrick, Dunoon, Dunstaffnage, Tarbert
 * Hereditary High Sheriff of Argyllshire
 * Member King's Body Guard for Scotland

The Duke of Argyll is also the chief of the Scottish clan of Campbell and in this capacity is known as MacCailein Mòr, which is Gaelic for "The Great MacColin" referring to Cailean Mór (Colin the Great) of Lochawe (Colin of Lochow) who was killed in fighting with Alexander, Lord of Lorne in 1296.

Since James IV's reign, the Duke has also held the position of Master of the Household of Scotland.

Coat of arms
The heraldic blazon for the coat of arms of the dukedom is: Quarterly: 1st and 4th gyronny of eight or and sable (for Campbell); 2nd and 3rd argent, a lymphad, sails furled and oars in action sable, flags and pennants flying gules (for Lorne).

Lords Campbell (1445)

 * Duncan Campbell, 1st Lord Campbell (d. 1453)
 * Archibald Campbell, Master of Campbell (d. c. 1431–1440) (eldest son of the 1st Lord, died before his father was created a Lord of Parliament)
 * Colin Campbell, 2nd Lord Campbell (c. 1433–1493) (created Earl of Argyll in 1457)

Earls of Argyll (1457)

 * Colin Campbell, 1st Earl of Argyll (c. 1433–1493) (only son of the Master)
 * Archibald Campbell, 2nd Earl of Argyll (d. 1513) (eldest son of the 1st Earl)
 * Colin Campbell, 3rd Earl of Argyll (c. 1486–1529) (eldest son of the 2nd Earl)
 * Archibald Campbell, 4th Earl of Argyll (c. 1507–1558) (only son of the 3rd Earl)
 * Archibald Campbell, 5th Earl of Argyll (c. 1537–1573) (elder son of the 4th Earl, died without issue)
 * Colin Campbell, 6th Earl of Argyll (c. 1541/1546–1584) (younger son of the 4th Earl)
 * Archibald Campbell, 7th Earl of Argyll (c. 1576–1638) (elder son of the 6th Earl)
 * Archibald Campbell, 8th Earl of Argyll (1607–1661) (created Marquess of Argyll in 1641)

Marquesses of Argyll (1641)

 * Archibald Campbell, 1st Marquess of Argyll (1607–1661) (elder son of the 7th Earl, was tried for high treason, attainted and all his honours forfeit in 1661)

Earls of Argyll (1457; restored 1663)

 * Archibald Campbell, 9th Earl of Argyll (c. 1629–1685) (elder son of the 1st Marquess, was restored in 1663 to his father's honours, excepting his Marquessate. He was later tried for high treason and all his honours forfeit in 1681)
 * Archibald Campbell, 10th Earl of Argyll (1658–1703) (Elder son of the 9th Earl, restored to his father's honours in 1685, created Duke of Argyll in 1701)

Dukes of Argyll (1701)

 * Archibald Campbell, 1st Duke of Argyll (1658–1703) (eldest son of the 9th Earl, was restored to his father's honours in 1685)
 * John Campbell, 2nd Duke of Argyll, 1st Duke of Greenwich (1680–1743) (eldest son of the 1st Duke, died without male issue)
 * Archibald Campbell, 3rd Duke of Argyll (1682–1761) (second and youngest son of the 1st Duke, died without legitimate issue)
 * John Campbell, 4th Duke of Argyll (1693–1770) (eldest son of John Campbell, second son of the 9th Earl and younger brother of the 1st Duke)
 * John Campbell, 5th Duke of Argyll (1723–1806) (elder/est son of the 4th Duke)
 * George John Campbell, Earl of Campbell (1763–1764) (eldest son of the 5th Duke, died in infancy)
 * George William Campbell, 6th Duke of Argyll (1768–1839) (second son of the 5th Duke, died without issue)
 * John Douglas Edward Henry Campbell, 7th Duke of Argyll (1777–1847) (third and youngest son of the 5th Duke)
 * John Henry Campbell (1821–1837) (eldest son of the 7th Duke, died young)
 * George John Douglas Campbell, 8th Duke of Argyll (1823–1900) (second and younger son of the 7th Duke, created Duke of Argyll in the Peerage of the United Kingdom in 1892)
 * John George Edward Henry Douglas Sutherland Campbell, 9th Duke of Argyll (S) and 2nd Duke of Argyll (UK) (1845–1914) (eldest son of the 8th Duke, married The Princess Louise, daughter of Queen Victoria, but died without issue)
 * Niall Diarmid Campbell, 10th Duke of Argyll (S) and 3rd Duke of Argyll (UK) (1872–1949) (only son of Lord Archibald Campbell, second son of the 8th Duke, died unmarried)
 * Ian Douglas Campbell, 11th Duke of Argyll (S) and 4th Duke of Argyll (UK) (1903–1973) (grandson of Lord Walter Campbell, third son of the 8th Duke)
 * Ian Campbell, 12th Duke of Argyll (S) and 5th Duke of Argyll (UK) (1937–2001) (eldest son of the 11th Duke)
 * Torquhil Ian Campbell, 13th Duke of Argyll (S) and 6th Duke of Argyll (UK) (b. 1968) (only son of the 12th Duke)

The heir apparent is the present holder's eldest son, Archibald Frederick Campbell, Marquess of Lorne (b. 2004).

Lords Kintyre (1626)
For further succession see above
 * James Campbell, 1st Earl of Irvine, 1st Lord Kintyre (1626–1645)
 * Archibald Campbell, 1st Marquess of Argyll, 2nd Lord Kintyre (1607–1661)

Campbell baronets, of Lundie (1627)
For further succession see above
 * Colin Campbell, 1st Baronet (b. 1599) (only son of the younger son of the 6th Earl)
 * Colin Campbell, 2nd Baronet (d. 1696) (only son of the 1st Baronet, died without issue)
 * Archibald Campbell, 1st Duke of Argyll (1658–1703)

Current line of succession

 * Coronet of a British Duke.svg Ian Campbell, 11th Duke of Argyll (1903-1973)
 * Coronet of a British Duke.svg Ian Campbell, 12th Duke of Argyll (1937-2001)
 * Coronet of a British Duke.svg Torquhil Campbell, 13th Duke of Argyll (b. 1968)
 * (1) Archibald Friedrich Campbell, Marquess of Lorne (b. 2004), known as Archie Lorne
 * (2) Lord Rory James Campbell (b. 2006)
 * (3) Lord Colin Ivar Campbell (b. 1946)

Literary references
The 2nd duke features prominently in novel The Heart of Midlothian by Sir Walter Scott

title: Duke of Argyll


text:
Epidemiological method

The science of epidemiology has matured significantly from the times of Hippocrates, Semmelweis and John Snow. The techniques for gathering and analyzing epidemiological data vary depending on the type of disease being monitored but each study will have overarching similarities.

Outline of the process of an epidemiological study

 * 1) Establish that a problem exists
 * 2) * Full epidemiological studies are expensive and laborious undertakings. Before any study is started, a case must be made for the importance of the research.
 * 3) Confirm the homogeneity of the events
 * 4) * Any conclusions drawn from inhomogeneous cases will be suspicious. All events or occurrences of the disease must be true cases of the disease.
 * 5) Collect all the events
 * 6) * It is important to collect as much information as possible about each event in order to inspect a large number of possible risk factors. The events may be collected from varied methods of epidemiological study or from censuses or hospital records.
 * 7) * The events can be characterized by Incidence rates and prevalence rates.
 * 8) * Often, occurrence of a single disease entity is set as an event.
 * 9) * Given inherent heterogeneous nature of any given disease (i.e., the unique disease principle ), a single disease entity may be treated as disease subtypes. This framework is well conceptualized in the interdisciplinary field of molecular pathological epidemiology (MPE).
 * 10) Characterize the events as to epidemiological factors
 * 11) Predisposing factors
 * 12) * Non-environmental factors that increase the likelihood of getting a disease. Genetic history, age, and gender are examples.
 * 13) Enabling/disabling factors
 * 14) * Factors relating to the environment that either increase or decrease the likelihood of disease. Exercise and good diet are examples of disabling factors. A weakened immune system and poor nutrition are examples of enabling factors.
 * 15) Precipitation factors
 * 16) * This factor is the most important in that it identifies the source of exposure. It may be a germ, toxin or gene.
 * 17) Reinforcing factors
 * 18) * These are factors that compound the likelihood of getting a disease. They may include repeated exposure or excessive environmental stresses.
 * 19) Look for patterns and trends
 * 20) * Here one looks for similarities in the cases which may identify major risk factors for contracting the disease. Epidemic curves may be used to identify such risk factors.
 * 21) Formulate a hypothesis
 * 22) * If a trend has been observed in the cases, the researcher may postulate as to the nature of the relationship between the potential disease-causing agent and the disease.
 * 23) Test the hypothesis
 * 24) * Because epidemiological studies can rarely be conducted in a laboratory the results are often polluted by uncontrollable variations in the cases. This often makes the results difficult to interpret. Two methods have evolved to assess the strength of the relationship between the disease causing agent and the disease.
 * 25) * Koch's postulates were the first criteria developed for epidemiological relationships. Because they only work well for highly contagious bacteria and toxins, this method is largely out of favor.
 * 26) * Bradford-Hill Criteria are the current standards for epidemiological relationships. A relationship may fill all, some, or none of the criteria and still be true.
 * 27) Publish the results.

Measures
Epidemiologists are famous for their use of rates. Each measure serves to characterize the disease giving valuable information about contagiousness, incubation period, duration, and mortality of the disease.

Measures of occurrence

 * 1) Incidence measures
 * 2) Incidence rate, where cases included are defined using a case definition
 * 3) Hazard rate
 * 4) Cumulative incidence
 * 5) Prevalence measures
 * 6) Point prevalence
 * 7) Period prevalence

Measures of association

 * 1) Relative measures
 * 2) Risk ratio
 * 3) Rate ratio
 * 4) Odds ratio
 * 5) Hazard ratio
 * 6) Absolute measures
 * 7) Absolute risk reduction
 * 8) Attributable risk
 * 9) Attributable risk in exposed
 * 10) Percent attributable risk
 * 11) Levin's attributable risk

Other measures

 * 1) Virulence and Infectivity
 * 2) Mortality rate and Morbidity rate
 * 3) Case fatality
 * 4) Sensitivity (tests) and Specificity (tests)

Limitations
Epidemiological (and other observational) studies typically highlight associations between exposures and outcomes, rather than causation. While some consider this a limitation of observational research, epidemiological models of causation (e.g. Bradford Hill criteria) contend that an entire body of evidence is needed before determining if an association is truly causal. Moreover, many research questions are impossible to study in experimental settings, due to concerns around ethics and study validity. For example, the link between cigarette smoke and lung cancer was uncovered largely through observational research; however research ethics would certainly prohibit conducting a randomized trial of cigarette smoking once it had already been identified as a potential health threat.

title: Epidemiological method


text:
The Art of Fugue

The Art of Fugue, or The Art of the Fugue (Die Kunst der Fuge), BWV 1080, is an incomplete musical work of unspecified instrumentation by Johann Sebastian Bach. Written in the last decade of his life, The Art of Fugue is the culmination of Bach's experimentation with monothematic instrumental works.

This work consists of fourteen fugues and four canons in D minor, each using some variation of a single principal subject, and generally ordered to increase in complexity. "The governing idea of the work", as put by Bach specialist Christoph Wolff, "was an exploration in depth of the contrapuntal possibilities inherent in a single musical subject." The word "contrapunctus" is often used for each fugue.

Mus. ms. autogr. P 200

The earliest extant source of the work is an autograph manuscript possibly written from 1740 to 1746, usually referred to by its call number as Mus. ms. autogr. P 200 in the Berlin State Library. Bearing the title Die / Kunst der Fuga  [sic] / di Sig[nore]'' Joh. Seb. Bach, which was written by Bach's son-in-law Johann Christoph Altnickol, followed by (in eigenhändiger Partitur) written by Georg Poelchau, the autograph contains twelve untitled fugues and two canons arranged in a different order than in the first printed edition, with the absence of Contrapunctus 4, Fuga a 2 clav (two-keyboard version of Contrapunctus 13), Canon alla decima, and Canon alla duodecima''.

The autograph manuscript presents the then-untitled Contrapuncti and canons in the following order: [Contrapunctus 1], [Contrapunctus 3], [Contrapunctus 2], [Contrapunctus 5], [Contrapunctus 9], an early version of [Contrapunctus 10], [Contrapunctus 6], [Contrapunctus 7], Canon in Hypodiapason with its two-stave solution Resolutio Canonis (entitled Canon alla Ottava in the first printed edition), [Contrapunctus 8], [Contrapunctus 11], Canon in Hypodiatesseron, al roversio  [sic] e per augmentationem, perpetuus presented in two staves and then on one, [Contrapunctus 12] with the inversus form of the fugue written directly below the rectus form, [Contrapunctus 13] with the same rectus–inversus format, and a two-stave Canon al roverscio et per augmentationem—a second version of Canon in Hypodiatesseron.

Mus. ms. autogr. P 200, Beilage
Bundled with the primary autograph are three supplementary manuscripts, each affixed to a composition that would appear in the first printed edition. Referred to as Mus. ms. autogr. P 200/Beilage 1, Mus. ms. autogr. P 200/Beilage 2, and Mus. ms. autogr. P 200/Beilage 3, they are written under the title  Die Kunst / der Fuga / von J.S.B.

Mus. ms. autogr. P 200, Beilage 1 contains a final preparatory revision of the Canon in Hypodiatesseron, under the title Canon p[er] Augmentationem contrario Motu crossed out. The manuscript contains line break and page break information for the engraving process, most of which was transcribed in the first printed edition. Written on the top region of the manuscript is a note written by Johann Christoph Friedrich Bach:

"N.B. Der seel. Papa hat auf die Platte diesen Titul stechen lassen, Canon per Augment: in Contrapuncto all octava, er hat es aber wieder ausgestrichen auf der Probe Platte und gesetzet wie forn stehet

N.B. The late father had written on the copper plate the following title, Canon per Augment: in Contrapuncto all octava, but had struck it out again on the proof sheet and restored the title as it was formerly"

Mus. ms. autogr. P 200, Beilage 2 contains two-keyboard arrangements of Contrapunctus 13 inversus and rectus, entitled ''Fuga a 2. Clav: and Alio modo Fuga a 2 Clav.'' in the first printed edition respectively. Like Beilage 1, the manuscript served as a preparatory edition for the first printed edition.

Mus. ms. autogr. P 200, Beilage 3 contains a fragment of a three-subject fugue, which would be later called Fuga a 3 Soggetti in the first printed edition. Unlike the fugues written in the primary autograph, the Fuga is presented in a two-stave keyboard system, instead of with individual staves for each voice. The fugue abruptly breaks off on the fifth page, specifically on the 239th measure and ends with the note written by Carl Philipp Emanuel Bach: "Ueber dieser Fuge, wo der Nahme BACH im Contrasubject angebracht worden, ist der Verfasser gestorben." ("At the point where the composer introduces the name BACH [for which the English notation would be B♭–A–C–B♮] in the countersubject to this fugue, the composer died.") The following page contains a list of errata by Carl Philipp Emanuel Bach for the first printed edition (pages 21–35).

First and second printed editions
The first printed version was published under the title Die / Kunst der Fuge / durch / Herrn Johann Sebastian Bach / ehemahligen Capellmeister und Musikdirector zu Leipzig in May 1751, slightly less than a year after Bach's death. In addition to changes in the order, notation, and material of pieces which appeared in the autograph, it contained two new fugues, two new canons, and three pieces of ostensibly spurious inclusion. A second edition was published in 1752, but differed only in its addition of a preface by Friedrich Wilhelm Marpurg.

In spite of its revisions, the printed edition of 1751 contained a number of glaring editorial errors. The majority of these may be attributed to Bach's relatively sudden death in the midst of publication. Three pieces were included that do not appear to have been part of Bach's intended order: an unrevised (and thus redundant) version of the second double fugue, Contrapunctus X; a two-keyboard arrangement of the first mirror fugue, Contrapunctus XIII; and an organ chorale prelude on "Vor deinen Thron tret ich hiermit" ("Herewith I come before Thy Throne"), derived from BWV 668a, and noted in the introduction to the edition as a recompense for the work's incompleteness, having purportedly been dictated by Bach on his deathbed.

The anomalous character of the published order and the Unfinished Fugue, have engendered a wide variety of theories which attempt to restore the work to the state originally intended by Bach.

Structure
The Art of Fugue is based on a single subject, which each canon and fugue employs in some variation:

The work divides into seven groups, according to each piece's prevailing contrapuntal device; in both editions, these groups and their respective components are generally ordered to increase in complexity. In the order in which they occur in the printed edition of 1751 (without the aforementioned works of spurious inclusion), the groups, and their components are as follows.

Simple fugues:

 * Contrapunctus 1: four-voice fugue on principal subject
 * Contrapunctus 2: four-voice fugue on principal subject, accompanied by a 'French' style dotted rhythm
 * Contrapunctus 3: four-voice fugue on principal subject in inversion, employing intense chromaticism
 * Contrapunctus 4: four-voice fugue on principal subject in inversion, employing counter-subjects

Stretto-fugues (counter-fugues), in which the subject is used simultaneously in regular, inverted, augmented, and diminished forms:

 * Contrapunctus 5: has many stretto entries, as do Contrapuncti 6 and 7
 * Contrapunctus 6, a 4 in Stylo Francese: adds both forms of the theme in diminution, (halving note lengths), with little rising and descending clusters of semiquavers in one voice answered or punctuated by similar groups in demisemiquavers in another, against sustained notes in the accompanying voices. The dotted rhythm, enhanced by these little rising and descending groups, suggests what is called "French style" in Bach's day, hence the name Stylo Francese.
 * Contrapunctus 7, a 4 per Augment[ationem] et Diminut[ionem]: uses augmented (doubling all note lengths) and diminished versions of the main subject and its inversion.

Double and triple fugues, employing two and three subjects respectively:

 * Contrapunctus 8, a 3: triple fugue with three subjects, having independent expositions
 * Contrapunctus 9, a 4, alla Duodecima: double fugue, with two subjects occurring dependently and in invertible counterpoint at the twelfth
 * Contrapunctus 10, a 4, alla Decima: double fugue, with two subjects occurring dependently and in invertible counterpoint at the tenth
 * Contrapunctus 11, a 4: triple fugue, employing the three subjects of Contrapunctus 8 in inversion

Mirror fugues, in which a piece is notated once and then with voices and counterpoint completely inverted, without violating contrapuntal rules or musicality:

 * Contrapunctus inversus 12 a 4 [forma inversa and recta]
 * Contrapunctus inversus 13 a 3 [forma recta and inversa]

Canons, labeled by interval and technique:

 * Canon per Augmentationem in Contrario Motu: Canon in which the following voice is both inverted and augmented. The following voice, running at half-speed, eventually lags the first voice by 20 bars, making the canon effect hard to hear. Three versions have appeared in the autograph Mus. ms. autogr. P 200: Canon in Hypodiatesseron, al roversio  [sic] e per augmentationem, perpetuus, Canon al roverscio et per augmentationem, and Canon p. Augmentationem contrario Motu, the third of which appears on the second supplemental Beilage.
 * Canon alla Ottava: canon in imitation at the octave; titled Canon in Hypodiapason in Mus. ms. autogr. P 200.
 * Canon alla Decima [in] Contrapunto alla Terza: canon in imitation at the tenth
 * Canon alla Duodecima in Contrapunto alla Quinta: canon in imitation at the twelfth

Alternate variants and arrangements:

 * Contra[punctus] a 4: alternate version of the last 22 bars of Contrapunctus 10.
 * Fuga a 2 Clav: and Alio modo. Fuga a 2 Clav.: two-keyboard arrangements of Contrapunctus inversus a 3, the forma inversa and recta, respectively.

Incomplete fugue:

 * [Contrapunctus 14] Fuga a 3 Soggetti: four-voice triple fugue (not completed by Bach, but likely to have become a quadruple fugue: see below), the third subject of which begins with the BACH motif, B♭–A–C–B♮ ('H' in German letter notation).

Instrumentation
Both editions of the Art of Fugue are written in open score, where each voice is written on its own staff. This has led some to conclude that the Art of Fugue was intended as an intellectual exercise, meant to be studied more than heard. The renowned keyboardist Gustav Leonhardt argued that the Art of Fugue was intended to be played on a keyboard instrument, and specifically the harpsichord. Leonhardt's arguments included the following:
 * 1) It was common practice in the 17th and early 18th centuries to publish keyboard pieces in open score, especially those that are contrapuntally complex. Examples include Frescobaldi's Fiori musicali (1635), Samuel Scheidt's Tabulatura Nova (1624), works by Johann Jakob Froberger (1616–1667), Franz Anton Maichelbeck (1702–1750), and others.
 * 2) The range of none of the ensemble or orchestral instruments of the period corresponds to any of the ranges of the voices in The Art of Fugue. Furthermore, none of the melodic shapes that characterize Bach's ensemble writing are found in the work, and there is no basso continuo.
 * 3) The fugue types used are reminiscent of the types in The Well-Tempered Clavier, rather than Bach's ensemble fugues; Leonhardt also shows an "optical" resemblance between the fugues of the two collections, and points out other stylistic similarities between them.
 * 4) Finally, since the bass voice in The Art of Fugue occasionally rises above the tenor, and the tenor becomes the "real" bass, Leonhardt deduces that the bass part was not meant to be doubled at 16-foot pitch, thus eliminating the pipe organ as the intended instrument, leaving the harpsichord as the most logical choice.

It is now generally accepted by scholars that the work was envisioned for keyboard.

Fuga a 3 Soggetti

Fuga a 3 Soggetti ("fugue in three subjects"), also called the "Unfinished Fugue" and Contrapunctus 14, was contained in a handwritten manuscript bundled with the autograph manuscript Mus. ms. autogr. P 200. It breaks off abruptly in the middle of its third section, with an only partially written measure 239. This autograph carries a note in the handwriting of Carl Philipp Emanuel Bach, stating "Über dieser Fuge, wo der Name B A C H im Contrasubject angebracht worden, ist der Verfasser gestorben." ("While working on this fugue, which introduces the name BACH [for which the English notation would be B♭–A–C–B♮] in the countersubject, the composer died.") This account is disputed by modern scholars, as the manuscript is clearly written in Bach's own hand, and thus dates to a time before his deteriorating health and vision would have prevented his ability to write, probably 1748–1749.

Attempts at completion
Several musicians and musicologists have composed conjectural completions of Contrapunctus XIV which include the fourth subject, including musicologists Donald Tovey (1931), Zoltán Göncz (1992), Yngve Jan Trede (1995), and Thomas Daniel (2010), organists Helmut Walcha, David Goode, Lionel Rogg, and Davitt Moroney (1989), conductor Rudolf Barshai (2010) and Daniil Trifonov (2021). Ferruccio Busoni's Fantasia contrappuntistica is based on Contrapunctus XIV, but it develops Bach's ideas to Busoni's own purposes in Busoni's musical style, rather than working out Bach's thoughts as Bach himself might have done.

Principles of construction
Loïc Sylvestre and Marco Costa reported a mathematical architecture of The Art of Fugue, based on bar counts, which shows that the whole work could have been conceived on the basis of the Fibonacci series and the golden ratio.

Dominic Florence proposes that a concept he calls "opposition" governs all the methods that Bach uses in Contrapuncti 1, 2, 3, and 5 to create variety. These include changes in "melody (contrary motion), polyphony (contrapuntal inversion), harmony (dissonance), [rhythmic] density (texture), rhythm (syncopation), and tonality (modulation}". For example, Contrapuncti 1 and 2 both switch repeatedly between the keys of A minor and D minor; Contrapuncti 2 and 3 in addition enter F major and G minor, Contrapunctus 2 also visiting B-flat major once in a further "tonal remove": all three begin and end in D minor. He concludes that "Analyses of fugues should focus on continuous, dynamic, organic processes that evolve over time, rather than on the static and discontinuous dismemberment into strictly delineated sections."

Religious interpretations
In 1984, the German musicologist Hans Heinrich Eggebrecht suggested a possible religious interpretation of The Art of Fugue, which he agreed was impossible to prove: that the work illustrated in musical terms the Christian doctrine of redemption by God's grace alone, sola gratia, rather than by any action an individual can take. Eggebrecht noted the presence in fugue 3 of the presence of the composer's surname Bach in its theme, the sequence of notes B-A-C-H-C#-D. In Eggebrecht's view, this could mean that the composer is not just signing the work, but is placing himself by his grave, accepting sola gratia as he reaches towards the tonic note, which marks the end of the fugue and symbolically the end of his life. Further, the six-note fragment is chromatic, denoting sinful humanity, whereas the work as a whole is diatonic, symbolising God's perfection.

The Russian musicologist Anatoly Milka suggests that the various numbers embedded in the work have a numerological significance related to the Book of Revelation, the last book of the Bible which describes the apocalypse.

Unfinished Contrapunctus 14
In her 2007 doctoral thesis about the unfinished ending of Contrapunctus 14, the New Zealand organist and conductor Indra Hughes proposed that the work was left unfinished not because Bach died, but as a deliberate choice by Bach to encourage independent efforts at a completion.

Douglas Hofstadter's book Gödel, Escher, Bach discusses the unfinished fugue and Bach's supposed death during composition as a tongue-in-cheek illustration of the Austrian logician Kurt Gödel's first incompleteness theorem. According to Gödel, the very power of a "sufficiently powerful" formal mathematical system can be exploited to "undermine" the system, by leading to statements that assert such things as "I cannot be proven in this system". In Hofstadter's discussion, Bach's great compositional talent is used as a metaphor for a "sufficiently powerful" formal system; however, Bach's insertion of his own name "in code" into the fugue is not, even metaphorically, a case of Gödelian self-reference; and Bach's failure to finish his self-referential fugue serves as a metaphor for the unprovability of the Gödelian assertion, and thus for the incompleteness of the formal system.

title: The Art of Fugue


text:
Hazard ratio

In survival analysis, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions characterised by two distinct levels of a treatment variable of interest. For example, in a clinical study of a drug, the treated population may die at twice the of the control population. The hazard ratio would be 2, indicating a higher hazard of death from the treatment.

For example, a scientific paper might use an HR to state something such as: "Adequate COVID-19 vaccination status was associated with significantly decreased risk for the composite of severe COVID-19 or mortality with a[n] HR of 0.20 (95% CI, 0.17–0.22)." In essence, the hazard for the composite outcome was 80% lower among the vaccinated relative to those who were unvaccinated in the same study. So, for a hazardous outcome (e.g., severe disease or death), an HR below 1 indicates that the treatment (e.g., vaccination) is protective against the outcome of interest. In other cases, an HR greater than 1 indicates the treatment is favorable. For example, if the outcome is actually favorable (e.g., accepting a job offer to end a spell of unemployment), an HR greater than 1 indicates that seeking a job is favorable to not seeking one (if "treatment" is defined as seeking a job).

Hazard ratios differ from relative risks (RRs) and odds ratios (ORs) in that RRs and ORs are cumulative over an entire study, using a defined endpoint, while HRs represent instantaneous risk over the study time period, or some subset thereof. Hazard ratios suffer somewhat less from selection bias with respect to the endpoints chosen and can indicate risks that happen before the endpoint.

Definition and derivation
Regression models are used to obtain hazard ratios and their confidence intervals.

The instantaneous hazard rate is the limit of the number of events per unit time divided by the number at risk, as the time interval approaches 0:
 * $$ h(t) = \lim_{\Delta t \to 0} \frac{\text{observed events in interval}\ [t, t + \Delta t]/N(t)}{\Delta t},$$

where N(t) is the number at risk at the beginning of an interval. A hazard is the probability that a patient fails between $$t$$ and $$t + \Delta t$$, given that they have survived up to time $$t$$, divided by $$\Delta t$$, as $$\Delta t$$ approaches zero.

The hazard ratio is the effect on this hazard rate of a difference, such as group membership (for example, treatment or control, male or female), as estimated by regression models that treat the logarithm of the HR as a function of a baseline hazard $$h_0(t)$$ and a linear combination of explanatory variables:
 * $$\log h(t) = f\big(h_0(t), \alpha + \beta_1 X_1 + \cdots + \beta_k X_k\big).$$

Such models are generally classed proportional hazards regression models; the best known being the Cox proportional hazards model, and the exponential, Gompertz and Weibull parametric models.

For two groups that differ only in treatment condition, the ratio of the hazard functions is given by $$e^\beta$$, where $$\beta$$ is the estimate of treatment effect derived from the regression model. This hazard ratio, that is, the ratio between the predicted hazard for a member of one group and that for a member of the other group, is given by holding everything else constant, i.e. assuming proportionality of the hazard functions.

For a continuous explanatory variable, the same interpretation applies to a unit difference. Other HR models have different formulations and the interpretation of the parameter estimates differs accordingly.

Interpretation
In its simplest form, the hazard ratio can be interpreted as the chance of an event occurring in the treatment arm divided by the chance of the event occurring in the control arm, or vice versa, of a study. The resolution of these endpoints are usually depicted using Kaplan–Meier survival curves. These curves relate the proportion of each group where the endpoint has not been reached. The endpoint could be any dependent variable associated with the covariate (independent variable), e.g. death, remission of disease or contraction of disease. The curve represents the odds of an endpoint having occurred at each point in time (the hazard). The hazard ratio is simply the relationship between the instantaneous hazards in the two groups and represents, in a single number, the magnitude of distance between the Kaplan–Meier plots.

Hazard ratios do not reflect a time unit of the study. The difference between hazard-based and time-based measures is akin to the difference between the odds of winning a race and the margin of victory. When a study reports one hazard ratio per time period, it is assumed that difference between groups was proportional. Hazard ratios become meaningless when this assumption of proportionality is not met.

If the proportional hazard assumption holds, a hazard ratio of one means equivalence in the hazard rate of the two groups, whereas a hazard ratio other than one indicates difference in hazard rates between groups. The researcher indicates the probability of this sample difference being due to chance by reporting the probability associated with some test statistic. For instance, the $$\beta$$ from the Cox-model or the log-rank test might then be used to assess the significance of any differences observed in these survival curves.

Conventionally, probabilities lower than 0.05 are considered significant and researchers provide a 95% confidence interval for the hazard ratio, e.g. derived from the standard deviation of the Cox-model regression coefficient, i.e. $$\beta$$. Statistically significant hazard ratios cannot include unity (one) in their confidence intervals.

The proportional hazards assumption
The proportional hazards assumption for hazard ratio estimation is strong and often unreasonable. Complications, adverse effects and late effects are all possible causes of change in the hazard rate over time. For instance, a surgical procedure may have high early risk, but excellent long term outcomes.

If the hazard ratio between groups remain constant, this is not a problem for interpretation. However, interpretation of hazard ratios become impossible when selection bias exists between groups. For instance, a particularly risky surgery might result in the survival of a systematically more robust group who would have fared better under any of the competing treatment conditions, making it look as if the risky procedure was better. Follow-up time is also important. A cancer treatment associated with better remission rates might on follow-up be associated with higher relapse rates. The researchers' decision about when to follow up is arbitrary and may lead to very different reported hazard ratios.

The hazard ratio and survival
Hazard ratios are often treated as a ratio of death probabilities. For example, a hazard ratio of 2 is thought to mean that a group has twice the chance of dying than a comparison group. In the Cox-model, this can be shown to translate to the following relationship between group survival functions: $$S_1(t) = S_0(t)^r$$ (where r is the hazard ratio). Therefore, with a hazard ratio of 2, if $$S_0(t) = 0.2$$ (20% survived at time t), $$S_1(t) = 0.2^2 = 0.04$$ (4% survived at t). The corresponding death probabilities are 0.8 and 0.96. It should be clear that the hazard ratio is a relative measure of effect and tells us nothing about absolute risk.

While hazard ratios allow for hypothesis testing, they should be considered alongside other measures for interpretation of the treatment effect, e.g. the ratio of median times (median ratio) at which treatment and control group participants are at some endpoint. If the analogy of a race is applied, the hazard ratio is equivalent to the odds that an individual in the group with the higher hazard reaches the end of the race first. The probability of being first can be derived from the odds, which is the probability of being first divided by the probability of not being first:

 * $$\mathit{HR} = \frac P{1 - P}$$; conversely, $$P = \frac{\mathit{HR}}{1 + \mathit{HR}}$$.

In the previous example, a hazard ratio of 2 corresponds to a 67% chance of an early death. The hazard ratio does not convey information about how soon the death will occur.

The hazard ratio, treatment effect and time-based endpoints
Treatment effect depends on the underlying disease related to survival function, not just the hazard ratio. Since the hazard ratio does not give us direct time-to-event information, researchers have to report median endpoint times and calculate the median endpoint time ratio by dividing the control group median value by the treatment group median value.

While the median endpoint ratio is a relative speed measure, the hazard ratio is not. The relationship between treatment effect and the hazard ratio is given as $$e^\beta$$. A statistically important, but practically insignificant effect can produce a large hazard ratio, e.g. a treatment increasing the number of one-year survivors in a population from one in 10,000 to one in 1,000 has a hazard ratio of 10. It is unlikely that such a treatment would have had much impact on the median endpoint time ratio, which likely would have been close to unity, i.e. mortality was largely the same regardless of group membership and clinically insignificant.

By contrast, a treatment group in which 50% of infections are resolved after one week (versus 25% in the control) yields a hazard ratio of two. If it takes ten weeks for all cases in the treatment group and half of cases in the control group to resolve, the ten-week hazard ratio remains at two, but the median endpoint time ratio is ten, a clinically significant difference.

title: Hazard ratio


text:
Taupō Volcanic Zone

The Taupō Volcanic Zone (TVZ) is a volcanic area in the North Island of New Zealand. It has been active for at least the past two million years and is still highly active.

Mount Ruapehu marks its south-western end and the zone runs north-eastward through the Taupō and Rotorua areas and offshore into the Bay of Plenty. It is part of a larger Central Volcanic Region that extends to the Coromandel Peninsula and has been active for four million years. The zone is contained within the tectonic intra-arc continental Taupō Rift and this rift volcanic zone is widening unevenly east–west, with the greatest rate of widening at the Bay of Plenty coast, the least at Mount Ruapehu and a rate of about 8 mm per year at Taupō. The zone is named after Lake Taupō, the flooded caldera of the largest volcano in the zone, the Taupō Volcano and contains a large central volcanic plateau as well as other landforms.

Activity
There are numerous volcanic vents and geothermal fields in the zone, with Mount Ruapehu, Mount Ngauruhoe and Whakaari / White Island erupting most frequently. Whakaari has been in continuous activity since 1826 if you count such as steaming fumaroles, but the same applies to say the Okataina volcanic centre. The Taupō Volcanic Zone has produced in the last 350,000 years over 3900 km3 material, more than anywhere else on Earth, from over 300 silicic eruptions, with 12 of these eruptions being caldera-forming. Detailed stratigraphy in the zone is only available from the Ōkataina Rotoiti eruption but including this event, the zone has been more productive than any other rhyolite predominant volcanic area over the last 50,000 odd years at 12.8 km3 per thousand years. Comparison of large events in the Taupō volcanic zone over the last 1.6 million years at 3.8 km3 per thousand years with Yellowstone Caldera's 2.1 million year productivity at 3.0 km3 per thousand years favours Taupo. Both the Taupō Volcano and the Ōkataina Caldera have had multiple eruptions in the last 25,000 years. The zone's largest eruption since the arrival of Europeans was that of Mount Tarawera (within the Ōkataina Caldera) in 1886, which killed over 100 people. Early Māori would also have been affected by the much larger Kaharoa eruption from Tarawera around 1315 CE.

The last major eruption from Lake Taupō, the Hatepe eruption, occurred in 232 CE. It is believed to have first emptied the lake, then followed that feat with a pyroclastic flow that covered about 20000 km2 of land with volcanic ash. A total of 120 km3 of material expressed as dense-rock equivalent (DRE) is believed to have been ejected, and over 30 km3 of material is estimated to have been ejected in just a few minutes. The date of this activity was previously thought to be 186 AD as the ash expulsion was thought to be sufficiently large to turn the sky red over Rome and China (as documented in Hou Han Shu), but this has since been disproven.

Whakaari / White Island had a major, edifice failure collapse of its volcano dated to 946 BCE ± 52 years. It has been suggested that this was the cause of the tsunami tens of metres tall that went up to 7 km inland in the Bay of Plenty at about this time. Although significant tsunami's can be associated with volcanic eruptions, it is unknown if the cause was a relatively small eruption of Whakaari or another cause such as a large local earthquake

Taupō erupted an estimated 1170 km3 of DRE material in its Oruanui eruption 25,580 years ago. This was Earth's most recent eruption reaching VEI-8, the highest level on the Volcanic Explosivity Index.

The Rotorua caldera has been dormant longer, with its main eruption occurring about 225,000 years ago, although lava dome extrusion has occurred within the last 25,000 years.

Extent and geological context
The Taupō volcanic zone is approximately 350 km long by 50 km wide. Mount Ruapehu marks its southwestern end, while Whakaari / White Island is considered its northeastern limit.

It forms a southern portion of the active Lau-Havre-Taupō back-arc basin, which lies behind the Kermadec-Tonga Subduction Zone. Mayor Island and Mount Taranaki are recently active back arc volcanoes on the New Zealand extension of this arc. Mayor Island / Tūhua is the northern-most shield volcano adjacent to the New Zealand coast, and is believed to have been active in the last 1000 years. It is formed from rhyolite magma. It has a quite complex eruptive history but only with one definite significant Plinian eruption. Mount Taranaki is an andesite cone and the most recent of four Taranaki volcanoes about 140 km west of the Taupō Volcanic Zone.

Associated with the Taupō volcanic zone, intra-arc extension is expressed as normal faulting within a zone known as the Taupō Rift. Volcanic activity continues to the north-northeast, along the line of the Taupō Volcanic Zone, through several undersea volcanoes in the South Kermadec Ridge Seamounts, then shifts eastward to the parallel volcanic arc of the Kermadec Islands and Tonga. Although the back-arc basin continues to propagate to the south-west, with the South Wanganui Basin forming an initial back-arc basin, volcanic activity has not yet begun in this region.

South of Kaikōura the plate boundary changes to a transform boundary with oblique continental collision uplifting the Southern Alps in the South Island. A subduction zone reappears south-west of Fiordland, at the south-western corner of the South Island, although here the subduction is in the opposite direction. Solander Island / Hautere is an extinct volcano associated with this subduction zone, and the only one that protrudes above the sea.

Tectonics
In the North Island rifting associated with plate tectonics has defined a Central Volcanic Region, that has been active for four million years and this extends westward from the Taupō volcanic zone through the western Bay of Plenty to the eastern side of the Coromandel Peninsula. The dominant rifting axis associated with the Central Volcanic Region has moved with time, from the back-arc associated Hauraki Rift to the intra-arc Taupō Rift. As there is presently no absolute consensus with regard to the cause of the Taupō Rift's extension or its exceptional current volcanic productivity, some of the discussion on this page has been simplified, rather than all possible models being presented.

Recent scientific work indicates that the Earth's crust below the Taupō Volcanic Zone may be as little as 16 kilometres thick. A film of magma 50 kilometres (30 mi) wide and 160 kilometres (100 mi) long lies 10 kilometres under the surface. The geological record indicates that some of the volcanoes in the area erupt infrequently but have large, violent and destructive eruptions when they do. Technically the zone is in the continental intraarc Taupō Rift, which is a continuation of oceanic plate structures associated with oblique Australian and Pacific Plate convergence in the Hikurangi subduction zone. At Taupō the rift volcanic zone is widening east–west at the rate of about 8 mm/year, while at Mount Ruapehu it is only 2 –/year and this increases at the north eastern end at the Bay of Plenty coast to 10 –/year. The rift has had three active stages of faulting in the last 2 million years with the modern Taupō rift evolving in the last 25,000 years after the massive Oruanui eruption and now being within two essentially inactive rift systems. These are the surrounding limits of the young Taupō Rift between 25,000 and 350,000 years and old Taupō Rift system whose northern boundary is now located well to the north of the other two being created before 350,000 years ago.

The Tauranga Volcanic Centre which was active between 2.95 to 1.9 million years ago, and was previously classified as part of the Central Volcanic Region, appears now to be in a tectonic continuum with the Taupō Volcanic Zone. Recent ocean floor tephra studies off the east coast of the North Island have shown an abrupt compositional change in these, from about 4.5 million years ago, that has been suggested to distinguish Coromandel Volcanic Zone activity from that of the Taupō Volcanic Zone. Further the distinctive Waiteariki ignimbrite that erupted 2.1 million years ago in a supereruption, presumably from the gravity anomaly defined Omanawa Caldera, is within the postulated borders of the old Taupō Rift.

Faults
The multiple intra-rift faults are some of the most active in the country and some have the potential to create over magnitude 7 events. The fault structures are perhaps most well characterised related to the Ruapehu and Tongariro grabens. The recent deposits from major eruptions and lake features mean many potentially significant faults are uncharacterised, either completely (for example the 6.5 MW 1987 Edgecumbe earthquake resulted in the mapping of the Edgecumbe fault for the first time) or frequency of events and their likely magnitude are not understood. It can not be assumed that just because the rate of expansion of the rift is greatest near the coast that this is where most significant tectonic earthquakes in terms of human risk will be. The Waihi Fault Zone south of Lake Taupō and associated with the Tongariro graben has a particular risk of inducing massive landslips which has caused significant loss of life and appears to be more active than many other faults in the zone.

Volcanism
The north (Whakatane Graben – Bay of Plenty) part of the zone is predominantly formed from andesitic magma and represented by the continuously active Whakaari / White Island andesite–dacite stratovolcano. Although Strombolian activity has occurred the explosive eruptions are typically phreatic or phreatomagmatic. The active emergent summit tops the larger, 16 km × 18 km, submarine volcano with a total volume of 78 km3.

The central part of the zone is composed of eight caldera centres the oldest of which is the Mangakino caldera which was active more than a million years ago (1.62–0.91 Ma). This produced ignimbrite that 170 km away in Auckland is up to 9 m thick. Other than the now buried Kapenga caldera there are five caldera centres, Rotorua, Ohakuri, Reporoa, Ōkataina and Taupō. These have resulted from massive infrequent eruptions of gaseous very viscous rhyolite magma which is rich in silicon, potassium, and sodium and created the ignimbrite sheets of the North Island Volcanic Plateau. The detailed composition suggests subduction erosion might play a predominant role in producing this rhyolite, as later assimilation and fractional crystallization of primary basalt magma, is difficult to model to explain the composition and volumes erupted. This central zone has had the largest number of very large silicic caldera-forming eruptions recently on earth as mentioned earlier.

During a period of less than 100,000 years commencing with the massive Whakamaru eruption about 335,000 years ago of greater than 2000 km3 dense-rock equivalent of material, just to the north of the present Lake Taupō, over 4000 km3 total was erupted. These eruptions essentially defined the limits of the present central volcanic plateau, although its current central landscape is mainly a product of later smaller events over the last 200,000 years than the Whakamaru eruption. The other volcanic plateau defining eruptions were to the west, the 150 km3 Matahina eruption of about 280,000 years ago, the mainly tephra 50 km3 Chimp (Chimpanzee) eruption between 320 and 275 ka, the central 50 km3 Pokai eruption of about 275 ka, and the paired Mamaku to the north and east central Ohakuri eruptions of about 240,000 years ago that together produced more than 245 km3 dense-rock equivalent of material. The southern Taupō Volcano Oruanui eruption about 25,600 years ago produced 530 km3 dense-rock equivalent of material and its recent Hatepe eruption of 232 CE ± 10 years had 120 km3 dense-rock equivalent. Since the Whakamaru eruption the central part of the zone has dominated, so that when the whole zone is considered there has been about 3000 km3 of rhyolite, 300 km3 of andesite, 20 km3 of dacite and 5 km3 of basalt erupted.

Less gaseous rhyolite magma dome building effusive eruptions have built features such as the Horomatangi Reefs or Motutaiko Island in Lake Taupō or the lava dome of Mount Tarawera. This later as part of the Ōkataina caldera complex is the highest risk volcanic field in New Zealand to man. Mount Tauhara adjacent to Lake Taupō is actually a dacitic dome and so intermediate in composition between andesite and rhyolite but still more viscous than basalt which is rarely found in the zone.

The southern part of the zone contain classic volcanic cone structure formed from andesite magma in effusive eruptions that cool to form dark grey lava if gas-poor or scoria if gas-rich of this part of the zone. Mount Ruapehu, the tallest mountain in the North Island, is a 150 km3 andesite cone surrounded by a 150 km3 ring-plain. This ring plain is formed from numerous volcanic deposits created by slope failure, eruptions, or lahars. Northwest of Ruapehu is Hauhungatahi, the oldest recorded volcano in the south of the plateau, with to the north the two prominent volcanic mountains in the Tongariro volcanic centre being Tongariro and Ngauruhoe which are part of a single composite stratovolcano.

Risks
The most likely risk is earthquake associated with multiple active faults, such as within the Taupō Fault Belt, but many faults will be uncharacterised as was the case with the 1987 Edgecumbe earthquake. Earthquakes can be associated with landslides and inland or coastal tsunami that can result in great loss of life and both have happened on the Waihi Fault Zone. The relative low grade volcanic activity of the andesite volcanoes at each end of the zone has resulted in recorded history in both direct loss of life and disrupted transport and tourism. The only high grade eruption in recorded history was atypically basaltic from Mount Tarawera and although very destructive is not likely to be a perfect model for the more typical and often larger rhyolitic events associated with the Taupō Volcano and the Ōkataina Caldera. As mentioned earlier the Ōkataina caldera complex is the highest risk volcanic field risk in New Zealand to man and the recent frequency of rhyolitic events there is not reassuring, along with the timescale of likely warning of such an event. These eruptions are associated with tephra production that results in deep ash fall over wide areas (e.g. the Whakatane eruption of ~ 5500 years ago had 5 mm ashfall 900 km away on the Chatham Islands) ` pyroclastic flows and surges, which rarely have covered large areas of the North Island in ignimbrite sheets, earthquakes, lake tsunamis, prolonged lava dome growth and associated block and ash flows with post-eruption lahars and flooding.

Volcanoes, lakes and geothermal fields
The following Volcanic Centres belong to the modern Taupō Volcanic Zone in what proved to be an evolving classification scheme:

Rotorua, Ōkataina, Maroa, Taupō, Tongariro and Mangakino. The old zone almost certainly contains volcanoes in the Tauranga Volcanic Centre.
 * Tauranga Volcanic Centre – Bay of Plenty
 * Activity commenced here over two million years ago and is now extinct.
 * Whakatāne Graben – Bay of Plenty
 * Submarine Whakatāne Seamount -36.8°N, 177.5°W
 * Mayor Island / Tūhua -37.28333°N, 176.25°W
 * Moutohora Island -37.85639°N, 176.97333°W
 * Whakaari / White Island -37.52°N, 177.18°W
 * Te Paepae o Aotea
 * Putauaki -38.1°N, 176.8°W
 * Geothermal field
 * Kawerau Power Station -38.0631°N, 176.7271°W
 * Rotorua Volcanic Centre
 * Rotorua Caldera, size: 22 km wide -38.08°N, 176.27°W
 * Mount Ngongotahā -38.11897°N, 176.1981°W
 * Lakes
 * Lake Rotorua -38.08°N, 176.27°W
 * Mokoia Island -38.08333°N, 176.3°W
 * Geothermal fields
 * Tikitere/Hell's Gate -38.065°N, 176.361°W
 * Whakarewarewa -38.16222°N, 176.25639°W
 * Pōhutu Geyser (Te Puia)
 * Ōkataina Volcanic Centre: The Haroharo and Tarawera complexes impounded the lakes against the outer margins of the Ōkataina Caldera. The Okareka Embayment and the Tarawera Volcanic Complex were placed inside the Haroharo Caldera which in turn is inside the Okataina Ring Structure, according to Newhall (1988), but this was reclassified by Cole (2009).
 * Ōkataina Caldera, size: roughly 27 x 20 km -38.22°N, 176.5°W
 * Haroharo volcanic complex, northern end of the Okataina Volcanic Centre with infilling of several sub-caldera's from caldera forming eruptions:
 * Rotoiti Caldera
 * Matahina Caldera
 * Utu Caldera
 * Mount Tarawera and Tarawera volcanic complex -38.22°N, 176.5°W
 * Okareka vent
 * Puhipuhi Embayment
 * Ōkareka Embayment -38.20215°N, 176.34829°W
 * Rotomā Caldera -38.08333°N, 176.58333°W
 * Lakes
 * Lake Ōkataina -38.11667°N, 176.41667°W
 * Lake Tarawera -38.2°N, 176.45°W
 * Lake Rotokākahi (Green Lake) -38.21667°N, 176.33333°W
 * Lake Tikitapu (Blue Lake) -38.2°N, 176.33333°W
 * Lake Ōkāreka -38.16667°N, 176.36667°W
 * Lake Rotomahana -38.26667°N, 176.45°W
 * Lake Rotoiti -38.039°N, 176.4277°W
 * Lake Rotomā -38.0476°N, 176.5878°W
 * Lake Rotoehu -38.01667°N, 176.53333°W
 * Geothermal fields
 * Waimangu Volcanic Rift Valley -38.2825°N, 176.39889°W
 * Frying Pan Lake -38.28359°N, 176.39487°W
 * Maroa Volcanic Centre: The Maroa Caldera formed in the north-east corner of the Whakamaru Caldera, and the Whakamaru Caldera partially overlaps with the Taupō Caldera on the South. The Waikato River course follows roughly the northern Maroa Caldera rim on one side. The town of Whakamaru and the artificial Lake Whakamaru, on the Waikato River, have the same name too. The paired single event eruption of the Ohakuri Caldera at its north western limits with the Rotorua Caldera added later complexity after this classification was developed. Accordingly, later naming, terms this the Whakamaru caldera complex.
 * Ohakuri Caldera -38.378°N, 176.019°W
 * Maroa Caldera, size: 16 x 25 km -38.42°N, 176.08°W
 * Puketarata volcanic complex -38.55057°N, 176.05452°W
 * Reporoa Caldera, size: 10 x 15 km -38.41667°N, 176.33333°W
 * Whakamaru Caldera, size: 30 x 40 km -38.42°N, 175.8°W
 * Geothermal fields
 * Waiotapu -38.35944°N, 176.36972°W
 * Wairakei -38.62669°N, 176.10349°W
 * Craters of the Moon (Karapiti) -38.64667°N, 176.06833°W
 * Orakei Korako -38.47333°N, 176.14833°W
 * Ngatamariki -38.54722°N, 176.19583°W
 * Rotokaua -38.62797°N, 176.19219°W
 * Ohaaki Power Station -38.527°N, 176.292°W
 * Taupō Volcanic Center
 * Taupō Caldera, size: roughly 35 km wide -38.82°N, 176°W
 * Mount Tauhara -38.69444°N, 176.16278°W
 * Ben Lomond rhyolite dome (contains obsidian) -38.595°N, 175.95333°W
 * Lake Taupō -38.82°N, 176°W
 * Horomatangi Reefs -38.8°N, 176°W
 * Motutaiko Island -38.854°N, 175.942°W
 * Geothermal fields
 * Tauhara-Taupō -38.96786°N, 175.76338°W
 * Tongariro Volcanic Centre: Lake Taupō, Kakaramea, Pihanga, Tongariro and Ruapehu are roughly aligned on the main fault.
 * Kakaramea-Tihia Massif -38.98889°N, 175.70833°W
 * Pihanga -39.04132°N, 175.76861°W
 * Mount Tongariro and Tongariro volcanic complex -39.13333°N, 175.65°W
 * Mount Ngauruhoe, a main Tongariro vent -39.15683°N, 175.63217°W
 * Tama crater lakes, main Tongariro vents
 * Upper Tama -39.1854°N, 175.6223°W
 * Lower Tama -39.2025°N, 175.6068°W
 * Mount Ruapehu -39.18°N, 175.35°W
 * Hauhungatahi -39.23°N, 175.44°W
 * Lakes
 * Lake Rotoaira -39.0545°N, 175.7143°W
 * Lake Rotopounamu -39.0267°N, 175.7382°W
 * Geothermal fields
 * Ketetahi Springs -39.10748°N, 175.64767°W
 * Mangakino Volcanic Centre: The Mangakino Volcanic Centre is the westernmost extinct rhyolitic caldera volcano in the Taupō Volcanic Zone and activity commenced at least 1.62 million years ago. The course of the Waikato River crosses this area, between the artificial Lake Ohakuri (-38.42273°N, 176.12547°W), the town of Mangakino (-38.38333°N, 175.78333°W) and Hamilton.
 * Artificial Lake Maraetai

Other important features of the TVZ include the Ngakuru and Ruapehu grabens.

Note
There is more recent, somewhat different classification, by some of the same authors, that uses the term caldera complex:
 * North part: Whakatane Graben – Bay of Plenty
 * Central part:
 * West of the main fault zone:
 * Mangakino caldera complex; may be transitional between Coromandel Volcanic Zone (CVZ) and Taupō Volcanic Zone (TVZ) (1.62 – 0.91 mio. years old)
 * Kapenga caldera; lies between the Maroa caldera and the Rotorua caldera, it is completely buried under more recent tephra (circa 700,000 years old)
 * Okareka Embayment, lies inside the northern end of the Kapenga caldera, but is now usually regarded as part of the recently active Okataina caldera complex
 * Rotorua single event caldera; Mamaku Ignimbrite (circa 225,000 years old)
 * Main fault zone:
 * Ōkataina caldera complex
 * Haroharo caldera complex
 * Tarawera volcanic complex
 * Whakamaru caldera complex
 * Maroa caldera
 * Ohakuri single event caldera
 * Taupō caldera complex
 * East of the main fault zone:
 * Reporoa single event caldera; Kaingaroa Ignimbrite (circa 240,000 years old)
 * South part: Tongariro Volcanic Centre

title: Taupō Volcanic Zone


text: Talk:Motivation

title: Talk:Motivation